Propagation Phenomena for Man–Environment Epidemic Model with Nonlocal Dispersals

Abstract

In this paper, we focus on the propagation phenomena of a bistable man–environment epidemic model with nonlocal dispersals, where there exists a positive feedback interaction between the concentration of infectious agent and infectious human population. The monostable and bistable traveling wave solutions and three-wave entire solutions are studied. First, by applying and developing the known results for monostable case, we give a summary of the existence and asymptotic behavior of all monostable traveling wave solutions in two different monostable intervals and further find some relationship between them. The existence of bistable traveling wave solutions is obtained by introducing the results about monotone semiflows with weak compactness. Second, we give twelve types of three-wave entire solutions, which contain all possibilities of three-wave entire solutions originating from three traveling wave solutions with different nonzero wave speeds, by constructing new auxiliary functions and super- and sub-solutions for every type. We also show that these entire solutions are globally Lipschitz continuous with respect to spatial variable. In addition, the nonexistence result of entire solutions originating from more than four traveling wave solutions is obtained.

1 Introduction

Dispersal, as a common and important phenomenon in ecology and epidemiology, describes the movement mechanism by which species move into a new area. Mathematically, dispersal can be modeled by the following nonlocal dispersal operator:

$$\begin{aligned} u\mapsto J*u-u=\int _{{\mathbb {R}}^N}J(x-y)u(y){\text {d}}y-u, \end{aligned}$$

which describes the movements of organisms between not only adjacent but also nonadjacent spatial locations. Compared with its local counterpart represented by Laplacian operator (i.e., $u\mapsto \Delta u$), nonlocal dispersal is more effective to study the long-range dispersal phenomena, such as the rapid spread of infectious disease across countries or continents by the travel of infected human (see Cannas et al. 2006; Hallatschek and Fisher 2014). Here, the kernel J is a nonnegative function satisfying $\int _{{\mathbb {R}}^N}J(x){\text {d}}x=1$, and $J(x-y)$ stands for the probability distribution of the movement jumping from location y to x. Over the past two decades, the following nonlocal dispersal equation with a reaction term

$$\begin{aligned} u_t=J*u-u+f(u),~x\in {\mathbb {R}}^N,~t\in {\mathbb {R}} \end{aligned}$$

has attracted extensive attention, and we refer to Bates et al. (1997), Carr and Chmaj (2004), Coville et al. (2008), and Schumacher (1980) for the works on traveling wave solutions, and Dong et al. (2021), Li et al. (2010), and Sun et al. (2011) for the works on entire solutions. We also refer to Andreu-Vaillo et al. (2010), Bates (2006), Fife (2003), Kao et al. (2010), and Murray (2003) for the works on other problems about nonlocal dispersals.

In this paper, we focus on an epidemic model where both the dispersals of infectious agent and infectious human population are nonlocal, and there is a positive feedback interaction between the concentration of infectious agent and infectious human population, that is, once the human hosts infected, they will promote the growth rate of bacteria, and in turn, an increase in the concentration of infectious agent can lead to an increase in the infection rate of human population. This model is written as

$$\begin{aligned} \left\{ \begin{aligned}&u_t(x,t)=J_1*u(x,t)-u(x,t)-\alpha u(x,t)+h(v(x,t)),~&x\in {\mathbb {R}},t\in {\mathbb {R}},\\&v_t(x,t)=J_2*v(x,t)-v(x,t)-\beta v(x,t)+g(u(x,t)),~&x\in {\mathbb {R}},t\in {\mathbb {R}}. \end{aligned}\right. \nonumber \\ \end{aligned}$$

(1.1)

Here, u(x, t) and v(x, t) biologically stand for the spatial densities of infectious agent and infectious human population at location x and time t, respectively; $\alpha$ denotes the natural death rate of infectious agent; $1/\beta$ is the infectious period of infectious human population; h(v) means the growth rate of infectious agent due to the infectious human population; g(u) stands for the infection rate of the human population under the assumption that the total susceptible human population is constant during the evolution of epidemic. In the system (1.1), the dispersals of infectious agent and infectious human population are described by

$$\begin{aligned} \begin{aligned}&J_1*u(x,t)-u(x,t)=\int _{{\mathbb {R}}} J_1(x-y)u(y,t) {\text {d}}y-u(x,t),\\&J_2*v(x,t)-v(x,t)=\int _{{\mathbb {R}}} J_2(x-y)v(y,t) {\text {d}}y-v(x,t), \end{aligned} \end{aligned}$$

respectively. The nonlocal dispersals describe the long-distance movements of infectious agent and infectious humans across cites or countries by air traffic and other long-distance transportation. The system (1.1) can model the spread of the epidemic by fecal–oral transmission such as typhoid fever, cholera, hepatitis A, and poliomyelitis. Fecal–oral transmission occurs when infectious agents in the fecal material from an infected individual contaminate food or water, which is ingested by a second individual. Hence, fecal–oral route transmission is classically associated with contamination of water by human or animal waste. In (1.1), the nonlocal dispersal of infectious agent can be regarded as the long-distance transportation of infected food and water.

If infectious human do not move during the infectious period (for example, they are in sickbeds or quarantined probably), (1.1) reduces to the following degenerate model:

$$\begin{aligned} \left\{ \begin{aligned}&u_t(x,t)=J_1*u(x,t)-u(x,t)-\alpha u(x,t)+h(v(x,t)),&x\in {\mathbb {R}},t\in {\mathbb {R}},\\&v_t(x,t)=-\beta v(x,t)+g(u(x,t)),&x\in {\mathbb {R}},t\in {\mathbb {R}}. \end{aligned}\right. \nonumber \\ \end{aligned}$$

(1.2)

In addition, when the movements of infectious agent and infectious human population happen only between adjacent spatial locations, the local dispersal is more realistic and the local dispersal model is written as

$$\begin{aligned} \left\{ \begin{aligned}&u_t(x,t)=\Delta u(x,t)-\alpha u(x,t)+h(v(x,t)),&x\in {\mathbb {R}},t\in {\mathbb {R}},\\&v_t(x,t)=\Delta v(x,t)-\beta v(x,t)+g(u(x,t)),&x\in {\mathbb {R}},t\in {\mathbb {R}}. \end{aligned}\right. \end{aligned}$$

(1.3)

The model (1.3) has been previously considered by Capasso and Maddalena (1981, 1982) to study the spread of cholera in the European Mediterranean region in 1973. With some suitable modifications, (1.3) was also used by Capasso and Wilson (1997) to model other epidemics with fecal–oral transmission.

In these epidemic models, the study of the wave propagation phenomena such as traveling wave solutions and entire solutions has attracted a number of researchers. A traveling wave solution is a special solution that keeps its shape while moving at a constant speed, and it can describe the spread of the infectious disease from an outbreak to an endemic disease. An entire solution means a classical solution defined in the whole space and at all time $t \in {\mathbb {R}}$. Mathematically, the study of entire solutions can help us to understand the structures of the global attractors, and biologically, different entire solutions represent different propagation patterns of disease. Hence, it is important and interesting to find more entire solutions. Since the pioneering works on reaction–diffusion (local dispersal) equations by Hamel and Nadirashvili (1999, 2001) for monostable case and Yagisita Yagisita (2003) for bistable case, there have been large amounts of research on entire solutions for various models, see, e.g., Chen and Guo (2005), Chen et al. (2006), Fukao et al. (2004), Li et al. (2008), and Morita and Ninomiya (2006). Particularly, for the local dispersal model (1.3), we refer to Hsu and Yang (2013), Volpert et al. (1994), Xu and Zhao (2005), and Zhao and Wang (2004) for the works on traveling wave solutions, and Wu and Hsu (2016) for the works on entire solutions. For the nonlocal dispersal model (1.1), we refer to Li et al. (2017) and Meng et al. (2019) for the works of monostable traveling wave solutions and entire solutions. We also refer to Hu et al. (2015) for the works of monostable traveling wave solutions in m-component cooperative systems which contain (1.1) as a special example.

Note that the aforementioned works for (1.1) have essentially considered only monostable traveling wave solutions and two-wave entire solutions. To the best of our knowledge, there is no result about bistable traveling wave solutions or three-wave (or more) entire solutions of (1.1). Here, we say (u, v) is a N-wave entire solution if it originates from the N traveling wave solutions $(c_j, \varvec{\phi }_j)_{j=1,\ldots , N}$, in the sense that,

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty } \sum \limits _{j=1}^{N}\left( \sup \limits _{\rho _j(t)\le x\le \rho _{j+1}(t)}\left\| (u,v)(x,t)-\varvec{\phi }_j(x+c_j t+\vartheta _j)\right\| \right) =0, \end{aligned}$$

(1.4)

where $\{\vartheta _j\}_{j=1,\ldots , N}$ denotes some constant and $\{\rho _j(t)\}_{j=1,\ldots , N+1}$ is some function satisfying $\rho _j(t)<\rho _{j+1}(t)$, $\rho _1(t)=-\infty$, and $\rho _{N+1}(t)=+\infty$. In fact, even for (1.2), there are almost no results on bistable traveling wave solutions and N-wave entire solutions except the works by Zhang et al. (2016) for two-wave entire solutions and Wu et al. (2018) for three-wave entire solutions. However, the study of (1.1) is much more difficult than (1.2), since (1.1) is essentially a two-component nonlocal dispersal (integral) system but (1.2) can be transformed into a scalar nonlocal dispersal equation by substituting $v(x,t)=e^{-\beta t}v(x,0)+\int _0^t e^{\beta (s-t)}g(u(x,s)){\text {d}}s$ (from the second equation) into the first equation of (1.2). We also refer to Chen et al. (2018) and Dong et al. (2021) for the three-wave entire solutions of scalar bistable equations, and Guo and Wu (2019) for Lotka–Volterra diffusion–competition models.

The aim of this paper is to study monostable and bistable traveling wave solutions and three-wave entire solutions in the system (1.1) with two stable equilibria $E_0=(0,0)$ and $E_2\in$ Int(${\mathbb {R}}_+^2$) and an unstable equilibrium $E_1\in (E_0, E_2)$. Throughout this paper, we always use the standard ordering in ${\mathbb {R}}^2$. That is, for any $u=(u_1,u_2),v=(v_1,v_2)$, we write $u\ge v$ if $u_i\ge v_i$, $i=1,2$; $u \gg v$ if $u_i> v_i$, $i=1,2$; and $u>v$ if $u_i\ge v_i$ with $i=1,2$ but $u\ne v$. For any $r_1,r_2\in {\mathbb {R}}^2$ with $r_1<r_2$, we define $[r_1,r_2]= \{u\in {\mathbb {R}}^2:r_1\le u\le r_2\}$ and $(r_1,r_2) = \{u\in {\mathbb {R}}^2:r_1< u< r_2\}$. For any $v \in {\mathbb {R}}^2$, $\Vert v\Vert$ means the Euclidean norm.

First, we study both monostable and bistable traveling wave solutions. Note that although (1.1) is bistable in the interval $[E_0,E_2]$, it also has two monostable intervals $[E_0,E_1]$ and $[E_1,E_2]$. In the intervals $[E_0,E_1]$ and $[E_1,E_2]$, we give a summary of the existence and asymptotic behavior of all monostable traveling wave solutions by applying and developing the known results for $[E_0,E_1]$ (see Li et al. 2017; Meng et al. 2019) into $[E_1, E_2]$. Furthermore, we find some relationship of monostable traveling wave solutions between these two monostable intervals, that is, the rightward (or leftward) traveling wave solution in $[E_0,E_1]$ has the same minimal wave speed as that in $[E_1,E_2]$, and when their wave speeds are equal, they also have the same exponential decay rate. In the interval $[E_0,E_2]$, we give the existence of bistable traveling wave solutions by introducing the result given by Fang and Zhao (2015) for monotone semiflows with weak compactness. The asymptotic behavior of these bistable traveling wave solutions is also obtained by studying the associated eigenvalue problems.

Second, we construct twelve types of three-wave entire solutions. In the construction, we only choose traveling wave solutions with nonzero wave speeds. Under this consideration, these twelve types contain all possibilities of three-wave entire solutions which originate from three traveling wave solutions $(c_j, \varvec{\phi }_j)_{j=1,2, 3}$ with different nonzero wave speeds. The sketchy profiles as $t\rightarrow -\infty$ of these entire solutions are depicted by Figs. 1 and  2. In fact, the construction of a N-wave entire solution originating from $(c_j, \varvec{\phi }_j)_{j=1,\ldots , N}$ is essentially a procedure linking the profiles of $\varvec{\phi }_j$ with $\varvec{\phi }_{j+1}$ for any $j=1,\ldots , N-1$. The tools are a pair of suitable super- and sub-solutions constructed by some appropriate auxiliary functions. Although the idea of linking two traveling wave solutions by super- and sub-solutions comes from Chen et al. (2018) and Morita and Ninomiya (2006), the difficulty is how to construct appropriate auxiliary functions and corresponding super- and sub-solutions for different types of entire solutions and different models. We construct new auxiliary functions and super- and sub-solutions only for the first six types of three-wave entire solutions (see Fig. 1), and the last six types (see Fig. 2) can be easily obtained from the first six types by a symmetric transformation on space ${\mathbb {R}}$. By applying these auxiliary functions and super- and sub-solutions, the existence of these twelve types of entire solutions is given. Under some appropriate conditions, we have that these twelve types of entire solutions are globally Lipschitz continuous with respect to x. In addition, the nonexistence result of entire solutions originating from more than four traveling wave solutions is obtained.

Fig. 1

The sketchy profiles of the first six entire solutions as $t\rightarrow -\infty$

Fig. 2

The sketchy profiles of the last six entire solutions as $t\rightarrow -\infty$

The rest of this paper is organized as follows. In Sect. 2, we give the existence and asymptotic behavior of all monostable and bistable traveling wave solutions of (1.1). In Sect. 3, we focus only on the construction of the three-wave entire solutions. The nonexistence result of entire solutions originating from more than four traveling wave solutions is also obtained. The proofs of three important lemmas in Sect. 3 are given in the Appendix.

2 Traveling Wave Solutions

In this section, we study monostable and bistable traveling wave solutions of (1.1). We first give the main assumptions. Let $g(\cdot )$ and $h(\cdot )$ be two functions satisfying

• (J1) $g\in C^{1+\gamma _1}({\mathbb {R}})$, $h\in C^{1+\gamma _2}({\mathbb {R}})$ for some $\gamma _1,\gamma _2\in (0,1)$, $g(0)=h(0)=0$, $g'(u)>0$ for $u\ge 0$ and $h'(v)>0$ for $v\ge 0$, and the function $f(u)\triangleq h(\frac{g(u)}{\beta })-\alpha u$ is bistable, in the sense that there exists $u_1^*\in {\mathbb {R}}^+$ and $u_2^*\in {\mathbb {R}}^+$ such that $f(u_1^*)=f(u_2^*)=0$, $f'(0)=g'(0)h'(0)/\beta -\alpha <0$, $f'(u_1^*)>0$, and $f'(u_2^*)<0$;

• (J2) $g'(u_1^*)\ge (g(u)-g(u_1^*))/(u-u_1^*)$ for $u\in [0,u_2^*]$ and $h'(v_1^*)\ge (h(v)-h(v_1^*))/(v-v_1^*)$ for $v\in [0,v_2^*]$, where $v_1^*\triangleq g(u_1^*)/\beta$ and $v_2^*\triangleq g(u_2^*)/\beta$.

By (J1), it is clear that (1.1) admits three nonnegative equilibria: two stable equilibria $E_0=(0,0)$ and $E_2=(u_2^*,v_2^*)$, and an unstable equilibrium $E_1=(u_1^*,v_1^*)$. Similar to the Fisher–KPP assumption (namely $f(u)\leqslant f'(0)u$) in a monostable reaction–diffusion equation, the purpose of (J2) is to ensure that the minimal wave speed is linearly selected. For example, according to Corollary 3.2 in Xu et al. (2020), we can obtain that for the monostable traveling wave solution connecting $E_1$ and $E_2$, the minimal wave speed $c_\mathrm{min}$ must satisfy $c_\mathrm{min} \ge c_L$, where $c_L$ is defined in (2.2) later. If $c_\mathrm{min}=c_L$, then we say that the minimal wave speed is linearly selected; otherwise, i.e., $c_\mathrm{min} > c_L$, we say that the minimal wave speed is nonlinearly selected. Next, we give an example of g(u) and h(v) as follows:

$$\begin{aligned} g(u)=\beta \left[ sin\left( \frac{\pi }{2}u-\frac{\pi }{2}\right) +1\right] , h(v)=\alpha v. \end{aligned}$$

Then, the model (1.1) has three equilibria (0, 0), (1, 1), and (2, 2). Some calculations show that $g'(0)h'(0)=g'(2)h'(2)=0<\alpha \beta$, which implies that (0, 0) and (2, 2) are stable, and $g'(1)h'(1)=\frac{\pi }{2}\alpha \beta >\alpha \beta$, which means that (1, 1) is unstable.

Suppose that $J_1(\cdot )$ and $J_2(\cdot )$ are two continuous and nonnegative functions satisfying

• (J3) For $i\in \{1,2\}$, $J_i\in C^1({\mathbb {R}})$, $J_i'\in L^1({\mathbb {R}})$, $\int _{{\mathbb {R}}}J_i(x) dx=1$ and $J_i(x)\ge 0$ for $x\in {\mathbb {R}}$. Moreover, there exist $x_i^-\in {\mathbb {R}}^-$ and $x_i^+\in {\mathbb {R}}^+$ such that $J_i(x_i^\pm )>0$;

• (J4) $J_i$ has compact support and denote $m:=\sup \{|y|:y\in \text {supp}(J_i),i=1,2\}>0$.

Note that $J_1$ and $J_2$ in this paper may be asymmetric on ${\mathbb {R}}$.

Let $(u(x, t), v(x, t)) = (\varphi (\xi ),\psi (\xi ))$ with $\xi = x + ct$ be a traveling wave solution of (1.1) connecting $E_i$ and $E_j$, where $(\varphi ,\psi )\in C^1({\mathbb {R}},{\mathbb {R}}^2)$. By substituting it into (1.1), we get that

$$\begin{aligned} \left\{ \begin{aligned} c\varphi '(\xi )=J_1*\varphi (\xi )-\varphi (\xi )-\alpha \varphi (\xi )+h(\psi (\xi )), \\ c\psi '(\xi )=J_2*\psi (\xi )-\psi (\xi )-\beta \psi (\xi )+g(\varphi (\xi )), \\ \end{aligned} \right. \end{aligned}$$

(2.1)

and the asymptotic boundary conditions are

$$\begin{aligned} \lim \limits _{\xi \rightarrow -\infty }(\varphi (\xi ), \psi (\xi ))=E_i, \lim \limits _{\xi \rightarrow +\infty }(\varphi (\xi ), \psi (\xi ))=E_j. \end{aligned}$$

The main results of this section consist of two parts. First, in Sect. 2.1, we summarize the results about the existence and asymptotic behavior of all monostable traveling wave solutions in intervals $[E_0, E_1]$ and $[ E_1, E_2]$, and further find some relationship between the traveling wave solutions in these two monostable intervals. Second, in Sect. 2.2, we introduce the theory by Fang and Zhao (2015) about the monotone semiflows with weak compactness and obtain the existence of bistable traveling wave solutions. The asymptotic behavior of bistable traveling wave solutions is also given by studying the corresponding eigenvalue problem.

2.1 Monostable Traveling Wave Solutions

Although the system (1.1) satisfies the bistable assumption (J1), it also admits some monostable traveling wave solutions, which play a very important role in the construction of entire solutions in Sect. 3. Indeed, if we restrict (u, v) in the interval $[E_0,E_1]$ or $[E_1,E_2]$, then (1.1) can be regarded as a monostable system. Define

$$\begin{aligned} c(\lambda )=\frac{1}{2\lambda }\left[ F_u(0, \lambda )+F_v(0, \lambda )+\sqrt{(F_u(0, \lambda )-F_v(0, \lambda ))^2+4 g'(u_1^*)h'(v_1^*)}\right] ~~\text {for}~\lambda \ne 0, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&F_u(c, \lambda )=\int _{{\mathbb {R}}}J_1(z)e^{-\lambda z}dz-1-c\lambda -\alpha ,\\&F_v(c, \lambda )=\int _{{\mathbb {R}}}J_2(z)e^{-\lambda z}dz-1-c\lambda -\beta .\\ \end{aligned} \end{aligned}$$

By Theorem 2.1 in Xu et al. (2020), we can define

$$\begin{aligned} c_L\triangleq \inf _{\lambda \in {\mathbb {R}}^+} c(\lambda )~~\text {and}~~c_R\triangleq \sup \limits _{\lambda \in \mathbb R^-} c(\lambda ), \end{aligned}$$

(2.2)

and it holds that

$$\begin{aligned} c_R<c_L. \end{aligned}$$

(2.3)

The existence of all monostable traveling wave solutions in (1.1) is given by the following two theorems, which are straightforward consequences of Hu et al. (2015, Theorem 3.6), Li et al. (2017, Theorem 2.1), and Xu et al. (2020, Corollary 3.2).

Theorem 2.1

Assume (J1)–(J4) hold. Then, for any $c_1\le c_R$, (1.1) admits a nondecreasing monostable traveling wave solution $\Phi _1(x+c_1t)$ satisfying

$$\begin{aligned} \Phi _1(-\infty )=E_0,~~\Phi _1(+\infty )=E_1. \end{aligned}$$

(2.4)

Moreover, (1.1) has no traveling wave solution $\Phi _1(x+c_1t)$ with $c_1> c_R$ satisfying (2.4). Similarly, for any ${\hat{c}}_1\ge c_L$, (1.1) admits a nonincreasing monostable traveling wave solution ${\hat{\Phi }}_1(x+{\hat{c}}_1t)$ satisfying

$$\begin{aligned} ~{\hat{\Phi }}_1(-\infty )=E_1,~~{\hat{\Phi }}_1(+\infty )=E_0. \end{aligned}$$

(2.5)

Moreover, (1.1) has no traveling wave solution ${\hat{\Phi }}_1(x+{\hat{c}}_1t)$ with ${{\hat{c}}}_1< c_L$ satisfying (2.5).

Theorem 2.2

Assume (J1)–(J4) hold. Then, for any $c_2\ge c_L$, (1.1) admits a nondecreasing monostable traveling wave solution $\Phi _2(x+c_2t)$ satisfying

$$\begin{aligned} \Phi _2(-\infty )=E_1,~~\Phi _2(+\infty )=E_2. \end{aligned}$$

(2.6)

Moreover, (1.1) has no traveling wave solution $\Phi _2(x+c_2t)$ with $c_2< c_L$ satisfying (2.6). Similarly, for any ${\hat{c}}_2\le c_R$, (1.1) admits a nonincreasing monostable traveling wave solution ${\hat{\Phi }}_2(x+{\hat{c}}_2t)$ satisfying

$$\begin{aligned} {\hat{\Phi }}_2(-\infty )=E_2,~~{\hat{\Phi }}_2(+\infty )=E_1. \end{aligned}$$

(2.7)

Moreover, (1.1) has no traveling wave solution ${\hat{\Phi }}_2(x+{\hat{c}}_2t)$ with ${\hat{c}}_2> c_R$ satisfying (2.7).

Remark 2.3

The traveling wave solution with nonzero wave speed in Theorems 2.1 and 2.2 is unique up to translation. When the wave speed is equal to zero, the continuous traveling wave solution is unique up to translation. We refer to Coville et al. (2008) for more details about the uniqueness of traveling wave solutions in a nonlocal dispersal equation.

Now, we consider the asymptotic behavior of these traveling wave solutions. For c and $\lambda \in {\mathbb {R}}$, define the functions

$$\begin{aligned} \begin{aligned}&\Gamma _0(c, \lambda )=F_u(c, \lambda )\cdot F_v(c, \lambda )-h'(0)g'(0),\\&\Gamma _1(c, \lambda )=F_u(c, \lambda )\cdot F_v(c, \lambda )-h'(v_1^*)g'(u_1^*),\\&\Gamma _2(c, \lambda )=F_u(c, \lambda )\cdot F_v(c, \lambda )-h'(v_2^*)g'(u_2^*).\\ \end{aligned} \end{aligned}$$

For any fixed $c\in {\mathbb {R}}$, we consider the equation $\Gamma _i(c,\lambda )=0$ with $i\in \{1,2,3\}$. The following two lemmas study the number of its roots.

Lemma 2.4

Assume (J1), (J3), and (J4) hold. Then, for any fixed $c\in \mathbb R$, $\Gamma _i(c,\lambda )=0$ with $i\in \{0,2\}$ has exactly two negative roots and two positive roots.

Proof

We only consider the equation $\Gamma _0(c,\lambda )=0$, since the other case is similar. Some simple calculations imply that

$$\begin{aligned} F_u(c,0)<0, \lim \limits _{\lambda \rightarrow \pm \infty }F_u(c,\lambda )=+\infty ,~\text {and}~ \frac{\partial ^2 F_u(c,\lambda )}{\partial \lambda ^2}>0~\text {for any}~~\lambda \in {\mathbb {R}}. \end{aligned}$$

Then, the equation $F_u(c,\lambda )=0$ has two real roots $\lambda _u^-(c)\in (-\infty ,0)$ and $\lambda _u^+(c)\in (0,+\infty )$ by the continuity of $F_u(c,\lambda )$ with respect to $\lambda$. Similarly, the equation $F_v(c,\lambda )=0$ has two real roots $\lambda _v^-(c)\in (-\infty ,0)$ and $\lambda _v^+(c)\in (0,+\infty )$. It is easy to check that

$$\begin{aligned} \begin{aligned}&\Gamma _0(c,\pm \infty )=+\infty ,~\Gamma _0(c,0)=\alpha \beta -h'(0)g'(0)>0,\\&\Gamma _0(c,\lambda )<0~\text {for}~\lambda \in [\min \{\lambda _u^+(c),\lambda _v^+(c)\},\max \{\lambda _u^+(c),\lambda _v^+(c)\}]. \end{aligned} \end{aligned}$$

Thus, the equation $\Gamma _0(c,\lambda )=0$ has a unique root in $(\max \{\lambda _u^+(c),\lambda _v^+(c)\},+\infty )$ and at least one root in $(0, \min \{\lambda _u^+(c),\lambda _v^+(c)\})$. Applying a similar method to the proof of Li et al. (2017, Lemma 2.6), we can obtain that $\Gamma _0(c,\lambda )=0$ has at most one root in $(0, \min \{\lambda _u^+(c),\lambda _v^+(c)\})$. Then, the equation $\Gamma _0(c,\lambda )=0$ has two positive roots. Similarly, the equation $\Gamma _0(c,\lambda )=0$ has two negative roots. It completes the proof. $\square$

Lemma 2.5

Assume (J1)–(J4) hold. For $c>c_L$, $\Gamma _1(c, \lambda )=0$ has and only has three different positive roots. When $c= c_L$, $\Gamma _1(c, \lambda )=0$ has two different positive roots. Similarly, for $c<c_R$, $\Gamma _1(c, \lambda )=0$ has and only has three different negative roots. When $c= c_R$, $\Gamma _1(c, \lambda )=0$ has two different negative roots.

Proof

For the case $c\ge c_L$, the proof can be obtained by a similar method to Li et al. (2017, Lemma 2.6). For the case $c<c_R$, consider

$$\begin{aligned} {\tilde{\Gamma }}(c,\lambda )\triangleq \Gamma _1(-c, -\lambda )=F_u(-c, -\lambda )\cdot F_v(-c, -\lambda )-h'(v_1^*)g'(u_1^*) \end{aligned}$$

with

$$\begin{aligned} \begin{aligned}&F_u(-c, -\lambda )=\int _{{\mathbb {R}}}J_1(z)e^{\lambda z}{\text {d}}z-1-c\lambda -\alpha ,\\&F_v(-c, -\lambda )=\int _{{\mathbb {R}}}J_2(z)e^{\lambda z}{\text {d}}z-1-c\lambda -\beta .\\ \end{aligned} \end{aligned}$$

By substituting $J_i(z)$ for $J_i(-z)$ with $i\in \{1,2\}$, we see that the equation ${\tilde{\Gamma }}(c,\lambda )=0$ has the same property of roots as $\Gamma _1(c, \lambda )=0$. Namely, ${\tilde{\Gamma }}(c, \lambda )=0$ has and only has three different positive roots for $c>{{\tilde{c}}}_L$ and two different positive roots for $c={{\tilde{c}}}_L$, where

$$\begin{aligned} {{\tilde{c}}}_L= & {} \inf \limits _{\lambda \in \mathbb R^+}\left\{ \frac{1}{2\lambda }\left[ F_u(0,-\lambda )+F_v(0,-\lambda )\right. \right. \\&\quad \left. \left. +\sqrt{(F_u(0,-\lambda )-F_v(0,-\lambda ))^2+4 g'(u_1^*)h'(v_1^*)}\right] \right\} . \end{aligned}$$

A simple calculation shows that ${{\tilde{c}}}_L=-c_R$, which completes the proof. $\square$

For $c\in {\mathbb {R}}$, we define

$$\begin{aligned} \begin{aligned}&\lambda _0^+(c)=\text {the smallest positive root of}~{ \Gamma _0(c,\lambda )=0},\\&\lambda _0^-(c)=\text {the largest negative root of}~{ \Gamma _0(c,\lambda )=0},\\&\lambda _2^+(c)=\text {the smallest positive root of}~{ \Gamma _2(c,\lambda )=0},\\&\lambda _2^-(c)=\text {the largest negative root of }~{\Gamma _2(c,\lambda )=0}, \\ \end{aligned} \end{aligned}$$

(2.8)

and

$$\begin{aligned} \begin{aligned}&\lambda _1^+(c)=\text {the smallest positive root of}~{ \Gamma _1(c, \lambda )=0}~ \text {for}~{ c\ge c_L>0},\\&\lambda _1^-(c)=\text {the largest negative root of }~{\Gamma _1(c, \lambda )=0}~ \text {for}~ {c\le c_R<0}. \end{aligned} \end{aligned}$$

The asymptotic behavior of all monostable traveling wave solutions is given by the following two theorems. Since the calculations for different cases in the proof are analogous and similar to Li et al. (2017, Theorem 2.2), we omit them.

Theorem 2.6

Assume (J1)–(J4) hold. Let $\Phi _1(x+c_1t)$ and ${\hat{\Phi }}_1(x+{\hat{c}}_1t)$ be the nondecreasing and nonincreasing traveling wave solutions with $c_1 \ne 0$ and ${{\hat{c}}}_1 \ne 0$ in (1.1) as given in Theorem 2.1. Then, all the following limits exist in ${\mathbb {R}}^+\times {\mathbb {R}}^+$ and

• (i) for $c_1<c_R$, $\lim \limits _{\xi \rightarrow +\infty }(E_1-\Phi _1(\xi ))/e^{\lambda _1^-(c_1)\xi }=-\lim \limits _{\xi \rightarrow +\infty }\Phi '_1(\xi )/(\lambda _1^-(c_1)e^{\lambda _1^-(c_1)\xi })$, for $c_1{=}c_R$, $\lim \limits _{\xi \rightarrow +\infty }(E_1-\Phi _1(\xi ))/(|\xi |e^{\lambda _1^-(c_1)\xi }){=}-\lim \limits _{\xi \rightarrow +\infty }\Phi '_1(\xi )/(\lambda _1^-(c_1)|\xi |e^{\lambda _1^-(c_1)\xi })$, for $c_1\le c_R$, $\lim \limits _{\xi \rightarrow -\infty }\Phi _1(\xi )/e^{\lambda _0^+(c_1)\xi }=\lim \limits _{\xi \rightarrow -\infty }\Phi '_1(\xi )/(\lambda _0^+(c_1)e^{\lambda _0^+(c_1)\xi })$,

• (ii) for ${{\hat{c}}}_1>c_L$, $\lim \limits _{\xi \rightarrow -\infty }(E_1-{{\hat{\Phi }}}_1(\xi ))/e^{\lambda _1^+({{\hat{c}}}_1)\xi }=-\lim \limits _{\xi \rightarrow -\infty }{{\hat{\Phi }}}'_1(\xi )/(\lambda _1^+({{\hat{c}}}_1)e^{\lambda _1^+({{\hat{c}}}_1)\xi })$, for ${{\hat{c}}}_1=c_L$, $\lim \limits _{\xi \rightarrow -\infty }( E_1-{{\hat{\Phi }}}_1(\xi ))/(|\xi |e^{\lambda _1^+({{\hat{c}}}_1)\xi })=-\lim \limits _{\xi \rightarrow -\infty }{{\hat{\Phi }}}'_1(\xi )/(\lambda _1^+({{\hat{c}}}_1)|\xi |e^{\lambda _1^+({{\hat{c}}}_1)\xi })$, for ${{\hat{c}}}_1\ge c_L$, $\lim \limits _{\xi \rightarrow +\infty }\hat{\Phi }_1(\xi )/e^{\lambda _0^-({{\hat{c}}}_1)\xi }=\lim \limits _{\xi \rightarrow +\infty }\hat{\Phi }'_1(\xi )/(\lambda _0^-({{\hat{c}}}_1)e^{\lambda _0^-({{\hat{c}}}_1)\xi })$.

Theorem 2.7

Assume (J1)–(J4) hold. Let $\Phi _2(x+c_2t)$ and ${\hat{\Phi }}_2(x+{\hat{c}}_2t)$ be the nondecreasing and nonincreasing traveling wave solutions with $c_2 \ne 0$ and ${{\hat{c}}}_2 \ne 0$ in (1.1) as given in Theorem  2.2. Then, all the following limits exist in ${\mathbb {R}}^+\times {\mathbb {R}}^+$ and

• (i) for $c_2>c_L$, $\lim \limits _{\xi \rightarrow -\infty }(\Phi _2(\xi )-E_1)/e^{\lambda _1^+(c_2)\xi }=\lim \limits _{\xi \rightarrow -\infty }\Phi '_2(\xi )/(\lambda _1^+(c_2)e^{\lambda _1^+(c_2)\xi })$, for $c_2{=}c_L$, $\lim \limits _{\xi \rightarrow -\infty }(\Phi _2(\xi )-E_1)/(|\xi |e^{\lambda _1^+(c_2)\xi }){=}\lim \limits _{\xi \rightarrow -\infty }\Phi '_2(\xi )/(\lambda _1^+(c_2)|\xi |e^{\lambda _1^+(c_2)\xi })$, for $c_2\ge c_L$, $\lim \limits _{\xi \rightarrow +\infty }(E_2-\Phi _2(\xi ))/e^{\lambda _2^-(c_2)\xi }=-\lim \limits _{\xi \rightarrow +\infty }\Phi '_2(\xi )/(\lambda _2^-(c_2)e^{\lambda _2^-(c_2)\xi })$,

• (ii) for ${{\hat{c}}}_2<c_R$, $\lim \limits _{\xi \rightarrow +\infty }({{\hat{\Phi }}}_2(\xi )-E_1)/e^{\lambda _1^-({{\hat{c}}}_2)\xi }=\lim \limits _{\xi \rightarrow +\infty }{{\hat{\Phi }}}'_2(\xi )/(\lambda _1^-({{\hat{c}}}_2)e^{\lambda _1^-({{\hat{c}}}_2)\xi })$, for ${{\hat{c}}}_2=c_R$, $\lim \limits _{\xi \rightarrow +\infty }({{\hat{\Phi }}}_2(\xi )-E_1)/(|\xi |e^{\lambda _1^-({{\hat{c}}}_2)\xi })=\lim \limits _{\xi \rightarrow +\infty }{{\hat{\Phi }}}'_2(\xi )/(\lambda _1^-({{\hat{c}}}_2)|\xi |e^{\lambda _1^-({{\hat{c}}}_2)\xi })$, for ${{\hat{c}}}_2\le c_R$, $\lim \limits _{\xi \rightarrow -\infty }(E_2-\hat{\Phi }_2(\xi ))/e^{\lambda _2^+({{\hat{c}}}_2)\xi }=-\lim \limits _{\xi \rightarrow -\infty }\hat{\Phi }'_2(\xi )/(\lambda _2^+({{\hat{c}}}_2)e^{\lambda _2^+({{\hat{c}}}_2)\xi })$.

From Theorems 2.1, 2.2, 2.6, and 2.7, we can get some relationship between the traveling wave solutions in the intervals $[E_0,E_1]$ and $[E_1,E_2]$. We see that the two rightward traveling wave solutions $\Phi _1$ and ${{\hat{\Phi }}}_2$ have the same minimal wave speed. And when their wave speeds are equal, they also have the same exponential decay rate as $\xi \rightarrow +\infty$. The difference is that $\Phi _1$ is in the interval $[E_0,E_1]$, while ${{\hat{\Phi }}}_2$ is in $[E_1,E_2]$, and hence the convergence $\Phi _1\rightarrow E_1$ is from below, while the convergence $\Phi _2\rightarrow E_1$ is from above. Similarly, for the two leftward traveling wave solutions $\hat{\Phi }_1$ and $\Phi _2$, although the convergence $\hat{\Phi }_1\rightarrow E_1$ is from below, while the convergence $\Phi _2\rightarrow E_1$ is from above as $\xi \rightarrow -\infty$, they have the same minimal wave speed, and when their wave speeds are equal, they also have the same exponential decay rate.

2.2 Bistable Traveling Wave Solutions

In this subsection, we introduce the theory developed by Fang and Zhao (2015) about the monotone semiflows with weak compactness, and study the existence and asymptotic behavior of bistable traveling wave solutions in (1.1).

First, we give some notations. Let ${\mathcal {X}}$ be an ordered Banach space with the norm $\Vert \cdot \Vert _{{\mathcal {X}}}$ and the cone ${\mathcal {X}}^+$. Assume that $\text {Int}({\mathcal {X}}^+)$ is not empty. Let ${\mathcal {C}}$ denote the set of all bounded and continuous functions from ${\mathbb {R}}$ to ${\mathcal {X}}$ and let ${\mathcal {M}}$ denote the set of all nondecreasing functions from ${\mathbb {R}}$ to ${\mathcal {X}}$. We equip ${\mathcal {C}}$ and ${\mathcal {M}}$ with the compact open topology and the standard cone consisting of all nonnegative functions. For any $u,v\in {\mathcal {H}}$ with ${\mathcal {H}}\in \{{\mathcal {X}},{\mathcal {C}},{\mathcal {M}}\}$, we write $u\ge v$ if $u-v\in {\mathcal {H}}^+$; $u>v$ if $u\ge v$ but $u\ne v$; and $u\gg v$ if $u-v\in \text {Int}({{\mathcal {H}}}^+)$. For any $r_1,r_2\in {\mathcal {X}}$ with $r_1<r_2$, we define ${{\mathcal {H}}}_{[r_1,r_2]}= \{u\in {\mathcal {H}}:r_1\le u\le r_2\}$ and write ${{\mathcal {H}}}_{[0,r]}$ as ${{\mathcal {H}}}_r$ for short. We say $\{Q_t\}_{t\ge 0}$ is a semiflow on ${\mathcal {M}}_{{\textbf {b}}}$ when it satisfies that $Q_0= I$ where I is a identity map, $Q_t\circ Q_s = Q_{t+s}$ for any $t,s > 0$, and $Q_{t_n}[\varphi _n](y) \rightarrow Q_t[\varphi ] (y)$ in ${\mathcal {X}}_{{{\textbf {b}}}}$ for almost all $y\in {\mathbb {R}}$ whenever $t_n \rightarrow t$ and $\varphi _n \rightarrow \varphi$ in ${\mathcal {M}}_{{{\textbf {b}}}}$.

Let Q be a map from ${\mathcal {M}}_{{{\textbf {b}}}}$ to ${\mathcal {M}}_{{{\textbf {b}}}}$ with two fixed points ${{\textbf {0}}}$ and ${{\textbf {b}}}$. We denote E by the set consisting of all fixed points of Q restricted on ${\mathcal {X}}_{{{\textbf {b}}}}$. The following definition states the stability of fixed points.

Definition 2.8

For $Q:{\mathcal {M}}_{{{\textbf {b}}}}\rightarrow {\mathcal {M}}_{{{\textbf {b}}}}$, a fixed point ${{\textbf {a}}}\in E$ is said to be strongly stable from below if there is a constant $\delta >0$ and a unit vector ${{\textbf {e}}}\in \text {Int}({\mathcal {X}}^+)$ such that for any $\eta \in (0,\delta ]$,

$$\begin{aligned} Q[{{\textbf {a}}}-\eta {{\textbf {e}}}]\gg {{\textbf {a}}}-\eta {{\textbf {e}}}. \end{aligned}$$

(2.9)

Strong instability from below is defined by reversing the inequality (2.9). Similarly, we can define strong stability and strong instability from above.

Now, we give some fundamental assumptions on Q.

• (B1) (Translation Invariance) $T_y \circ Q[\phi ] = Q \circ T_y[\phi ]$ for all $\phi \in {\mathcal {M}}_{{{\textbf {b}}}}$ and $y \in {\mathbb {R}}$, where $T_y[\phi ](x):=\phi (x-y)$.

• (B2) (Continuity) $Q:{\mathcal {M}}_{{{\textbf {b}}}}\rightarrow {\mathcal {M}}_{{{\textbf {b}}}}$ is continuous in the sense that if $\phi _n\rightarrow \phi$ in ${\mathcal {M}}_{{{\textbf {b}}}}$, then $Q[\phi _n](x) \rightarrow Q[\phi ](x)$ in ${\mathcal {X}}_{{{\textbf {b}}}}$ for almost all $x \in {\mathbb {R}}$.

• (B3) (Monotonicity) Q is order preserving in the sense that $Q[\phi ] \ge Q[\varphi ]$ whenever $\phi \ge \varphi$.

• (B4) (Weak Compactness) For any fixed $y \in {\mathbb {R}}$, the set $Q[{\mathcal {M}}_{{{\textbf {b}}}}](y)$ is precompact in ${\mathcal {X}}_{{{\textbf {b}}}}$.

• (B5) (Bistability) Fixed points ${{\textbf {0}}}$ and ${{\textbf {b}}}$ are strongly stable from above and below, respectively, for the map $Q:{\mathcal {X}}_{{{\textbf {b}}}}\rightarrow {\mathcal {X}}_{{{\textbf {b}}}}$. The set $E\setminus \{{{\textbf {0}}},{{\textbf {b}}}\} \subset {\mathcal {X}}_{{{\textbf {b}}}}$ is totally unordered.

For each ${{\textbf {a}}}\in E\setminus \{{{\textbf {0}}},{{\textbf {b}}}\}$, it follows from the assumption (B5) that Q has no other fixed point in both ${\mathcal {M}}_{[{{\textbf {a}}},{{\textbf {b}}}]}\setminus \{{{\textbf {a}}},{{\textbf {b}}}\}$ and ${\mathcal {M}}_{[{{\textbf {0}}},{{\textbf {a}}}]}\setminus \{{\textbf {0}},{{\textbf {a}}}\}$. In view of the assumption (B5), there are constants $\delta _1,\delta _2>0$ and unit vectors ${{\textbf {e}}}_1,{{\textbf {e}}}_2\in \text {Int}({\mathcal {X}}^+)$ such that $Q[\eta {{\textbf {e}}}_1]\ll \eta {{\textbf {e}}}_1$ for $\eta \in (0,\delta _1]$ and $Q[{{\textbf {b}}}-\eta {{\textbf {e}}}_2]\gg {{\textbf {b}}}-\eta {{\textbf {e}}}_2$ for $\eta \in (0,\delta _2]$. Denote $\theta ^-=\sup \{\theta \in [0,1]:\theta {{\textbf {a}}}+(1-\theta ){{\textbf {b}}}\in {\mathcal {X}}_{[{{\textbf {b}}}-\delta _2{{\textbf {e}}}_2,{{\textbf {b}}}]}\}$ and $\theta ^+= \sup \{\theta \in [0,1]:\theta {{\textbf {a}}}\in {\mathcal {X}}_{[0,\delta _1{{\textbf {e}}}_1]}\}$. Let $\nu ^-=\theta ^-{{\textbf {a}}}+(1-\theta ^-){{\textbf {b}}}$ and $\nu ^+=\theta ^+{{\textbf {a}}}$. Define

$$\begin{aligned}&c_-^*({{\textbf {a}}},{{\textbf {b}}})= \sup \left\{ c\in {\mathbb {R}}:\lim \limits _{n\rightarrow \infty ,x\ge -cn }Q^n[\phi _{{{\textbf {a}}}}^-](x)={{\textbf {b}}}\right\} {,}\\&c_+^*({{\textbf {0}}},{{\textbf {a}}})= \sup \left\{ c\in {\mathbb {R}}:\lim \limits _{n\rightarrow \infty ,x\le cn }Q^n[\phi _{{{\textbf {a}}}}^+](x)={{\textbf {0}}}\right\} , \end{aligned}$$

(2.10)

where the continuous initial functions $\phi _{{{\textbf {a}}}}^\pm \in {\mathcal {M}}_{{{\textbf {b}}}}$ satisfy that

$$\begin{aligned} \begin{aligned}&\phi _{{{\textbf {a}}}}^-(x)={{\textbf {a}}}~\text {for}~x\le -1,~\phi _{{{\textbf {a}}}}^-(x)=\nu ^-~\text {for}~x\ge 0,\\&\phi _{{{\textbf {a}}}}^+(x)={{\textbf {a}}}~\text {for}~x\ge 1,~\phi _{{{\textbf {a}}}}^+(x)=\nu ^+~\text {for}~ x\le 0. \end{aligned} \end{aligned}$$

We further assume that

• (B6) (Counter-propagation) For each ${{\textbf {a}}}\in E\setminus \{{{\textbf {0}}},{{\textbf {b}}}\}$, $c_-^*({{\textbf {a}}},{{\textbf {b}}})+c_+^*({{\textbf {0}}},{{\textbf {a}}})>0$.

As stated in Fang and Zhao (2015), assumption (B6) can ensure that the spatial propagation directions of the two monostable systems restricted on ${\mathcal {M}}_{[{{\textbf {a}}},{{\textbf {b}}}]}$ and ${\mathcal {M}}_{[{{\textbf {0}}},{{\textbf {a}}}]}$ with front-like initial data are opposite.

Let $C(M,{\mathbb {R}}^d)$ denote the set of all continuous functions from the compact metric space M to the d-dimensional Euclidean space ${\mathbb {R}}^d$ with the maximum norm and the standard cone consisting of all nonnegative functions. The following theorem shows the existence of nondecreasing bistable traveling wave solutions in monotone semiflows with weak compactness. It is a key tool to study bistable traveling wave solutions in (1.1).

Theorem 2.9

(see Fang and Zhao 2015, Theorem 5.3) Let ${\mathcal {X}}=C(M,{\mathbb {R}}^d)$. Assume that $\{Q_t\}_{t\ge 0}$ is a semiflow on ${\mathcal {M}}_{{{\textbf {b}}}}$ and for any $t > 0$, the map $Q_t$ satisfies (B1) and (B3)–(B6). Then, there is a constant $c \in {\mathbb {R}}$ and a nondecreasing function $\phi \in {\mathcal {M}}_{{{\textbf {b}}}}$ connecting ${{\textbf {0}}}$ to ${{\textbf {b}}}$ such that $Q_t[\phi ](x)=\phi (x+c t)$ for $x\in {\mathbb {R}}$.

In what follows, we apply Theorem 2.9 and prove the existence of bistable traveling solutions connecting $E_0$ and $E_2$ in (1.1). Note that we usually choose ${\mathcal {X}}=C(M,{\mathbb {R}}^d)$ to study some more complex systems, such as the time-periodic reaction–diffusion systems and the nonlocal dispersal equations with time delay (see, e.g., Fang and Zhao 2014, 2015). However, in (1.1), we only need to choose ${\mathcal {X}}={\mathbb {R}}^2$ with the Euclidean norm $\Vert \cdot \Vert$, which means that M is a singleton. Let P(t) be the solution semigroup of the following nonlocal dispersal equation:

$$\begin{aligned}\left\{ \begin{aligned}&u_t =J*u-u,~x\in {\mathbb {R}},t>0,\\&u(x,0)=\varphi (x),~x\in {\mathbb {R}}. \end{aligned}\right. \end{aligned}$$

As stated in Weng and Zhao (2006), P(t) can be written as

$$\begin{aligned} P(t)[\varphi ](x)=e^{-t}\sum \limits _{m=0}^{\infty }\frac{t^m}{m!}a_m(\varphi )(x), \end{aligned}$$

(2.11)

where $a_0(\varphi )=\varphi$ and $a_m(\varphi )=J*a_{m-1}(\varphi )$ for $m\ge 1$. Now, we consider the Cauchy problem of the system (1.1) as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} u_t(x,t)=J_1*u(x,t)-u(x,t)-\alpha u(x,t)+h(v(x,t)),&{}\quad ~(x,t)\in {\mathbb {R}}\times {\mathbb {R}}^+,\\ v_t(x,t)=J_2*v(x,t)-v(x,t)-\beta v(x,t)+g(u(x,t)),&{}\quad ~(x,t)\in {\mathbb {R}}\times {\mathbb {R}}^+,\\ (u(x,0),v(x,0))=\phi (x),&{}\quad x\in {\mathbb {R}}. \end{array}\right. \end{aligned}$$

(2.12)

Let

$$\begin{aligned}\pmb {\omega }=(u,v)^T,~~{{\textbf {F}}}(\pmb {\omega })=\begin{pmatrix}-\alpha u+h(v)\\ beta v+g(u)\end{pmatrix}~~~{\text {and}}~~~{{\textbf {P}}}(t)=\text {diag}\left( P_1(t),P_2(t)\right) {,}\end{aligned}$$

where $P_i(t)$ with $i\in \{1,2\}$ is defined by (2.12) with substituting $J_i$ for J. Then, (2.13) can be written as

$$\begin{aligned} \pmb {\omega }(x,t)={{\textbf {P}}}(t)[\phi ](x)+\int _0^t {{\textbf {P}}}(t-s)[{{\textbf {F}}}(\pmb {\omega }(\cdot ,s))](x){\text {d}}s,~~x\in {\mathbb {R}},~t\ge 0. \end{aligned}$$

By applying the standard contracting mapping theorem or the theory of abstract functional–differential equations in Martin and Smith (1990), the existence and uniqueness of the solution of (2.13) can be obtained as follows.

Lemma 2.10

Assume (J1) and (J3) hold. For any $\phi \in {\mathcal {M}}_{E_2}$ (or ${\mathcal {C}}_{E_2}$) with $E_0 \le \phi \le E_2$, the system (1.1) has a unique solution $\pmb {\omega }(x, t; \phi )$ on ${\mathbb {R}} \times [0,\infty )$ satisfying $\pmb {\omega }(x, 0; \phi ) =\phi (x)$, and moreover, $E_0 \le \pmb {\omega }(x, t; \phi )\le E_2$.

For $t>0$, let $Q_t:{\mathcal {M}}_{E_2}\rightarrow {\mathcal {M}}_{E_2}$ satisfy $Q_t(\phi )=\pmb {\omega }(\cdot ,t;\phi )$. It is easy to check that $Q_t$ satisfies assumptions (B1) and (B4) for any $t>0$. Next, we show that $\{Q_t\}_{t\ge 0}$ is a semiflow on ${\mathcal {M}}_{E_2}$ and $Q_t$ satisfies assumption (B3) for any $t>0$.

Lemma 2.11

Assume (J1) and (J3) hold. Then, $\{Q_t\}_{t\ge 0}$ is a monotone semiflow on ${\mathcal {M}}_{E_2}$.

Proof

For any $t>0$, we first show the monotonicity of $Q_t$ on ${\mathcal {M}}_{E_2}$. By (J1), the function ${{\textbf {F}}}:{\mathcal {X}}_{E_2}\rightarrow {\mathbb {R}}^2$ is Lipschitz continuous and cooperative. Thus, we can choose a sufficiently large constant $a>0$ such that the function ${{\textbf {F}}}_a(\pmb {\omega }):={{\textbf {F}}}(\pmb {\omega })+a\pmb {\omega }$ is nondecreasing on ${\mathcal {X}}_{E_2}$. Then, (2.13) can be written as an abstract integral equation on ${\mathcal {M}}_{E_2}$, namely

$$\begin{aligned} \pmb {\omega }(t)=e^{-at}{{\textbf {P}}}(t)[\phi ]+\int _0^t e^{-a(t-s)}{{\textbf {P}}}(t-s)[{{\textbf {F}}}_a(\pmb {\omega }(s))]{\text {d}}s,~ t\ge 0. \end{aligned}$$

By a similar argument to Thieme (1979, Lemma 3.2), we can prove that $Q_t:{\mathcal {M}}_{E_2}\rightarrow {\mathcal {M}}_{E_2}$ is monotone for any $t>0$.

It suffices to prove that $Q_{t}$ is continuous for $(t,\phi )\in {\mathbb {R}}^+\times {\mathcal {M}}_{E_2}$ with respect to the compact open topology. For any given $\phi \in {\mathcal {M}}_{E_2}$, it follows from (1.1) that $\frac{\partial }{\partial t} \pmb {\omega }(x,t;\phi )$ is bounded in ${\mathbb {R}}^2$ for any $(x,t)\in {\mathbb {R}}\times {\mathbb {R}}^+$. Then, there exists a constant $L=L(\phi )>0$ such that

$$\begin{aligned} \Vert \pmb {\omega }(x,t_1;\phi )-\pmb {\omega }(x,t_2;\phi )\Vert \le L|t_1-t_2|~~\text {for any}~ x\in {\mathbb {R}}~\text {and}~t_1,t_2\ge 0. \end{aligned}$$

Therefore, for any $\phi \in {\mathcal {M}}_{E_2}$, the function $t\mapsto Q_t(\phi )=\pmb {\omega }(\cdot ,t;\phi )$ from ${\mathbb {R}}^+$ to ${\mathcal {M}}_{E_2}$ is continuous with respect to the compact open topology.

For any $\phi _1$, $\phi _2\in {\mathcal {M}}_{E_2}$, $\epsilon >0$, and $t_0>0$, we define

$$\begin{aligned}&\gamma (x,t)=(\gamma _1(x,t),\gamma _2(x,t))=(u(x,t;\phi _1)-u(x,t;\phi _2),v(x,t;\phi _1)-v(x,t;\phi _2)),\\&k_1= \sup \limits _{x\in {\mathbb {R}},t\in [0,t_0]}\gamma _1(x,t),\quad k_2= \sup \limits _{x\in {\mathbb {R}},t\in [0,t_0]}\gamma _2(x,t),\\&N_1=\max \limits _{[0,u_2^*]}g'(u),\quad N_2=\max \limits _{[0,v_2^*]}h'(v),\\&\epsilon '=\frac{\sqrt{2}\epsilon }{2(1+\max \{N_1,N_2\}t_0)e^{N_1N_2t_0^2}},\quad \varepsilon = \frac{\epsilon '}{2+e^{t_0}+\max \{N_1,N_2\}t_0},\\&\Omega _{\rho }(z)= [-\rho +z,\rho +z]~\text {for}~ \rho >0,z\in {\mathbb {R}},\quad |\varphi |_{\Omega _{\rho }(z)}=\sup \limits _{x\in \Omega _{\rho }(z)}|\varphi (x)|. \end{aligned}$$

Then, there exist $(x_1^*,t_1^*)$ and $(x_2^*,t_2^*)$ in ${\mathbb {R}}\times [0,t_0]$ such that

$$\begin{aligned} \gamma _1(x,t)\le \gamma _1(x_1^*,t_1^*)+\varepsilon ,~\gamma _2(x,t)\le \gamma _2(x_2^*,t_2^*)+\varepsilon ,~(x,t)\in {\mathbb {R}}\times [0,t_0]. \end{aligned}$$

We choose $M_i$ with $i=1,2$ sufficiently large such that

$$\begin{aligned} \int _{|x_i^*-y|\ge M_i}J_i(y){\text {d}}y\le \frac{\varepsilon }{\max \{k_1,k_2\}},~i=1,2. \end{aligned}$$

First, we consider the case $\phi _1\ge \phi _2$. The monotonicity of $Q_t$ implies $\pmb {\omega }(x,t;\phi _1)\ge \pmb {\omega }(x,t;\phi _2)$. It follows that

$$\begin{aligned}\left\{ \begin{aligned} \gamma _1(x,t)\le P_1(t)[\gamma _1(\cdot ,0)](x)+\int _0^tP_1(t-s)[N_2\gamma _2(\cdot ,s)](x){\text {d}}s,\\ \gamma _2(x,t)\le P_2(t)[\gamma _2(\cdot ,0)](x)+\int _0^tP_2(t-s)[N_1\gamma _1(\cdot ,s)](x){\text {d}}s. \end{aligned}\right. \end{aligned}$$

According to Weng and Zhao (2006, Lemma 3.1), there is a sufficiently large number $M_i'$ in $(M_i,+\infty )$ and a small number $\delta >0$ such that ${{\textbf {P}}}(t)[\phi ](x_i^*)<(1+e^{t_0})\mathbf {\varepsilon }$ for $t\in [0,t_0]$, provided that $\phi \in {\mathcal {M}}$ satisfies $\Vert \phi (x)\Vert <\delta$ for any $x\in \Omega _{M_i'}(x_i^*)$. Choose $M>\max \{M_1,M_2\}$ sufficiently large such that $\Omega _{M_i'}(x_i^*)\subset \Omega _{M}(0)$. If $\Vert \phi _1(x)-\phi _2(x)\Vert <\delta$ for $x\in \Omega _{M}(0)$, then

$$\begin{aligned} \begin{aligned} |\gamma _1(\cdot ,t)|_{\Omega _M(0)}&\le \varepsilon +\gamma _1(x_1^*,t_1^*)\le \varepsilon +P_1(t_1^*) [\gamma _1(\cdot ,0)](x_1^*)\\&\quad +N_2\int _0^{t_1^*}P_1(t_1^*-s) [\gamma _2(\cdot ,s)](x_1^*){\text {d}}s\\&\le \varepsilon +(1+e^{t_0})\varepsilon +N_2t_1^* \varepsilon +N_2\int _0^{t_1^*}|\gamma _2(\cdot ,s)|_{\Omega _{M_1'}(x_1^*)}{\text {d}}s\\&\le \epsilon '+N_2\int _0^{t_0}|\gamma _2(\cdot ,s)|_{\Omega _{M}(0)}{\text {d}}s. \end{aligned}\end{aligned}$$

Similarly, we have

$$\begin{aligned} |\gamma _2(\cdot ,t)|_{\Omega _M(0)}\le \epsilon '+N_1\int _0^{t_0}|\gamma _1(\cdot ,s)|_{\Omega _{M}(0)}{\text {d}}s. \end{aligned}$$

For any $t\in (0,t_0)$, it follows that

$$\begin{aligned}\left\{ \begin{aligned}&|\gamma _1(\cdot ,t)|_{\Omega _M(0)}\le (1+N_2t_0)\epsilon '+N_1N_2t_0\int _0^{t_0} |\gamma _1(\cdot ,s)|_{\Omega _M(0)}{\text {d}}s,\\&|\gamma _2(\cdot ,t)|_{\Omega _M(0)}\le (1+N_1t_0)\epsilon '+N_1N_2t_0\int _0^{t_0} |\gamma _2(\cdot ,s)|_{\Omega _M(0)}{\text {d}}s. \end{aligned}\right. \end{aligned}$$

By Gronwall’s inequality, it is easy to check that

$$\begin{aligned}\left\{ \begin{aligned}&|\gamma _1(\cdot ,t)|_{\Omega _M(0)}\le (1+N_2t_0)\epsilon ' e^{N_1N_2t_0^2},\\&|\gamma _2(\cdot ,t)|_{\Omega _M(0)}\le (1+N_1t_0)\epsilon ' e^{N_1N_2t_0^2}. \end{aligned}\right. \end{aligned}$$

We get that

$$\begin{aligned} \Vert \pmb {\omega }(0,t;\phi _1)-\pmb {\omega }(0,t;\phi _2)\Vert \le \epsilon ~\text {for}~ t\in [0,t_0]. \end{aligned}$$

Then, the spatial translation invariance in (1.1) implies that

$$\begin{aligned} \Vert \pmb {\omega }(z,t;\phi _1)-\pmb {\omega }(z,t;\phi _2)\Vert \le \epsilon ~\text {for}~ t\in [0,t_0], \end{aligned}$$

provided $\Vert \phi _1(x)-\phi _2(x)\Vert <\delta$ for any $x\in \Omega _M(z)$.

Second, we consider the case $\phi _1\ngeq \phi _2$. Define

$$\begin{aligned} \Psi _1(x)= \max \{\phi _1(x),\phi _2(x)\},~~ \Psi _2(x)= \min \{\phi _1(x),\phi _2(x)\}. \end{aligned}$$

We have that

$$\begin{aligned} \begin{aligned}&\Psi _1(x)-\Psi _2(x)=|\phi _1(x)-\phi _2(x)|,\\&\pmb {\omega }(x,t;\phi _1)\le \pmb {\omega }(x,t;\Psi _1),~\text {and}~\pmb {\omega }(x,t;\Psi _2)\le \pmb {\omega }(x,t;\phi _2). \end{aligned} \end{aligned}$$

It follows that

$$\begin{aligned} \Vert \pmb {\omega }(x,t;\phi _1)-\pmb {\omega }(x,t;\phi _2)\Vert \le \Vert \pmb {\omega }(x,t;\Psi _1)-\pmb {\omega }(x,t;\Psi _2)\Vert . \end{aligned}$$

Then, we have that $Q_{t}$ is continuous in $\phi$ with respect to the compact open topology, uniformly for t in any bounded interval. It completes the proof. $\square$

Similarly, we have that the map $Q_t: {\mathcal {C}}_{E_2}\rightarrow {\mathcal {C}}_{E_2}$ is continuous for $(t,\phi )\in \mathbb R^+\times {\mathcal {C}}_{E_2}$ with respect to the compact open topology. Next, we verify the stability of fixed points in (B5).

Lemma 2.12

Assume (J1) and (J3) hold. For $Q_t:{\mathcal {M}}_{E_2}\rightarrow {\mathcal {M}}_{E_2}$ with $t>0$, we have that

• (1) $E_0$ and $E_2$ are strongly stable from above and below, respectively;

• (2) $E_1$ is strongly unstable from both above and below.

Proof

We write the restriction of $Q_t$ on ${\mathcal {X}}_{E_2}$ as ${\bar{Q}}_t$. Let ${\bar{M}}_{0,t}$ be the derivation $D{\bar{Q}}_t[E_0]$ of ${\bar{Q}}_t$. Then, ${\bar{M}}_{0,t}$ is the solution map of the following system:

$$\begin{aligned} \nonumber \left\{ \begin{aligned}&\frac{{\text {d}}u}{{\text {d}}t}=-\alpha u+h'(0)v,\\&\frac{{\text {d}}v}{{\text {d}}t}=g'(0)u-\beta v. \end{aligned} \right. \end{aligned}$$

(2.13)

Consider the following eigenvalue problem:

$$\begin{aligned} \left\{ \begin{aligned}&-\alpha \varphi +h'(0)\psi =\lambda \varphi ,\\&g'(0)\varphi -\beta \psi =\lambda \psi . \end{aligned} \right. \end{aligned}$$

(2.14)

It is obvious that (2.14) admits two negative eigenvalues $\lambda _1$ and $\lambda _2$. Without loss of generality, we assume that $\lambda _2<\lambda _1<0$. Then, there exists an eigenfunction ${{\textbf {e}}}_{0}\gg E_0$ associated with the eigenvalue $\lambda _1$, namely ${\bar{M}}_{0,t}[{{\textbf {e}}}_{0}]=e^{\lambda _1t}{{\textbf {e}}}_{0}$. For any $t>0$, there is $\delta _{0}(t)>0$ such that

$$\begin{aligned} \begin{aligned} {\bar{Q}}_t[\delta {{\textbf {e}}}_{0}]&={\bar{Q}}_t[E_0]+D{\bar{Q}}_t[E_0][\delta {{\textbf {e}}}_{0}]+{{\textbf {o}}}(\delta ^2)=\delta {\bar{M}}_{0,t}[{{\textbf {e}}}_{0}]+{{\textbf {o}}}(\delta ^2)\\&=\delta e^{\lambda _1t}{{\textbf {e}}}_{0}+{{\textbf {o}}}(\delta ^2)=\delta {{\textbf {e}}}_{0}+\delta [e^{\lambda _1t}-1]{{\textbf {e}}}_{0}+{{\textbf {o}}}(\delta ^2)\\&\ll \delta {{\textbf {e}}}_{0}, \delta \in (0,\delta _{0}(t)]. \end{aligned} \end{aligned}$$

Similarly, there exist $\delta _{1}(t)$, $\delta _{2}(t)$, ${{\textbf {e}}}_{1}$, and ${{\textbf {e}}}_{2}$ such that

$$\begin{aligned} {\bar{Q}}_t[E_2-\delta {{\textbf {e}}}_{2}]\gg E_2-\delta {{\textbf {e}}}_{2},\delta \in (0,\delta _{2}(t)] \end{aligned}$$

and

$$\begin{aligned} {\bar{Q}}_t[E_1+\delta {{\textbf {e}}}_{1}]\gg E_1+\delta {{\textbf {e}}}_{1},{\bar{Q}}_t[E_1-\delta {{\textbf {e}}}_{1}]\ll E_1-\delta {{\textbf {e}}}_{1},\delta \in (0,\delta _{1}(t)]. \end{aligned}$$

It completes the proof. $\square$

Next, we study some propagating properties of the two monostable systems restricted on ${\mathcal {M}}_{[E_0,E_1]}$ and ${\mathcal {M}}_{[E_1,E_2]}$, respectively, to verify the assumption (B6).

Lemma 2.13

Assume (J1)–(J4) hold. Then, $c_-^*(E_1,E_2)+c_+^*(E_0,E_1)>0$.

Proof

For the rightward propagation dynamics of $\{Q_t\}_{t\ge 0}$, we restrict $\{Q_t\}_{t\ge 0}$ on ${\mathcal {M}}_{E_1}$. Define $({\tilde{u}}(x,t),{\tilde{v}}(x,t))=(u_1^*,v_1^*)-(u(x,t),v(x,t))$ and then the rightward propagation dynamics is equivalent to that of the following system restricted on ${\mathcal {M}}_{E_1}$

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial {\tilde{u}}}{\partial t}=J_1*{\tilde{u}}-{\tilde{u}}-\alpha {\tilde{u}}-h(v_1^*-{\tilde{v}})+h(v_1^*),\\&\frac{\partial {\tilde{v}}}{\partial t}=J_2*{\tilde{v}}-{\tilde{v}}-\beta {\tilde{v}}-g(u_1^*-{\tilde{u}})+g(u_1^*). \end{aligned} \right. \end{aligned}$$

(2.15)

Define a family of operators $\{{\tilde{Q}}_t\}_{t\ge 0}$ on ${\mathcal {M}}_{E_1}$ by ${\tilde{Q}}_t(\phi )= \tilde{\pmb {\omega }}(\cdot ,t;\phi )$, where $\tilde{\pmb {\omega }}(\cdot ,t;\phi )$ is the solution of (2.15) satisfying $\tilde{\pmb {\omega }}(\cdot ,0;\phi )=\phi \in {\mathcal {M}}_{E_1}$. We have that ${\tilde{Q}}_t(E_0)=E_0$ and ${\tilde{Q}}_t(E_1)=E_1$ for all $t\ge 0$. It is easy to check that ${\tilde{Q}}_t:{\mathcal {C}}_{E_1}\rightarrow {\mathcal {C}}_{E_1}$ is continuous and satisfies the assumptions (A1), (A3)–(A5) in Fang and Zhao (2014) with ${\mathcal {M}}_{E_1}$ replaced by ${\mathcal {C}}_{E_1}$. According to Fang and Zhao (2014, Remark 3.7), we have that ${\tilde{Q}}_1:{\mathcal {M}}_{E_1}\rightarrow {\mathcal {M}}_{E_1}$ admits a rightward spreading speed, which is no more than $c_+^{*}(E_0,E_1)$. For (1.1), Theorem 3.1 in Xu et al. (2020) shows that the rightward spreading speed is equal to $-c_R$, which is defined by (2.2). Then, we have that

$$\begin{aligned} c_+^{*}(E_0,E_1)\ge -c_R. \end{aligned}$$

Similarly, for the leftward propagation dynamics of $\{Q_t\}_{t\ge 0}$, we restrict $\{Q_t\}_{t\ge 0}$ on ${\mathcal {M}}_{[E_1,E_2]}$. Define $({\hat{u}}(x,t),{\hat{v}}(x$ $,t))=(u(x,t),v(x,t))-(u_1^*,v_1^*)$ and consider

$$\begin{aligned} \left\{ \begin{aligned} \frac{\partial {\hat{u}}}{\partial t}=J_1*{\hat{u}}-{\hat{u}}-\alpha {\hat{u}}+h({\hat{v}}+v_1^*)-h(v_1^*),\\ \frac{\partial {\hat{v}}}{\partial t}=J_2*{\hat{v}}-{\hat{v}}-\beta {\hat{v}}+g({\hat{u}}+u_1^*)-g(u_1^*). \end{aligned} \right. \end{aligned}$$

(2.16)

Let $E_1^*:=E_2-E_1$. We define $\{{\hat{Q}}_t\}_{t\ge 0}$ on ${\mathcal {M}}_{E_1^*}$ by ${\hat{Q}}_t(\phi )= \hat{\pmb {\omega }}(\cdot ,t;\phi )$, where $\hat{\pmb {\omega }}(\cdot ,t;\phi )$ is the solution of (2.16) satisfying $\hat{\pmb {\omega }}(\cdot ,0;\phi )=\phi \in {\mathcal {M}}_{E_1^*}$. Then, ${\hat{Q}}_1:{\mathcal {M}}_{E_1^*}\rightarrow {\mathcal {M}}_{E_1^*}$ admits a leftward spreading speed $c_L$ defined by (2.2) and it holds that

$$\begin{aligned} c_-^{*}(E_1,E_2)\ge c_L. \end{aligned}$$

Finally, the proof is completed by (2.3). $\square$

Now, we are ready to give the existence of bistable traveling wave solutions of (1.1).

Theorem 2.14

Assume (J1)–(J4) hold. Then, there is a constant $c_0\in {\mathbb {R}}$ such that (1.1) admits an increasing traveling wave solution $\Phi (x+c_0t)$ satisfying

$$\begin{aligned} \Phi (-\infty )=E_0~\text {and}~\Phi (+\infty )=E_2. \end{aligned}$$

Moreover, $\Phi (\cdot )$ is strictly increasing.

Proof

In view of Lemmas 2.11, 2.12, and 2.13 , the existence of traveling wave solutions in Theorem 2.9 implies that there is $c_0\in {\mathbb {R}}$ and $\Phi \in {\mathcal {M}}_{E_2}$ such that $Q_t(\Phi )(x)=\Phi (x+c_0t)$ for any $x\in {\mathbb {R}}$. Next, we prove that $\Phi$ is a classical solution of (1.1). It is easy to verify that

$$\begin{aligned} \frac{\partial }{\partial t}{{\textbf {P}}}(t)[\Phi ](x)=-{\textbf {P}}(t)[\Phi ](x)+J*({{\textbf {P}}}(t)[\Phi ])(x). \end{aligned}$$

(2.17)

Note that $\pmb {\omega }(x,t) := \Phi (x + c_0t)$ satisfies the following integral equation:

$$\begin{aligned} \pmb {\omega }(x,t)={{\textbf {P}}}(t)[\pmb {\omega }(\cdot ,0)](x)+\int _0^t {{\textbf {P}}}(t-s)[{{\textbf {F}}}(\pmb {\omega }(\cdot ,s))](x){\text {d}}s,~~x\in {\mathbb {R}},~t\ge 0.\nonumber \\ \end{aligned}$$

(2.18)

Since the right side of (2.18) is differentiable with respect to variable t, it follows from (2.17) that $\pmb {\omega }(x,t)$ is a classical solution of (1.1).

It remains to prove that the nondecreasing function $\Phi (\cdot )=(\varphi ,\psi )(\cdot )$ is strictly increasing. By a contradiction argument, without loss of generality, we assume that $\varphi (\cdot )$ is not strictly increasing; namely, there exist $x_1<x_2$ such that $\varphi (\xi )=\varphi (x_1)$ for any $\xi \in [x_1,x_2]$. It follows that $\varphi '(\xi )=0$ for $\xi \in [x_1,x_2]$. By (2.1), we have that

$$\begin{aligned} 0= & {} \int _{\mathbb {R}}J_1(y)(\varphi (x_1-y)-\varphi (x_2-y)){\text {d}}y-\alpha (\varphi (x_1)-\varphi (x_2))\\&\quad +h(\psi (x_1))-h(\psi (x_2)). \end{aligned}$$

It follows that

$$\begin{aligned} 0\ge & {} \int _{\mathbb {R}}J_1(y)(\varphi (x_1-y)-\varphi (x_2-y)){\text {d}}y=\alpha (\varphi (x_1)-\varphi (x_2))\\&\quad -h(\psi (x_1))+h(\psi (x_2))\ge 0, \end{aligned}$$

which implies $\int _{\mathbb {R}}J_1(y)(\varphi (x_1-y)-\varphi (x_2-y)){\text {d}}y=0$. Then, $\varphi (x_1-y)=\varphi (x_2-y)$ for any $y\in \text {supp}(J_1)$. Rechoose $x_1$ and $x_2$ and repeat the process above, we can obtain that $\varphi (\cdot )$ is a constant function, which contradicts $\varphi (-\infty )=0$ and $\varphi (+\infty )=u_2^*$. It completes the proof. $\square$

Since $J_1$ and $J_2$ are not required to be symmetric, the corresponding nondecreasing and nonincreasing traveling wave solutions can be different in shape. The following theorem gives the existence of nonincreasing traveling wave solution.

Theorem 2.15

Assume (J1)–(J4) hold. Then, there is a constant ${\hat{c}}_0\in {\mathbb {R}}$ such that (1.1) admits a decreasing traveling wave solution ${\hat{\Phi }}(x+{\hat{c}}_0t)$ satisfying

$$\begin{aligned} {\hat{\Phi }}(-\infty )=E_2~~\text {and}~~{\hat{\Phi }}(+\infty )=E_0. \end{aligned}$$

Moreover, ${\hat{\Phi }}(\cdot )$ is strictly decreasing.

Proof

According to Fang and Zhao (2015, Remark 2.1), when we consider the nonincreasing traveling waves, assumption (B6) should be replaced by

$$\begin{aligned} c_+^*(E_1,E_2)+c_-^*(E_0,E_1)>0. \end{aligned}$$

(2.19)

Here, $c_+^*(E_1,E_2)$ and $c_-^*(E_0,E_1)$ are defined similarly to (2.10) and (2.11), respectively. By the same argument as the proof of Lemma 2.13, we have that $c_+^*(E_1,E_2)\ge -c_R$ and $c_-^*(E_0,E_1)\ge c_L$. Then, (2.19) follows from $c_L-c_R>0$. The rest of the proof is similar to Theorem 2.14. $\square$

Now, we give the result about the asymptotic behavior of bistable traveling wave solutions. The proof is omitted since we can use the same techniques as in Theorems 2.6 and 2.7.

Theorem 2.16

Assume (J1)–(J4) hold. Let $\Phi (x+c_0t)$ and ${\hat{\Phi }}(x+{{\hat{c}}}_0t)$ be the nondecreasing and nonincreasing traveling wave solutions of (1.1) as in Theorems 2.14 and 2.15 with $c_0\ne 0$ and ${{\hat{c}}}_0\ne 0$, respectively. Then, all of the following limits exist in ${\mathbb {R}}^+\times {\mathbb {R}}^+$ and

$$\begin{aligned} \begin{aligned}&\lim \limits _{\xi \rightarrow -\infty }\Phi (\xi )/e^{\lambda _0^+(c_0)\xi } =\lim \limits _{\xi \rightarrow -\infty }\Phi '(\xi )/(\lambda _0^+(c_0)e^{\lambda _0^+(c_0)\xi }),\\&\lim \limits _{\xi \rightarrow +\infty }(E_2-\Phi (\xi ))/e^{\lambda _2^-(c_0)\xi } =-\lim \limits _{\xi \rightarrow +\infty }\Phi '(\xi )/(\lambda _2^-(c_0)e^{\lambda _2^-(c_0)\xi }),\\&\lim \limits _{\xi \rightarrow +\infty }{\hat{\Phi }}(\xi )/e^{\lambda _0^-({{\hat{c}}}_0)\xi }=\lim \limits _{\xi \rightarrow +\infty }{\hat{\Phi }}'(\xi )/(\lambda _0^-({{\hat{c}}}_0)e^{\lambda _0^-({{\hat{c}}}_0)\xi }),\\&\lim \limits _{\xi \rightarrow -\infty }(E_2-{\hat{\Phi }}(\xi ))/e^{\lambda _2^+({{\hat{c}}}_0)\xi }=-\lim \limits _{\xi \rightarrow -\infty }{\hat{\Phi }}'(\xi )/(\lambda _2^+({{\hat{c}}}_0)e^{\lambda _2^+({{\hat{c}}}_0)\xi }), \end{aligned} \end{aligned}$$

where $\lambda _i^+(c)$ and $\lambda _i^-(c)$ with $i\in \{0,2\}$ are defined by (2.8).

3 Entire Solutions

In this section, we focus on the construction of entire solutions of (1.1). As stated in Sect. 1, we only choose traveling wave solutions with nonzero wave speeds $c_0$, ${{\hat{c}}}_0$, $c_i$, and ${{\hat{c}}}_i$ $(i = 1, 2)$. By super- and sub-solutions method, we construct six types of entire solutions, and each of them originates from three traveling wave solutions. Figure 1 in Section 1 depicts the sketchy profiles of these entire solutions as $t\rightarrow -\infty$. The existence results of these six types of entire solutions are given in Sects. 3.2–3.7, respectively. Since the construction methods for these six types are similar, we give details only for the first type. And for the other five types, the main differences are shown in the corresponding subsections. Moreover, we prove that the entire solutions established in Sects. 3.2–3.7 are globally Lipschitz continuous with respect to x under the following assumption:

• (J5)  $\sup \limits _{u\in [0,u_2^*]}g'(u)\cdot \sup \limits _{v\in [0,v_2^*]}h'(v)<(1+\alpha )(1+\beta )$.

Finally, when $c_0 \ne {{\hat{c}}}_0$, we obtain the nonexistence of entire solutions originating from more than four traveling wave solutions in Sect. 3.8.

3.1 Preliminaries

This subsection gives some fundamental properties of (1.1), including the comparison principle, the Lipschitz continuity of solution, and the study of some ordinary differential equations. First, we give the definition of super- and sub-solutions of (1.1).

Definition 3.1

For $\tau ,T\in {\mathbb {R}}$ with $\tau <T$, a function $\pmb {\omega }= (u, v)$ is called a super-solution (sub-solution) of (1.1) on ${\mathbb {R}}\times [\tau , T )$, if $\pmb {\omega }\in C^{0,1}({\mathbb {R}}\times [\tau , T ),{\mathbb {R}}^2)$ and

$$\begin{aligned} \left\{ \begin{aligned}&u_t(x,t)\ge (\le )J_1*u(x,t)-u(x,t)-\alpha u(x,t)+h(v(x,t)),\\&v_t(x,t)\ge (\le )J_2*v(x,t)-v(x,t)-\beta v(x,t)+g(u(x,t)) \end{aligned}\right. \end{aligned}$$

for any $(x,t)\in {\mathbb {R}}\times [\tau , T)$. Moreover, $\pmb {\omega }=(u,v)$ is called a super-solution (sub-solution) of (1.1) on ${\mathbb {R}}\times (-\infty , T )$, if for any $\tau <T$, it is a super-solution (sub-solution) on ${\mathbb {R}}\times [\tau , T )$.

The following lemma states the comparison principle proven in Li et al. (2017, Theorem 3.6).

Lemma 3.2

Assume (J1) and (J3) hold. Let $\pmb {\omega }^+(x, t)$ and $\pmb {\omega }^-(x, t)$ be the super- and sub-solutions of (1.1) on ${\mathbb {R}}\times [0,\infty )$, respectively. If $\pmb {\omega }^+(x, 0) \ge \pmb {\omega }^-(x, 0)$ for $x\in {\mathbb {R}}$ and $E_0 \le \pmb {\omega }^-(x, t),\pmb {\omega }^+(x, t) \le E_2$, then $\pmb {\omega }^-(x, t)\le \pmb {\omega }^+(x, t)$ for $t\in [0,\infty )(x \text { is fixed})$.

In the study of entire solutions, the Lipschitz continuity of solution as given by the following lemma is a very important property. We refer to Li et al. (2017, Lemma 3.8) for its proof.

Lemma 3.3

Assume (J1)–(J5) hold. Let (u(x, t), v(x, t)) be a solution of the corresponding Cauchy problem of (1.1) with bounded initial data. Then, there is a constant $M>0$ (independent of initial data) such that

$$\begin{aligned} |u_t(x,t)|,|u_{tt}(x,t)|,|v_t(x,t)|,|v_{tt}(x,t)|\le M\quad \text {for any}~ x \in {\mathbb {R}},~t>0. \end{aligned}$$

In addition, if the initial data are globally Lipschitz continuous, namely there is $L_0>0$ such that

$$\begin{aligned} |u(x+{\tilde{x}},0)-u(x,0)|\le L_0{\tilde{x}},~|v(x+{\tilde{x}},0)-v(x,0)|\le L_0{\tilde{x}}~\quad ~\text {for any}~x\in {\mathbb {R}},~{\tilde{x}}>0, \end{aligned}$$

then for any $t>0$, the solution is also globally Lipschitz continuous and

$$\begin{aligned}&|u(x+{\tilde{x}}, t)-u(x, t)|, ~|v(x+{\tilde{x}}, t)-v(x, t)|\leqslant M'{\tilde{x}}, \\&|\frac{\partial u}{\partial t}(x+{\tilde{x}}, t)-\frac{\partial u}{\partial t}(x, t)|,~ |\frac{\partial v}{\partial t}(x+{\tilde{x}}, t)-\frac{\partial v}{\partial t}(x, t)|\leqslant M''{\tilde{x}}, \end{aligned}$$

where $M'$ and $M''$ are two positive constants independent of x and ${\tilde{x}}$.

The construction of entire solutions also involves the solutions of some ordinary differential equations. For any real numbers $s_1<s_2<s_3$, define

$$\begin{aligned} \nu _1:=(s_2-s_1)/2>0,~\nu _2:=[2s_3-(s_1+s_2)]/2>\nu _1. \end{aligned}$$

(3.1)

Consider the following ordinary differential equations: (c.f. Chen et al. 2018)

$$\begin{aligned} \left\{ \begin{aligned}&p_1' = \nu _1-Le^{\kappa p_1},-\infty<t<0,~p_1(0)=p_0,\\&p_2' = \nu _2+Le^{\kappa p_1},-\infty<t<0,~p_2(0)=r_0, \\&r_1' = \nu _1+Le^{\kappa r_1},-\infty<t<0,~r_1(0)=r_0, \\&r_2' = \nu _2-Le^{\kappa r_1},-\infty<t<0,~r_2(0)=p_0,\\ \end{aligned}\right. \end{aligned}$$

(3.2)

and

$$\begin{aligned} \left\{ \begin{aligned}&{\tilde{p}}_1' = \nu _1+Le^{\kappa {\tilde{p}}_1},-\infty<t<0,~{\tilde{p}}_1(0)=p_0, \\&{\tilde{p}}_2' = \nu _2+Le^{\kappa {\tilde{p}}_1},-\infty<t<0,~{\tilde{p}}_2(0)=p_0,\\&{\tilde{r}}_1' = \nu _1-Le^{\kappa {\tilde{r}}_1},-\infty<t<0,~{\tilde{r}}_1(0)=r_0, \\&{\tilde{r}}_2' = \nu _2-Le^{\kappa {\tilde{r}}_1},-\infty<t<0,~{\tilde{r}}_2(0)=r_0, \end{aligned}\right. \end{aligned}$$

(3.3)

where L, $p_0$, and $r_0$ are some constants determined later and

$$\begin{aligned} \kappa :=\min \left\{ \eta _1,\eta _2,\frac{(\nu _2-\nu _1)\eta _1}{4\nu _1},\frac{(\nu _2-\nu _1)\eta _2}{4\nu _1}\right\} . \end{aligned}$$

(3.4)

We can solve (3.2) and (3.3) explicitly as

$$\begin{aligned} \begin{aligned}&p_1(t)=\nu _1t-\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}-\frac{L(1-e^{\kappa \nu _1t})}{\nu _1}\right] ,~ p_2(t)=\nu _2t+\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}-\frac{L(1-e^{\kappa \nu _1t})}{\nu _1}\right] +p_0+r_0,\\&r_1(t)=\nu _1t-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}+\frac{L(1-e^{\kappa \nu _1t})}{\nu _1}\right] ,~ r_2(t)=\nu _2t+\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}+\frac{L(1-e^{\kappa \nu _1t})}{\nu _1}\right] +p_0+r_0,\\&{\tilde{p}}_1(t)=\nu _1t-\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}+\frac{L(1-e^{\kappa \nu _1t})}{\nu _1}\right] ,~ {\tilde{p}}_2(t)=\nu _2t-\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}+\frac{L(1-e^{\kappa \nu _1t})}{\nu _1}\right] ,\\&{\tilde{r}}_1(t)=\nu _1t-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}-\frac{L(1-e^{\kappa \nu _1t})}{\nu _1}\right] ,~ {\tilde{r}}_2(t)=\nu _2t-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}-\frac{L(1-e^{\kappa \nu _1t})}{\nu _1}\right] . \end{aligned} \end{aligned}$$

Now, for any given sufficiently large $\delta$, we take the initial values $p_0$ and $r_0$ satisfying

$$\begin{aligned} p_0=-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}+\frac{2L}{\nu _1}\right]<-\delta ~ \text {and}~ r_0<-\frac{1}{\kappa }\ln \left[ e^{\kappa \delta }+\frac{2L}{\nu _1}\right] <-\delta . \end{aligned}$$

Then, it follows that

$$\begin{aligned} \begin{aligned}&\lim \limits _{t\rightarrow -\infty }(r_1(t)-p_1(t))=\lim \limits _{t\rightarrow -\infty }(p_2(t)-r_2(t))=0,\\&\lim \limits _{t\rightarrow -\infty }(p_1(t)-\nu _1t)=-\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}-\frac{L}{\nu _1}\right] ,\\&\lim \limits _{t\rightarrow -\infty }(p_2(t)-\nu _2t)=\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}-\frac{L}{\nu _1}\right] +p_0+r_0,\\&\lim \limits _{t\rightarrow -\infty }(r_1(t)-\nu _1t)=-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}+\frac{L}{\nu _1}\right] ,\\&\lim \limits _{t\rightarrow -\infty }(r_2(t)-\nu _2t)=\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}+\frac{L}{\nu _1}\right] +p_0+r_0,\\&\lim \limits _{t\rightarrow -\infty }({\tilde{p}}_1(t)-{\tilde{r}}_1(t))=\lim \limits _{t\rightarrow -\infty }({\tilde{p}}_2(t)-{\tilde{r}}_2(t))=0,\\&\lim \limits _{t\rightarrow -\infty }({\tilde{p}}_1(t)-\nu _1t)=\lim \limits _{t\rightarrow -\infty }({\tilde{p}}_2(t)-\nu _2t)=-\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}+\frac{L}{\nu _1}\right] ,\\&\lim \limits _{t\rightarrow -\infty }({\tilde{r}}_1(t)-\nu _1t)=\lim \limits _{t\rightarrow -\infty }({\tilde{r}}_2(t)-\nu _2t)=-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}-\frac{L}{\nu _1}\right] . \end{aligned}\end{aligned}$$

(3.5)

In addition, there exists a positive constant N (dependent on L, $p_0$, and $r_0$) such that

$$\begin{aligned}&0<r_1(t)-p_1(t)=p_2(t)-r_2(t)\le Ne^{\kappa \nu _1t}~\text {for}~t\le 0,\nonumber \\&0<{\tilde{p}}_1(t)-{\tilde{r}}_1(t)={\tilde{p}}_2(t)-{\tilde{r}}_2(t)\le Ne^{\kappa \nu _1t}~\text {for}~t\le 0, \end{aligned}$$

(3.6)

and

$$\begin{aligned} p_1(t), p_2(t), r_1(t), r_2(t), {\tilde{p}}_1(t), {\tilde{p}}_2(t), {\tilde{r}}_1(t), {\tilde{r}}_2(t)\le -\delta ~\text {for}~t\le 0. \end{aligned}$$

Remark 3.4

In the next six subsections, the values of $\nu _1$ and $\nu _2$ are different for the six types of entire solutions, since $s_1$, $s_2$, and $s_3$ will be changed for every type. However, the above results about the solutions of Eqs. (3.2) and (3.3) are always true.

3.2 I-Type Entire Solution

In this subsection, we construct the first type of entire solution, which originates from one monostable and two bistable traveling wave solutions.

Theorem 3.5

Assume (J1)–(J4) hold and $c_0>{\hat{c}}_0>c_2$. Then, (1.1) admits an entire solution ${\mathcal {W}}_1:{\mathbb {R}}^2\rightarrow [0,u_2^*]\times [0,v_2^*]$ satisfying

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }&\left\{ \sup \limits _{x\le \rho _1(t)}\Vert {\mathcal {W}}_1(x,t)-\Phi _2(x+c_2t-\vartheta _1)\Vert \right. \nonumber \\&\quad +\sup \limits _{\rho _1(t)\le x\le \rho _2(t)}\Vert {\mathcal {W}}_1(x,t)-{\hat{\Phi }}(x+{\hat{c}}_0t+\vartheta _1)\Vert \nonumber \\&\quad \left. +\sup \limits _{\rho _2(t)\le x}\Vert {\mathcal {W}}_1(x,t)-\Phi (x+c_0t+\vartheta _2)\Vert \right\} =0, \end{aligned}$$

(3.7)

where $\vartheta _1$ and $\vartheta _2$ are some constants satisfying

$$\begin{aligned} \vartheta _1:=-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}+\frac{L}{\nu _1}\right] ,\quad \vartheta _2:=\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}+\frac{L}{\nu _1}\right] +p_0+r_0, \end{aligned}$$

and $\rho _1,\rho _2:{\mathbb {R}} \rightarrow {\mathbb {R}}$ are defined by

$$\begin{aligned} \rho _1(t):=\frac{-(c_2+{\hat{c}}_0)t}{2},\rho _2(t):=\frac{-({\hat{c}}_0+c_0)t}{2}. \end{aligned}$$

Moreover, when (J5) holds, there are two positive constants $D_1$ and $D_2$ such that for any $(x,t)\in {\mathbb {R}}^2$ and $\eta >0$,

$$\begin{aligned} \Vert {\mathcal {W}}_1(x+\eta ,t)-{\mathcal {W}}_1(x,t)\Vert \le D_1\eta ,\left\| \frac{\partial {\mathcal {W}}_1}{\partial t}(x+\eta ,t)-\frac{\partial {\mathcal {W}}_1}{\partial t}\right\| \le D_2\eta . \end{aligned}$$

(3.8)

The entire solution ${\mathcal {W}}_1(x,t)$ given by Theorem 3.5 can be regarded as a combination of the traveling wave solutions $(c_2,\Phi _2)$, $({\hat{c}}_0,{\hat{\Phi }})$, and $(c_0,\Phi )$ with $c_0>{\hat{c}}_0>c_2$. When $t\rightarrow -\infty$, its profile is found in Fig. 1a. Obviously, ${\mathcal {W}}_1(x,t)$ is differentiable with respect to t, and (3.8) further shows that it is globally Lipschitz continuous with respect to x.

As stated in Sect. 1, the construction of ${\mathcal {W}}_1$ is essentially a procedure that we link the profile of $\Phi _2$ with ${\hat{\Phi }}$ and link the profile of ${\hat{\Phi }}$ with $\Phi$ successively, and the main difficulty is to construct appropriate auxiliary functions and a pair of super- and sub-solutions. For ${\mathcal {W}}_1$ in Theorem 3.5, the construction procedure is quite complicated and it will be provided in the rest of this subsection. We first define the auxiliary functions as follows:

$$\begin{aligned}&P(x,y,z)=y\nonumber \\&\quad +\frac{(x-u_1^*)(u_2^*-y)z(u_2^*-y)+(u_2^*-x)y(u_2^*-z)(u_1^*-y)}{u_2^*(x-u_1^*)(u_2^*-y)+(u_2^*-u_1^*)y(u_2^*-z)},~~ (x,y,z)\in D_1\nonumber \\ \end{aligned}$$

(3.9)

and

$$\begin{aligned}&Q(x,y,z)=y\nonumber \\&\quad +\frac{(x-v_1^*)(v_2^*-y)z(v_2^*-y)+(v_2^*-x)y(v_2^*-z)(v_1^*-y)}{v_2^*(x-v_1^*)(v_2^*-y)+(v_2^*-v_1^*)y(v_2^*-z)},~~ (x,y,z)\in D_2,\nonumber \\ \end{aligned}$$

(3.10)

where

$$\begin{aligned} \begin{aligned}&D_1:=\left\{ (x,y,z)\in [u_1^*,u_2^*]\times [0,u_2^*]\times [0,u_2^*]~\big |~u_2^*(x-u_1^*)(u_2^*-y)+(u_2^*-u_1^*)y(u_2^*-z)>0\right\} ,\\&D_2:=\left\{ (x,y,z)\in [v_1^*,v_2^*]\times [0,v_2^*]\times [0,v_2^*]~\big |~v_2^*(x-v_1^*)(v_2^*-y)+(v_2^*-v_1^*)y(v_2^*-z)>0\right\} . \end{aligned} \end{aligned}$$

Lemma 3.6

There exists a positive constant $C_3$ such that

$$\begin{aligned}&\frac{|P_{xx}(x,y,z)|}{|y|},\frac{|P_{xx}(x,y,z)|}{|u_2^*-y|},\frac{|P_{xx}(x,y,z)|}{|u_2^*-z|},\frac{|P_{yy}(x,y,z)|}{|x-u_1^*|},\frac{|P_{yy}(x,y,z)|}{|u_2^*-z|},\frac{|P_{yy}(x,y,z)|}{|u_2^*-x|+|z|}\le C_3,\\&\frac{|P_{zz}(x,y,z)|}{|x-u_1^*|},\frac{|P_{zz}(x,y,z)|}{|u_2^*-y|},\frac{|P_{zz}(x,y,z)|}{|y|},\frac{|P_{xy}(x,y,z)|}{|u_2^*-z|},\frac{|P_{yz}(x,y,z)|}{|x-u_1^*|}\le C_3,\\&\frac{|P_{xz}(x,y,z)|}{|u_2^*-y|},\frac{|P_{xz}(x,y,z)|}{|y|},\frac{|P_{xz}(x,y,z)|}{|x-u_1^*|},\frac{|P_{xz}(x,y,z)|}{|u_2^*-z|}\le C_3,\\&\frac{|Q_{xx}(x,y,z)|}{|y|},\frac{|Q_{xx}(x,y,z)|}{|v_2^*-y|},\frac{|Q_{xx}(x,y,z)|}{|v_2^*-z|},\frac{|Q_{yy}(x,y,z)|}{|x-v_1^*|},\frac{|Q_{yy}(x,y,z)|}{|v_2^*-z|},\frac{|Q_{yy}(x,y,z)|}{|v_2^*-x|+|z|}\le C_3,\\&\frac{|Q_{zz}(x,y,z)|}{|x-v_1^*|},\frac{|Q_{zz}(x,y,z)|}{|v_2^*-y|},\frac{|Q_{zz}(x,y,z)|}{|y|},\frac{|Q_{xy}(x,y,z)|}{|v_2^*-z|},\frac{|Q_{yz}(x,y,z)|}{|x-v_1^*|}\le C_3,\\&\frac{|Q_{xz}(x,y,z)|}{|v_2^*-y|},\frac{|Q_{xz}(x,y,z)|}{|y|},\frac{|Q_{xz}(x,y,z)|}{|x-v_1^*|},\frac{|Q_{xz}(x,y,z)|}{|v_2^*-z|}\le C_3. \end{aligned}$$

Proof

Note that P can be rewritten as

$$\begin{aligned}&P(x,y,z)=x+(x-u_1^*)(u_2^*-y)\frac{u_2^*(z-x)}{u_2^*(x-u_1^*)(u_2^*-y)+(u_2^*-u_1^*)y(u_2^*-z)}, \\&P(x,y,z)=z+y(u_2^*-z)\frac{(u_2^*-u_1^*)(x-z)}{u_2^*(x-u_1^*)(u_2^*-y)+(u_2^*-u_1^*)y(u_2^*-z)}. \end{aligned}$$

Similarly, Q can be rewritten as

$$\begin{aligned}&Q(x,y,z)=x+(x-v_1^*)(v_2^*-y)\frac{v_2^*(z-x)}{v_2^*(x-v_1^*)(v_2^*-y)+(v_2^*-v_1^*)y(v_2^*-z)}, \\&Q(x,y,z)=z+y(v_2^*-z)\frac{(v_2^*-v_1^*)(x-z)}{v_2^*(x-v_1^*)(v_2^*-y)+(v_2^*-v_1^*)y(v_2^*-z)}. \end{aligned}$$

With the help of the above formulas, the calculations of derivatives can be simplified and we omit the details of calculations in the rest of the proof. $\square$

In order to obtain a pair of suitable super- and sub-solutions, we denote some notations. As stated in Remark 3.4, throughout this subsection we need to choose that

$$\begin{aligned} s_1:=c_2,~s_2:={\hat{c}}_0,~s_3:=c_0. \end{aligned}$$

(3.11)

Recall that $(c_2,\Phi _2)$, $({\hat{c}}_0,{\hat{\Phi }})$, and $(c_0,\Phi )$ are the traveling wave solutions of (1.1). For convenience, we denote

$$\begin{aligned} \begin{aligned}&\varvec{\phi }_1=(\phi _{11},\phi _{12}):=\Phi _2=(\varphi _2,\psi _2),\\&\varvec{\phi }_2=(\phi _{21},\phi _{22}):={\hat{\Phi }}=({\hat{\varphi }},{\hat{\psi }}),\\&\varvec{\phi }_3=(\phi _{31},\phi _{32}):=\Phi =(\varphi ,\psi ). \end{aligned} \end{aligned}$$

Then, $\varvec{\phi }_i=(\phi _{i1},\phi _{i2})$ with $i\in \{1,2,3\}$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&s_i\phi _{i1}'(\xi )=J_1*\phi _{i1}(\xi )-\phi _{i1}(\xi )-\alpha \phi _{i1}(\xi )+h(\phi _{i2}(\xi )), \\&s_i\phi _{i2}'(\xi )=J_2*\phi _{i2}(\xi )-\phi _{i2}(\xi )-\beta \phi _{i2}(\xi )+g(\phi _{i1}(\xi )){,}\\ \end{aligned} \right. \end{aligned}$$

and

$$\begin{aligned}&\left( \varvec{\phi }_1(-\infty ),\varvec{\phi }_1(+\infty ),\varvec{\phi }_2(-\infty ),\varvec{\phi }_2(+\infty ),\varvec{\phi }_3(-\infty ),\varvec{\phi }_3(+\infty )\right) \\&\quad =(E_1,E_2,E_2,E_0,E_0,E_2). \end{aligned}$$

By a translation if necessary, we may assume

$$\begin{aligned} \phi _{11}(0)= & {} \frac{u_1^*+u_2^*}{2},~(\phi _{21}(0),\phi _{22}(0))\ge \left( \frac{u_1^*+u_2^*}{2},\frac{v_1^*+v_2^*}{2}\right) ,~\text {and}~(\phi _{31}(0),\phi _{32}(0))\nonumber \\&\quad \le (u_1^*,v_1^*). \end{aligned}$$

(3.12)

By a transformation (u(x, t), v(x, t)) $=(U(\xi ,t),V(\xi ,t))$ with $\xi :=x+{\bar{c}}t$ and ${\bar{c}}:=\frac{s_1+s_2}{2}$, (1.1) becomes

$$\begin{aligned} \left\{ \begin{aligned} \frac{\partial U(\xi ,t)}{\partial t}=(J_1*U-U)(\xi ,t)-{\bar{c}}U_{\xi }(\xi ,t)+f_1(U(\xi ,t), V(\xi ,t)),~(\xi ,t)\in {\mathbb {R}}^2, \\ \frac{\partial V(\xi ,t)}{\partial t}=(J_2*V-V)(\xi ,t)-{\bar{c}}V_{\xi }(\xi ,t)+f_2(U(\xi ,t), V(\xi ,t)),~(\xi ,t)\in {\mathbb {R}}^2,\\ \end{aligned} \right. \nonumber \\ \end{aligned}$$

(3.13)

where

$$\begin{aligned} f_1(U, V):=-\alpha U+h(V),\quad f_2(U, V):=-\beta V+g(U). \end{aligned}$$

(3.14)

It is easy to check that (3.13) has three traveling wave solutions, namely

$$\begin{aligned} \varvec{\phi }_1(\xi -\nu _1t),~\varvec{\phi }_2(\xi +\nu _1t),~\text {and}~\varvec{\phi }_3(\xi +\nu _2t), \end{aligned}$$

where $\nu _1$ and $\nu _2$ are defined by (3.1) and $\nu _2>\nu _1>0$. We define the super- and sub-solutions of (3.13) similarly to Definition 3.1 for (1.1).

Now, we construct

$$\begin{aligned} \left\{ \begin{aligned}&\overline{U}(\xi ,t)= P(\phi _{11}(\xi -p_1(t)),\phi _{21}(\xi +p_1(t)),\phi _{31}(\xi +p_2(t))),\\&\overline{V}(\xi ,t)= Q(\phi _{12}(\xi -p_1(t)),\phi _{22}(\xi +p_1(t)),\phi _{32}(\xi +p_2(t))) \end{aligned}\right. \end{aligned}$$

(3.15)

and

$$\begin{aligned} \left\{ \begin{aligned}&{\underline{U}}(\xi ,t)= P(\phi _{11}(\xi -r_1(t)),\phi _{21}(\xi +r_1(t)),\phi _{31}(\xi +r_2(t))),\\&{\underline{V}}(\xi ,t)= Q(\phi _{12}(\xi -r_1(t)),\phi _{22}(\xi +r_1(t)),\phi _{32}(\xi +r_2(t))), \end{aligned}\right. \end{aligned}$$

(3.16)

where $(p_1(t),p_2(t),r_1(t),$ $r_2(t))$ is the solution of (3.2). In what follows, our purpose is to prove that $(\overline{U}(\xi ,t),\overline{V}(\xi ,t))$ and $({\underline{U}}(\xi ,t),{\underline{V}}(\xi ,t))$ are a pair of super- and sub-solution of (3.13), and apply it to complete the proof of Theorem 3.5. We accomplish this purpose (see Lemma 3.15) by showing lots of estimates for some related functions (see Lemmas 3.7–3.14).

For convenience, we first restate the results in Theorems 2.6, 2.7, and 2.16 about the asymptotic behavior of $\varvec{\phi }_1$, $\varvec{\phi }_2$, and $\varvec{\phi }_3$ in a uniform format, and hence, the proof of the following lemma is omitted.

Lemma 3.7

Assume (J1)–(J4) hold. Let

$$\begin{aligned} \eta _1=\min \{\lambda _1^+(c_2),\lambda _2^+({{\hat{c}}}_0),\lambda _0^+(c_0)\}, ~ \eta _2=\min \{\lambda _2^-(c_2),\lambda _0^-({{\hat{c}}}_0),\lambda _2^-(c_0)\}. \end{aligned}$$

There are three positive numbers $C_0$, $C_1$, and $C_2$ such that

• (i) for $x\le m$,

  $$\begin{aligned}&|\phi _{11}'(x)|,|\phi _{12}'(x)|,|\phi _{21}'(x)|,|\phi _{22}'(x)|,|\phi _{31}'(x)|,|\phi _{32}'(x)|\le C_0e^{\eta _1x},C_1\\&\le \frac{|\phi _{11}'(x)|}{|\phi _{11}(x)-u_1^*|},\frac{|\phi _{12}'(x)|}{|\phi _{12}(x)-v_1^*|},\frac{|\phi _{21}'(x)|}{|u_2^*-\phi _{21}(x)|},\frac{|\phi _{22}'(x)|}{|v_2^*-\phi _{22}(x)|},\frac{|\phi _{31}'(x)|}{|\phi _{31}(x)|},\frac{|\phi _{32}'(x)|}{|\phi _{32}(x)|}\\&\le C_2; \end{aligned}$$

• (ii) for $x\ge -m$,

  $$\begin{aligned}&|\phi _{11}'(x)|,|\phi _{12}'(x)|,|\phi _{21}'(x)|,|\phi _{22}'(x)|,|\phi _{31}'(x)|,|\phi _{32}'(x)|\le C_0e^{-\eta _2x},C_1\\&\le \frac{|\phi _{11}'(x)|}{|u_2^*-\phi _{11}(x)|},\frac{|\phi _{12}'(x)|}{|v_2^*-\phi _{12}(x)|},\frac{|\phi _{21}'(x)|}{|\phi _{21}(x)|},\frac{|\phi _{22}'(x)|}{|\phi _{22}(x)|},\frac{|\phi _{31}'(x)|}{|u_2^*-\phi _{31}(x)|},\frac{|\phi _{32}'(x)|}{|v_2^*-\phi _{32}(x)|}\\&\le C_2, \end{aligned}$$

where m is defined in (J4). Moreover, there is a constant $C>0$ such that for any $x\in {\mathbb {R}}$ and $y\in \text {supp}(J)$,

$$\begin{aligned}&\frac{\phi _{11}(x+y)-u_1^*}{\phi _{11}(x)-u_1^*},\frac{\phi _{12}(x+y)-v_1^*}{\phi _{12}(x)-v_1^*},\frac{u_2^*-\phi _{11}(x+y)}{u_2^*-\phi _{11}(x)},\frac{v_2^*-\phi _{12}(x+y)}{v_2^*-\phi _{12}(x)}\le C,\\&\frac{u_2^*-\phi _{21}(x+y)}{u_2^*-\phi _{21}(x)},\frac{v_2^*-\phi _{22}(x+y)}{v_2^*-\phi _{22}(x)},\frac{\phi _{21}(x+y)}{\phi _{21}(x)},\frac{\phi _{22}(x+y)}{\phi _{22}(x)}\le C,\\&\frac{|\phi _{31}(x+y)|}{|\phi _{31}(x)|},\frac{|\phi _{32}(x+y)|}{|\phi _{32}(x)|}\frac{|u_2^*-\phi _{31}(x+y)|}{|u_2^*-\phi _{31}(x)|},\frac{|v_2^*-\phi _{32}(x+y)|}{|v_2^*-\phi _{32}(x)|}\le C. \end{aligned}$$

Next, we give estimates for some related functions defined below. To increase the readability of the paper, their proofs are given in the Appendix at the end of the paper. Suppose that $q_1$ and $q_2:{\mathbb {R}}^-\rightarrow {\mathbb {R}}^-$ are two smooth functions satisfying

$$\begin{aligned} q_2(t)\le q_1(t)\le 0\quad \text {for any}~ t<0. \end{aligned}$$

(3.17)

Define four functions from ${\mathbb {R}}\times {\mathbb {R}}^-$ to $\mathbb R$ as follows:

$$\begin{aligned} \begin{aligned}&P^{0}(\xi ,t):=P\left( \phi _{11}(\xi -q_1(t)),\phi _{21}(\xi +q_1(t)),\phi _{31}(\xi +q_2(t))\right) ,\quad (\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-,\\&P^{x}(\xi ,t):=P_{x}\left( \phi _{11}(\xi -q_1(t)),\phi _{21}(\xi +q_1(t)),\phi _{31}(\xi +q_2(t))\right) ,\quad (\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-,\\&P^{y}(\xi ,t):=P_{y}\left( \phi _{11}(\xi -q_1(t)),\phi _{21}(\xi +q_1(t)),\phi _{31}(\xi +q_2(t))\right) ,\quad (\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-,\\&P^{z}(\xi ,t):=P_{z}\left( \phi _{11}(\xi -q_1(t)),\phi _{21}(\xi +q_1(t)),\phi _{31}(\xi +q_2(t))\right) ,\quad (\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-. \end{aligned} \end{aligned}$$

Note that all the functions $P^0$, $P^x$, $P^y$, and $P^z$ are dependent on $q_1$, $q_2$, and $\phi _{i1}$ with $i\in \{1,2,3\}$. The next lemma gives some lower bounds of $P^{x}$, $P^{y}$, and $P^{z}$.

Lemma 3.8

Assume (J1)–(J4) hold. There are some positive constants $\mu _1$, $\mu _2$, and $\mu _3$ such that for any $t<0$,

$$\begin{aligned} \begin{aligned}&P^{x}(\xi ,t)\ge \mu _1&\text {when}~\xi \le -q_1(t),\\&P^{y}(\xi ,t)\ge \mu _2&\text {when}~q_1(t)\le \xi \le -q_2(t),\\&P^{z}(\xi ,t)\ge \mu _3&\text {when}~\xi \ge -q_1(t). \end{aligned} \end{aligned}$$

For given $\delta \in {\mathbb {R}}^+$, we further assume that the functions $q_1,q_2:{\mathbb {R}}^-\rightarrow {\mathbb {R}}^-$ satisfy

$$\begin{aligned} q_1(t)<-\delta ,~q_2(t)<-\delta \quad \text {for any}~t\in {\mathbb {R}}^-. \end{aligned}$$

(3.18)

For $(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-$, define

$$\begin{aligned} A_1(\xi ,t):= & {} P^{x}(\xi ,t)\phi _{11}'(\xi -q_1(t))-P^{y}(\xi ,t)\phi _{21}'(\xi +q_1(t))\\&\quad +P^{z}(\xi ,t)\phi _{31}'(\xi +q_2(t)). \end{aligned}$$

Obviously, $A_1$ is bounded from above. The next lemma shows that under some appropriate conditions on $q_1$ and $q_2$, the function $A_1$ also has a positive bound from below.

Lemma 3.9

Assume (J1)–(J4) hold. There exists a sufficiently large constant $\delta >0$ such that if $q_1$ and $q_2$ satisfy (3.18) and $q_2(t)-q_1(t)<-\delta$ for any $t<0$, then

$$\begin{aligned} A_1(\xi ,t)>0\quad \text {for} ~(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-, \end{aligned}$$

and for any $t<0$, we have that

(i):

when $\xi \le q_1(t)$, $A_1(\xi ,t) \ge \frac{1}{2}P^{x}(\xi ,t)|\phi _{11}'(\xi -q_1(t))|$;

(ii):

when $q_1(t)\le \xi \le -q_1(t)$,

$$\begin{aligned}A_1(\xi ,t) \ge \frac{1}{2}\left[ P^{x}(\xi ,t)|\phi _{11}'(\xi -q_1(t))|+P^{y}(\xi ,t)|\phi _{21}'(\xi +q_1(t))|\right] ; \end{aligned}$$

(iii):

when $-q_1(t)\le \xi \le -q_2(t)$,

$$\begin{aligned} A_1(\xi ,t) \ge \frac{1}{2}\left[ P^{y}(\xi ,t)|\phi _{21}'(\xi +q_1(t))|+P^{z}(\xi ,t)|\phi _{31}'(\xi +q_2(t))|\right] ; \end{aligned}$$

(iv):

when $\xi \ge -q_2(t)$, $A_1(\xi ,t) \ge \frac{1}{2}P^{z}(\xi ,t)|\phi _{31}'(\xi +q_2(t))|$.

For $(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-$, we define

$$\begin{aligned} \begin{aligned} H_1(\xi ,t):=&(J_1*P^0(\xi ,t)-P^0(\xi ,t))-P^x(\xi ,t)(J_1*\phi _{11}(\xi -q_1(t))-\phi _{11}(\xi -q_1(t)))\\&\quad -P^y(\xi ,t)(J_1*\phi _{21}(\xi +q_1(t))-\phi _{21}(\xi +q_1(t)))\\&\quad -P^{z}(\xi ,t)(J_1*\phi _{31}(\xi +q_2(t))-\phi _{31}(\xi +q_2(t))).\\ \end{aligned} \end{aligned}$$

The following lemma gives some estimates for $H_1(\xi ,t)/A_1(\xi ,t)$.

Lemma 3.10

Assume (J1)–(J4) hold. For the $\delta$ given by Lemma 3.9, if $q_1$ and $q_2$ satisfy (3.18) and $q_2(t)-q_1(t)<-\delta$ for any $t\in {\mathbb {R}}^-$, then there is a positive constant ${\tilde{M}}$ such that for any $t<0$,

$$\begin{aligned} \left| \frac{H_1(\xi ,t)}{A_1(\xi ,t)}\right| \le \left\{ \begin{aligned}&{\tilde{M}}\left( e^{\eta _1q_1}+e^{\eta _1q_2}\right) ~&\text {when}~\xi \le 0, \\&{\tilde{M}}\left( e^{\eta _2q_1}+e^{\eta _1(q_2-q_1)/2}\right) ~&\text {when}~0\le \xi \le \frac{-q_1(t)-q_2(t)}{2},\\&{\tilde{M}}\left( e^{\eta _2q_1}+e^{\eta _2(q_2-q_1)/2}\right) ~&\text {when}~\xi \ge \frac{-q_1(t)-q_2(t)}{2}. \end{aligned} \right. \nonumber \\ \end{aligned}$$

(3.19)

For $(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-$, we define

$$\begin{aligned} \begin{aligned} F_1(\xi ,t):=&f_1(P^0(\xi ,t),Q^0(\xi ,t))-P^{x}(\xi ,t)f_1(\phi _{11}(\xi -q_1(t)),\phi _{12}(\xi -q_1(t)))\\&-P^{y}(\xi ,t) f_1(\phi _{21}(\xi +q_1(t)),\phi _{22}(\xi +q_1(t)))\\&-P^{z}(\xi ,t)f_1(\phi _{31}(\xi +q_2(t)),\phi _{32}(\xi +q_2(t))), \end{aligned} \end{aligned}$$

where $f_1$ is defined by (3.14), and

$$\begin{aligned} Q^{0}(\xi ,t):=Q\left( \phi _{12}(\xi -q_1(t)),\phi _{22}(\xi +q_1(t)),\phi _{32}(\xi +q_2(t))\right) ,\quad (\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-. \end{aligned}$$

By a similar argument to Wu et al. (2018, Lemma 4.4), there is some positive constant $C_5$ such that

$$\begin{aligned} \left\{ \begin{aligned} F_1(\xi ,t)\le&C_5(\phi _{11}-u_1^*+\phi _{12}-v_1^*)(u_2^*-\phi _{21}+v_2^*-\phi _{22})[(u_2^*-\phi _{11}+v_2^*-\phi _{12})+(\phi _{31}+\phi _{32})],\\ F_1(\xi ,t)\le&C_5(\phi _{21}+\phi _{22})(u_2^*-\phi _{31}+v_2^*-\phi _{32})[(u_2^*-\phi _{11}+v_2^*-\phi _{12})+(\phi _{31}+\phi _{32})]. \end{aligned} \right. \end{aligned}$$

(3.20)

With the help of (3.20), we can obtain the following result similar to Lemma 3.10, and the details of the proof are omitted.

Lemma 3.11

Assume (J1)–(J4) hold. For the $\delta$ given by Lemma 3.9, if $q_1$ and $q_2$ satisfy (3.18) and $q_2(t)-q_1(t)<-\delta$ for any $t\in {\mathbb {R}}^-$, then there is a positive constant ${\bar{M}}$ such that for any $t<0$,

$$\begin{aligned} \left| \frac{F_1(\xi ,t)}{A_1(\xi ,t)}\right| \le \left\{ \begin{aligned}&{\bar{M}}\left( e^{\eta _1q_1}+e^{\eta _1q_2}\right) ~\text {when}~\xi \le 0, \\&{\bar{M}}\left( e^{\eta _2q_1}+e^{\eta _1(q_2-q_1)/2}\right) ~\text {when}~0\le \xi \le \frac{-q_1(t)-q_2(t)}{2},\\&{\bar{M}}\left( e^{\eta _2q_1}+e^{\eta _2(q_2-q_1)/2}\right) ~\text {when}~\xi \ge \frac{-q_1(t)-q_2(t)}{2}. \end{aligned} \right. \end{aligned}$$

Note that Lemmas 3.9–3.11 provide the estimates for $A_1(\xi ,t)$, $F_1(\xi ,t)$, and $H_1(\xi ,t)$, which are related to $P^0(\xi ,t)$, $P^x(\xi ,t)$, $P^y(\xi ,t)$, and $P^z(\xi ,t)$. For $(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-$, we define

$$\begin{aligned} \begin{aligned} A_2(\xi ,t)&:=Q^{x}(\xi ,t)\phi _{12}'(\xi -q_1(t))-Q^{y}(\xi ,t)\phi _{22}'(\xi +q_1(t))+Q^{z}(\xi ,t)\phi _{32}'(\xi +q_2(t)),\\ H_2(\xi ,t)&:=(J_2*Q^0(\xi ,t)-Q^0(\xi ,t))-Q^x(\xi ,t)(J_2*\phi _{12}(\xi -q_1(t))-\phi _{12}(\xi -q_1(t)))\\&\quad -Q^y(\xi ,t)(J_2*\phi _{22}(\xi +q_1(t))-\phi _{22}(\xi +q_1(t)))\\&\quad -Q^{z}(\xi ,t)(J_2*\phi _{32}(\xi +q_2(t))-\phi _{32}(\xi +q_2(t))),\\ F_2(\xi ,t)&:=f_2(P^0(\xi ,t),Q^0(\xi ,t))-Q^{x}(\xi ,t)f_2(\phi _{11}(\xi -q_1(t)),\phi _{12}(\xi -q_1(t)))\\&\quad -Q^{y}(\xi ,t)f_2(\phi _{21}(\xi +q_1(t)),\phi _{22}(\xi +q_1(t)))\\&\quad -Q^{z}(\xi ,t)f_2(\phi _{31}(\xi +q_2(t)),\phi _{32}(\xi +q_2(t))), \end{aligned} \end{aligned}$$

where $f_2$ is defined by (3.14), and

$$\begin{aligned} \begin{aligned}&Q^{x}(\xi ,t):=Q_{x}\left( \phi _{12}(\xi -q_1(t)),\phi _{22}(\xi +q_1(t)),\phi _{32}(\xi +q_2(t))\right) ,\quad (\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-,\\&Q^{y}(\xi ,t):=Q_{y}\left( \phi _{12}(\xi -q_1(t)),\phi _{22}(\xi +q_1(t)),\phi _{32}(\xi +q_2(t))\right) ,\quad (\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-,\\&Q^{z}(\xi ,t):=Q_{z}\left( \phi _{12}(\xi -q_1(t)),\phi _{22}(\xi +q_1(t)),\phi _{32}(\xi +q_2(t))\right) ,\quad (\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-. \end{aligned} \end{aligned}$$

The next three lemmas give estimates for $Q^{x}(\xi ,t)$, $Q^{y}(\xi ,t)$, $Q^{z}(\xi ,t)$, $A_2(\xi ,t)$, $F_2(\xi ,t)$, and $H_2(\xi ,t)$. The proofs of these three lemmas, which are omitted, are similar to Lemmas 3.8, 3.9, and the combination of Lemmas 3.10 and 3.11, respectively.

Lemma 3.12

Assume (J1)–(J4) hold. There are some positive constants $\mu _1$, $\mu _2$, and $\mu _3$ such that for any $t<0$,

$$\begin{aligned} \begin{aligned}&Q^{x}(\xi ,t)\ge \mu _1&\text {when}~\xi \le -q_1(t),\\&Q^{y}(\xi ,t)\ge \mu _2&\text {when}~q_1(t)\le \xi \le -q_2(t),\\&Q^{z}(\xi ,t)\ge \mu _3&\text {when}~\xi \ge -q_1(t). \end{aligned} \end{aligned}$$

Lemma 3.13

Assume (J1)–(J4) hold. There exists a sufficiently large constant $\delta >0$ such that if $q_1$ and $q_2$ satisfy (3.18) and $q_2(t)-q_1(t)<-\delta$ for any $t<0$, then

$$\begin{aligned} A_2(\xi ,t)>0\quad \text {for} ~(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-, \end{aligned}$$

and for any $t<0$, we have that

(i):

when $\xi \le q_1(t)$, $A_2(\xi ,t) \ge \frac{1}{2}Q^{x}(\xi ,t)|\phi _{12}'(\xi -q_1(t))|$;

(ii):

when $q_1(t)\le \xi \le -q_1(t)$,

$$\begin{aligned}A_2(\xi ,t) \ge \frac{1}{2}\left[ Q^{x}(\xi ,t)|\phi _{12}'(\xi -q_1(t))|+Q^{y}(\xi ,t)|\phi _{22}'(\xi +q_1(t))|\right] ; \end{aligned}$$

(iii):

when $-q_1(t)\le \xi \le -q_2(t)$,

$$\begin{aligned} A_2(\xi ,t) \ge \frac{1}{2}\left[ Q^{y}(\xi ,t)|\phi _{22}'(\xi +q_1(t))|+Q^{z}(\xi ,t)|\phi _{32}'(\xi +q_2(t))|\right] ; \end{aligned}$$

(iv):

when $\xi \ge -q_2(t)$, $A_2(\xi ,t) \ge \frac{1}{2}Q^{z}(\xi ,t)|\phi _{32}'(\xi +q_2(t))|$.

Lemma 3.14

Assume (J1)–(J4) hold. For the $\delta$ given by Lemma 3.13, if $q_1$ and $q_2$ satisfy (3.18) and $q_2(t)-q_1(t)<-\delta$ for any $t<0$, then there is a positive constant ${\hat{M}}$ such that for any $t<0$,

$$\begin{aligned} \begin{aligned} \left| \frac{H_2(\xi ,t)}{A_2(\xi ,t)}\right| +\left| \frac{F_2(\xi ,t)}{A_2(\xi ,t)}\right| \le \left\{ \begin{aligned}&{\hat{M}}\left( e^{\eta _1q_1}+e^{\eta _1q_2}\right) ~\text {when}~\xi \le 0, \\&{\hat{M}}\left( e^{\eta _2q_1}+e^{\eta _1(q_2-q_1)/2}\right) ~\text {when}~0\le \xi \le \frac{-q_1(t)-q_2(t)}{2},\\&{\hat{M}}\left( e^{\eta _2q_1}+e^{\eta _2(q_2-q_1)/2}\right) ~\text {when}~\xi \ge \frac{-q_1(t)-q_2(t)}{2}. \end{aligned}\right. \end{aligned} \end{aligned}$$

Finally, we can give a pair of suitable super- and sub-solutions for the first type of entire solution and complete the proof of Theorem 3.5 by applying them.

Lemma 3.15

Assume that (J1)–(J4) hold, and $c_0>{\hat{c}}_0>c_2$. Let $(p_1(t),p_2(t),r_1(t),$ $r_2(t))$ be the solution of (3.2). Then, the functions defined by (3.15) and (3.16), namely,

$$\begin{aligned}\left\{ \begin{aligned}&\overline{U}(\xi ,t)= P(\phi _{11}(\xi -p_1(t)),\phi _{21}(\xi +p_1(t)),\phi _{31}(\xi +p_2(t))),\\&\overline{V}(\xi ,t)= Q(\phi _{12}(\xi -p_1(t)),\phi _{22}(\xi +p_1(t)),\phi _{32}(\xi +p_2(t))) \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned}\left\{ \begin{aligned}&{\underline{U}}(\xi ,t)= P(\phi _{11}(\xi -r_1(t)),\phi _{21}(\xi +r_1(t)),\phi _{31}(\xi +r_2(t))),\\&{\underline{V}}(\xi ,t)= Q(\phi _{12}(\xi -r_1(t)),\phi _{22}(\xi +r_1(t)),\phi _{32}(\xi +r_2(t))) \end{aligned}\right. \end{aligned}$$

are a pair of super- and sub-solutions of (3.13) for $t\le t_0<0$ with some $t_0<0$, when the constant L in (3.2) and (3.3) is sufficiently large. Moreover, there exists $\mu >0$ such that

$$\begin{aligned} {\underline{U}}(\xi ,t)\le \overline{U}(\xi ,t),~\sup \limits _{\xi \in {\mathbb {R}}}\{\overline{U}(\xi ,t)-{\underline{U}}(\xi ,t)\}\le \mu e^{\kappa \nu _1t}~\text {for}~\xi \in {\mathbb {R}},~t\le t_0, \qquad \end{aligned}$$

(3.21)

$$\begin{aligned} {\underline{V}}(\xi ,t)\le \overline{V}(\xi ,t),~\sup \limits _{\xi \in {\mathbb {R}}}\{\overline{V}(\xi ,t)-{\underline{V}}(\xi ,t)\}\le \mu e^{\kappa \nu _1t}~\text {for}~\xi \in {\mathbb {R}},~t\le t_0.\qquad \end{aligned}$$

(3.22)

where $\nu _1$ and $\kappa$ are defined by (3.1) and (3.4), respectively.

Proof

Let $\delta$ be a positive constant such that Lemmas 3.9 and 3.13 hold. By (3.2), it is easy to get that $p_2(t)-p_1(t)\rightarrow -\infty$ and $r_2(t)-r_1(t)\rightarrow -\infty$ as $t\rightarrow -\infty$. By (3.4), there exists a constant $t_0<0$ such that

$$\begin{aligned} p_2(t)-p_1(t)<-\delta ,\quad r_2(t)-r_1(t)<-\delta ~\text {when}~ t\le t_0, \end{aligned}$$

and

$$\begin{aligned}&\max \left\{ \frac{\eta _1(p_2(t)-p_1(t))}{2}, \frac{\eta _2(p_2(t)-p_1(t))}{2}\right\}<\kappa p_1(t)<0~\text {when}~ t\le t_0, \nonumber \\\end{aligned}$$

(3.23)

$$\begin{aligned}&\max \left\{ \frac{\eta _1(r_2(t)-r_1(t))}{2}, \frac{\eta _2(r_2(t)-r_1(t))}{2}\right\}<\kappa r_1(t)<0~\text {when}~ t\le t_0.\nonumber \\ \end{aligned}$$

(3.24)

Recall that $s_1$, $s_2$, and $s_3$ satisfy (3.11). For simplicity, we denote

$$\begin{aligned} \begin{aligned}&\Gamma _1(U,V)(\xi ,t):=\frac{\partial U}{\partial t}-(J_1*U-U)+{\bar{c}}U_{\xi }-f_1(U,V), \\&\Gamma _2(U,V)(\xi ,t):=\frac{\partial V}{\partial t}-(J_2*V-V)+{\bar{c}}V_{\xi }-f_2(U,V), \end{aligned} \end{aligned}$$

where $\xi = x+{\bar{c}}t$, ${\bar{c}}=\frac{s_1+s_2}{2}$, and $f_1,f_2$ are defined by (3.14). To prove the first conclusion of this lemma, it is sufficient to show that

$$\begin{aligned} \Gamma _i(\overline{U},\overline{V})(\xi ,t)\ge 0~~\text {and}~~\Gamma _i({\underline{U}},{\underline{V}})(\xi ,t)\le 0,~\text {for any } \xi \in {\mathbb {R}},t\le t_0, i=1,2. \end{aligned}$$

Next, we only prove $\Gamma _i(\overline{U},\overline{V})(\xi ,t)\ge 0$ with $i=1,2$ for $\xi \in {\mathbb {R}}$ and $t\le t_0$, since the proof of $\Gamma _i({\underline{U}},{\underline{V}})(\xi ,t)\le 0$ is similar. Direct calculations give that

$$\begin{aligned} \begin{aligned} \Gamma _1(\overline{U},\overline{V})&= -P^{x}\phi _{11}'(p_1'-\nu _1)+P^{y}\phi _{21}'(p_1'-\nu _1)+P^{z}\phi _{31}'(p_2'-\nu _2)\\&\quad -(J_1*\overline{U}-\overline{U})+P^x(J_1*\phi _{11}-\phi _{11})+ P^y(J_1*\phi _{21}-\phi _{21})\\&\quad +P^z(J_1*\phi _{31}-\phi _{31})\\&\quad -f_1(\overline{U},\overline{V})+P^xf_1(\phi _{11},\phi _{12})+P^yf_1(\phi _{21},\phi _{22})+P^zf_1(\phi _{31},\phi _{32}), \end{aligned} \end{aligned}$$

where $\nu _1$ and $\nu _2$ are defined in (3.1). By (3.2) and the definitions of the functions $A_1(\xi ,t)$, $F_1(\xi ,t)$, and $H_1(\xi ,t)$, we get that

$$\begin{aligned} \begin{aligned} \Gamma _1(\overline{U},\overline{V})(\xi ,t) =&(P^x\phi _{11}'-P^y\phi _{21}'+P^z\phi _{31}')Le^{\kappa p_1(t)}-H_1(\xi ,t)-F_1(\xi ,t)\\ =&A_1(\xi ,t)Le^{\kappa p_1(t)}-H_1(\xi ,t)-F_1(\xi ,t). \end{aligned} \end{aligned}$$

From Lemmas 3.10, 3.11, and (3.23), it follows that

$$\begin{aligned} |H_1(\xi ,t)+F_1(\xi ,t)|\le A_1(\xi ,t)({\tilde{M}}+{\bar{M}})e^{\kappa p_1(t)}, ~\text {for any } \xi \in {\mathbb {R}},t\le t_0. \end{aligned}$$

By choosing $L>{\tilde{M}}+{\bar{M}}$, we have

$$\begin{aligned} \Gamma _1(\overline{U},\overline{V})(\xi ,t)\ge A_1(\xi ,t)[Le^{\kappa p_1(t)}-({\tilde{M}}+{\bar{M}})e^{\kappa p_1(t)}]\ge 0,~\text {for any } \xi \in {\mathbb {R}},t\le t_0. \end{aligned}$$

By a similar argument to $\Gamma _1(\overline{U},\overline{V})(\xi ,t)$ (with replacing Lemmas 3.8, 3.9, 3.10, 3.11, and (3.23) by Lemmas 3.12, 3.13, 3.14, and (3.24)), we can take $L\ge {\hat{M}}$ such that

$$\begin{aligned} \begin{aligned} \Gamma _2(\overline{U},\overline{V})(\xi ,t)&=(Q^{x}\phi _{12}'-Q^{y}\phi _{22}'+Q^{z}\phi _{32}')Le^{\kappa p_1(t)}-H_2(\xi ,t)-F_2(\xi ,t)\\&\ge A_2(\xi ,t)(L-{\hat{M}})e^{\kappa p_1(t)}\ge 0,~\text {for any } \xi \in {\mathbb {R}},t\le t_0. \end{aligned} \end{aligned}$$

Finally, we prove (3.21), and the proof of (3.22) is similar to (3.21). We see that

$$\begin{aligned} \begin{aligned}&\overline{U}(\xi ,t)-{\underline{U}}(\xi ,t)\\ =&P_1(\phi _{11}(\xi -p_1),\phi _{21}(\xi +p_1),\phi _{31}(\xi +p_2))-P_1(\phi _{11}(\xi -r_1),\phi _{21}(\xi +r_1),\phi _{31}(\xi +r_2)). \end{aligned} \end{aligned}$$

From Newton–Leibniz formula, it follows that

$$\begin{aligned} \begin{aligned}&\overline{U}(\xi ,t)-{\underline{U}}(\xi ,t)\\&=(r_1-p_1)\int _0^1P_x(\phi _{11}(\xi -\theta p_1-(1-\theta )r_1),\phi _{21}(\xi +p_1),\phi _{31}(\xi +p_2))\phi _{11}'{\text {d}}\theta \\&\quad -(r_1-p_1)\int _0^1P_y(\phi _{11}(\xi - r_1),\phi _{21}(\xi +\theta p_1+(1-\theta )r_1),\phi _{31}(\xi +p_2))\phi _{21}'{\text {d}}\theta \\&\quad +(p_2-r_2)\int _0^1P_z(\phi _{11}(\xi -r_1),\phi _{21}(\xi +r_1),\phi _{31}(\xi +\theta p_2+(1-\theta )r_2)\phi _{31}'{\text {d}}\theta . \end{aligned}\nonumber \\ \end{aligned}$$

(3.25)

Obviously, the integral terms in (3.25) are bounded and positive. Then, by (3.6), we can easily obtain that

$$\begin{aligned} {\underline{U}}(\xi ,t)\le \overline{U}(\xi ,t)~\text {and}~\sup \limits _{\xi \in {\mathbb {R}}}\{\overline{U}(\xi ,t)-{\underline{U}}(\xi ,t)\}\le \mu e^{\kappa \nu _1t},~\text {for any } \xi \in {\mathbb {R}},t\le t_0. \end{aligned}$$

It completes the proof. $\square$

Proof of Theorem 3.5

We define $(\overline{U},\overline{V})$ and $({\underline{U}},{\underline{V}})$ by the same functions as in Lemma 3.15. Let ${\bar{c}}=(c_2+{{\hat{c}}}_0)/2$, and denote

$$\begin{aligned} \begin{aligned}&\overline{\varvec{\omega }}(x,t):=(\overline{u},\overline{v})(x,t)= (\overline{U},\overline{V})(x+{\bar{c}}t,t),\\&\underline{\varvec{\omega }}(x,t):=({\underline{u}},{\underline{v}})(x,t) =({\underline{U}},{\underline{V}})(x+{\bar{c}}t,t). \end{aligned} \end{aligned}$$

By Lemma 3.15, $\overline{\varvec{\omega }}(x,t)$ and $\underline{\varvec{\omega }}(x,t)$ are a pair of super- and sub-solutions of (1.1) for $x\in {\mathbb {R}}$ and $t\le t_0$. Now, we consider the following Cauchy problem

$$\begin{aligned} \left\{ \begin{aligned}&\partial _tu_n(x,t)=(J_1*u_n-u_n)(x,t)-\alpha u_n(x,t)+h(v_n(x,t)),\\&\partial _tv_n(x,t)=(J_2*v_n-v_n)(x,t)-\beta v_n(x,t)+g(u_n(x,t)),\\&(u_n,v_n)(x,-n)=\underline{\varvec{\omega }}(x,-n),~x\in {\mathbb {R}},t>-n. \end{aligned}\right. \end{aligned}$$

(3.26)

By Lemma 2.10, (3.26) has a unique solution $\varvec{\omega }_n(x,t)=(u_n(x,t),v_n(x,t))$ satisfying

$$\begin{aligned} E_0 \le \varvec{\omega }_n(x,t) \le E_2,~\text {for any~} x\in {\mathbb {R}},t>-n. \end{aligned}$$

Since $\underline{\varvec{\omega }}(x,t)$ is also a sub-solution of (3.26), we have

$$\begin{aligned} \underline{\varvec{\omega }}(x,-n)=\varvec{\omega }_n(x,-n)\le \varvec{\omega }_{n+1}(x,-n)\le E_2,~\text {for any~} x\in {\mathbb {R}},t>-n. \end{aligned}$$

It follows from Lemma 3.2 that

$$\begin{aligned} E_0\le \underline{\varvec{\omega }}(x,t)\le \varvec{\omega }_n(x,t)\le \varvec{\omega }_{n+1}(x,t)\le \min \{E_2,\overline{\varvec{\omega }}(x,t)\},~\text {for any~} x\in {\mathbb {R}},t>-n. \end{aligned}$$

Therefore, $\{ \varvec{\omega }_n(x,t)\}_{n=1}^\infty$ is bounded and nondecreasing with respect to n for any $(x, t) \in {\mathbb {R}}\times (-n,+\infty )$. Then, there exists a function $\varvec{\omega }(x,t)=(u(x,t),v(x,t))$ with $E_0 \le \varvec{\omega }(x,t) \le E_2$ such that for any $(x,t)\in {\mathbb {R}}^2$,

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }(u_n(x,t),v_n(x,t))=(u(x,t),v(x,t)). \end{aligned}$$

Note that for any given $t_0\in {\mathbb {R}}$, there exists $n\in {\mathbb {N}}$ such that $t_0>-n$ and

$$\begin{aligned} \pmb {\omega }_n(x,t)={{\textbf {P}}}(t)[\pmb {\omega }_n(\cdot ,t_0)](x)+\int _{t_0}^t {{\textbf {P}}}(t-s)[{{\textbf {F}}}(\pmb {\omega }_n(\cdot ,s))](x){\text {d}}s,~~x\in {\mathbb {R}},~t\ge t_0. \end{aligned}$$

By Lebesgue dominated convergence theorem, we have that

$$\begin{aligned} \pmb {\omega }(x,t)={{\textbf {P}}}(t)[\pmb {\omega }(\cdot ,t_0)](x)+\int _{t_0}^t {{\textbf {P}}}(t-s)[{{\textbf {F}}}(\pmb {\omega }(\cdot ,s))](x){\text {d}}s. \end{aligned}$$

We easily check that $\pmb {\omega }(x, t)$ is continuous and differentiable with respect to t. Thus, we obtain

$$\begin{aligned}\begin{aligned} u_t(x,t)&=\frac{\partial }{\partial t}P_1(t)[u(\cdot ,t_0)](x)+\int _{t_0}^t\frac{\partial }{\partial t}P_1(t-s)[-\alpha u(\cdot ,s)+h(v(\cdot ,s))](x){\text {d}}s\\&\quad +[-\alpha u(x,t)+h(v(x,t))]. \end{aligned}\end{aligned}$$

From (2.17), it follows that

$$\begin{aligned}\begin{aligned} u_t(x,t)&= -P_1(t)[u(\cdot ,t_0)](x)+J_1*P_1(t)[u(\cdot ,t_0)](x)+[-\alpha u(x,t)+h(v(x,t))]\\&\quad +\int _{t_0}^t\left\{ -P_1(t)[-\alpha u(\cdot ,s)+h(v(\cdot ,s))](x)+J_1*P_1(t)\right. \\&\quad \left. [-\alpha u(\cdot ,s)+h(v(\cdot ,s))](x)\right\} {\text {d}}s\\&=J_1*u(x,t)-u(x,t)-\alpha u(x,t)+h(v(x,t)). \end{aligned}\end{aligned}$$

Similarly, we have that

$$\begin{aligned} v_t(x,t)=J_2*v(x,t)-v(x,t)-\beta v(x,t)+g(u(x,t)). \end{aligned}$$

Thus, $\pmb {\omega }(x, t)=(u(x,t),v(x,t))$ is an entire solution of (1.1). The asymptotic behavior and the smooth property of $\pmb {\omega }(x, t)$ can be obtained by similar arguments to Chen et al. (2018, Theorem 4.3) and Zhang et al. (2016, Theorem 1.8), respectively, and we only give an outline of the proof here. According to Lemma 3.7, (3.5), (3.21), and (3.22), we can prove (3.7). By applying Lemma 3.3, Arzela–Ascoli theorem, a diagonal extraction process, and the uniqueness of the solution of (3.26), we can obtain the last assertion of this theorem. Therefore, the proof is completed. $\square$

3.3 II-Type Entire Solution

In this subsection, we construct the second type of entire solution, which originates from one monostable and two bistable traveling wave solutions, namely $({\hat{c}}_1,{\hat{\Phi }}_1)$, $(c_0,\Phi )$, and $({\hat{c}}_0,{\hat{\Phi }})$ with ${\hat{c}}_1<c_0<{\hat{c}}_0$. We see Fig. 1b for the profile of this entire solution as $t\rightarrow -\infty$. As stated in Sect. 3.2, the difficulty in the study of entire solutions is how to construct appropriate auxiliary functions and super- and sub-solutions for different types of entire solutions. Hence, we only provide the construction of auxiliary functions and super- and sub-solutions and omit the other details, which are similar to Theorem 3.5. The second type of entire solution is considered in the following theorem.

Theorem 3.16

Assume (J1)–(J4) hold and ${\hat{c}}_0>c_0>{\hat{c}}_1$. Then, (1.1) admits an entire solution ${\mathcal {W}}_2(x,t):{\mathbb {R}}^2\rightarrow [0,u_2^*]\times [0,v_2^*]$ satisfying

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }&\left\{ \sup \limits _{x\le \rho _1(t)}\Vert {\mathcal {W}}_2(x,t)-{\hat{\Phi }}_1(x+{\hat{c}}_1t-\vartheta _3)\Vert \right. \nonumber \\&\quad +\sup \limits _{\rho _1(t)\le x\le \rho _2(t)}\Vert {\mathcal {W}}_2(x,t)-\Phi (x+c_0t+\vartheta _3)\Vert \nonumber \\&\left. \quad +\sup \limits _{\rho _2(t)\le x}\Vert {\mathcal {W}}_2(x,t)-{\hat{\Phi }}(x+{\hat{c}}_0t+\vartheta _4)\Vert \right\} =0, \end{aligned}$$

where $\vartheta _3$ and $\vartheta _4$ are some constants satisfying

$$\begin{aligned} \vartheta _3:=-\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}-\frac{L}{\nu _1}\right] ,\quad \vartheta _4:=\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}-\frac{L}{\nu _1}\right] +p_0+r_0, \end{aligned}$$

and $\rho _1,\rho _2:{\mathbb {R}}\mapsto {\mathbb {R}}$ are defined by

$$\begin{aligned} \rho _1(t):=\frac{-({\hat{c}}_1+c_0)t}{2},\rho _2(t):=\frac{-(c_0+{\hat{c}}_0)t}{2}. \end{aligned}$$

Moreover, when (J5) holds, there are two positive constants $D_1$ and $D_2$ such that for any $(x,t)\in {\mathbb {R}}^2$ and $\eta >0$,

$$\begin{aligned} \Vert {\mathcal {W}}_2(x+\eta ,t)-{\mathcal {W}}_2(x,t)\Vert \le D_1\eta ,\left\| \frac{\partial {\mathcal {W}}_2}{\partial t}(x+\eta ,t)-\frac{\partial {\mathcal {W}}_2}{\partial t}\right\| \le D_2\eta . \end{aligned}$$

Now, we show the construction of auxiliary functions and super- and sub-solutions for ${\mathcal {W}}_2$. Throughout this subsection, choose the values of $s_1$, $s_2$, and $s_3$ as follows:

$$\begin{aligned} s_1:={\hat{c}}_1,~s_2:=c_0,~s_3:={\hat{c}}_0. \end{aligned}$$

For convenience, we denote

$$\begin{aligned} \varvec{\phi }_1(\xi ):={\hat{\Phi }}_1(\xi ),~\varvec{\phi }_2(\xi ):=\Phi (\xi ),~\varvec{\phi }_3(\xi ):={\hat{\Phi }}(\xi ), \end{aligned}$$

and assume that

$$\begin{aligned} \phi _{11}(0)= & {} \frac{u_1^*}{2},~(\phi _{21}(0),\phi _{22}(0))\ge \left( \frac{u_1^*+u_2^*}{2},\frac{v_1^*+v_2^*}{2}\right) ,~\text {and}~(\phi _{31}(0),\phi _{32}(0))\\&\quad \ge \left( \frac{u_1^*+u_2^*}{2},\frac{v_1^*+v_2^*}{2}\right) . \end{aligned}$$

Corresponding to (3.9) and (3.10) for the first type of entire solution, two auxiliary functions for this case to link traveling wave solutions are defined as follows:

$$\begin{aligned} P(x,y,z)=y+\frac{(u_1^*-x)y(u_2^*-z)(-y)+x(u_2^*-y)z(u_1^*-y)}{(u_1^*-x)yu_2^*+u_1^*(u_2^*-y)z},(x,y,z)\in D_1 \end{aligned}$$

and

$$\begin{aligned} Q(x,y,z)=y+\frac{(v_1^*-x)y(v_2^*-z)(-y)+x(v_2^*-y)z(v_1^*-y)}{(v_1^*-x)yu_2^*+v_1^*(v_2^*-y)z},(x,y,z)\in D_2, \end{aligned}$$

where

$$\begin{aligned} D_1:=\left\{ (x,y,z)\in [0,u_1^*]\times [0,u_2^*]\times [0,u_2^*]\big |(u_1^*-x)yu_2^*+u_1^*(u_2^*-y)z>0\right\} \end{aligned}$$

and

$$\begin{aligned} D_2:=\left\{ (x,y,z)\in [0,v_1^*]\times [0,v_2^*]\times [0,v_2^*]\big |(v_1^*-x)yu_2^*+v_1^*(v_2^*-y)z>0\right\} . \end{aligned}$$

The following lemma constructs a pair of super- and sub-solutions, and by this lemma we can prove Theorem 3.16. We omit the proofs, which are similar to Lemma 3.15 and Theorem 3.5.

Lemma 3.17

Assume (J1)–(J4) hold and ${\hat{c}}_0>c_0>{\hat{c}}_1$. Let $(p_1(t),p_2(t),r_1(t),$ $r_2(t))$ be the solution of (3.2), and let ${\bar{c}}:=({\hat{c}}_1+c_0)/2$. Then, the functions defined by

$$\begin{aligned}\left\{ \begin{aligned}&\overline{u}(x,t)= P(\phi _{11}(x+{\bar{c}}t-r_1(t)),\phi _{21}(x+{\bar{c}}t+r_1(t)),\phi _{31}(x+{\bar{c}}t+r_2(t))),\\&\overline{v}(x,t)= Q(\phi _{12}(x+{\bar{c}}t-r_1(t)),\phi _{22}(x+{\bar{c}}t+r_1(t)),\phi _{32}(x+{\bar{c}}t+r_2(t))) \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{aligned}&{\underline{u}}(x,t)= P(\phi _{11}(x+{\bar{c}}t-p_1(t)),\phi _{21}(x+{\bar{c}}t+p_1(t)),\phi _{31}(x+{\bar{c}}t+p_2(t))),\\&{\underline{v}}(x,t)= Q(\phi _{12}(x+{\bar{c}}t-p_1(t)), \phi _{22}(x+{\bar{c}}t+p_1(t)),\phi _{32}(x+{\bar{c}}t+p_2(t))) \end{aligned}\right. \end{aligned}$$

are a pair of super- and sub-solutions of (1.1) for $(x,t)\in {\mathbb {R}}\times (-\infty ,t_1]$ with some $t_1<0$. Moreover, (3.21) and (3.22) hold for $(\overline{u}(x,t),\overline{v}(x,t))$ and $({\underline{u}}(x,t),{\underline{v}}(x,t))$.

3.4 III-Type Entire Solution

The third type of entire solution originates from two monostable and one bistable traveling wave solutions, namely $(c_1,\Phi _1)$, $({\hat{c}}_1,{\hat{\Phi }}_1)$, and $(c_0,\Phi )$ with $c_1<{\hat{c}}_1<c_0$. We see Fig. 1c for the profile of this entire solution as $t\rightarrow -\infty$.

Theorem 3.18

Assume (J1)–(J4) hold and $c_0>{\hat{c}}_1>c_1$. Then, (1.1) admits an entire solution ${\mathcal {W}}_3(x,t):{\mathbb {R}}^2\rightarrow [0,u_2^*]\times [0,v_2^*]$ satisfying

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }&\left\{ \sup \limits _{x\le \rho _1(t)}\Vert {\mathcal {W}}_3(x,t)-\Phi _1(x+c_1t-\vartheta _{1})\Vert \right. \nonumber \\&\quad +\sup \limits _{\rho _1(t)\le x\le \rho _2(t)}\Vert {\mathcal {W}}_3(x,t)-{\hat{\Phi }}_1(x+{\hat{c}}_1t+\vartheta _{1})\Vert \nonumber \\&\left. \quad +\sup \limits _{\rho _2(t)\le x}\Vert {\mathcal {W}}_3(x,t)-\Phi (x+c_0t+\vartheta _{2})\Vert \right\} =0, \end{aligned}$$

where $\vartheta _1$ and $\vartheta _2$ are some constants satisfying

$$\begin{aligned} \vartheta _1:=-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}+\frac{L}{\nu _1}\right] ,\quad \vartheta _2:=\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}+\frac{L}{\nu _1}\right] +p_0+r_0, \end{aligned}$$

and $\rho _1,\rho _2:{\mathbb {R}}\mapsto {\mathbb {R}}$ are defined by

$$\begin{aligned} \rho _1(t):=\frac{-(c_1+{\hat{c}}_1)t}{2},\rho _2(t):=\frac{-({\hat{c}}_1+c_0)t}{2}. \end{aligned}$$

Moreover, when (J5) holds, there are two positive constants $D_1$ and $D_2$ such that for any $(x,t)\in {\mathbb {R}}^2$ and $\eta >0$,

$$\begin{aligned} \Vert {\mathcal {W}}_3(x+\eta ,t)-{\mathcal {W}}_3(x,t)\Vert \le D_1\eta ,\left\| \frac{\partial {\mathcal {W}}_3}{\partial t}(x+\eta ,t)-\frac{\partial {\mathcal {W}}_3}{\partial t}\right\| \le D_2\eta . \end{aligned}$$

Similar to the case of ${\mathcal {W}}_2$, we only provide the construction of auxiliary functions and super- and sub-solutions for ${\mathcal {W}}_3$ and omit the other details in the proof of Theorem 3.18. Throughout this subsection, we choose the values of $s_1$, $s_2$, and $s_3$ as follows:

$$\begin{aligned} s_1:=c_1,~s_2:={\hat{c}}_1,~s_3:=c_0. \end{aligned}$$

Denote

$$\begin{aligned} \varvec{\phi }_1(\xi ):=\Phi _1(\xi ),~\varvec{\phi }_2(\xi ):={\hat{\Phi }}_1(\xi ),~\varvec{\phi }_3(\xi ):=\Phi (\xi ), \end{aligned}$$

and assume that

$$\begin{aligned} (\phi _{11}(0),\phi _{21}(0))\ge & {} \left( \frac{u_1^*}{2},\frac{v_1^*}{2}\right) ,~\phi _{21}(0)=\frac{u_1^*}{2},~\text {and}~(\phi _{31}(0),\phi _{32}(0))\\\le & {} \left( \frac{u_1^*}{4},\frac{v_1^*}{4}\right) . \end{aligned}$$

The two auxiliary functions are defined as follows:

$$\begin{aligned} {P}(x,y,z)=y+\frac{x(u_1^*-y)z(u_2^*-y)+(u_1^*-x)y(u_2^*-z)(-y)}{x(u_1^*-y)u_2^*+u_1^*y(u_2^*-z)},(x,y,z)\in {D}_1 \end{aligned}$$

and

$$\begin{aligned} {Q}(x,y,z)=y+\frac{x(v_1^*-y)z(v_2^*-y)+(v_1^*-x)y(v_2^*-z)(-y)}{x(v_1^*-y)v_2^*+v_1^*y(v_2^*-z)},(x,y,z)\in {D}_2, \end{aligned}$$

where

$$\begin{aligned} {D}_1:=\left\{ (x,y,z)\in [0,u_1^*]\times [0,u_1^*]\times [0,u_2^*]\big |x(u_1^*-y)u_2^*+u_1^*y(u_2^*-z)>0\right\} \end{aligned}$$

and

$$\begin{aligned} {D}_2:=\left\{ (x,y,z)\in [0,v_1^*]\times [0,v_1^*]\times [0,v_2^*]\big |x(v_1^*-y)v_2^*+v_1^*y(v_2^*-z)>0\right\} . \end{aligned}$$

The super- and sub-solutions are given by the following lemma.

Lemma 3.19

Assume (J1)–(J4) hold and $c_0>{\hat{c}}_1>c_1$. Let $(p_1(t),p_2(t),r_1(t),$ $r_2(t))$ be the solution of (3.2), and ${{\bar{c}}}:=(c_1+{\hat{c}}_1)/2$. Then, the functions defined by

$$\begin{aligned}\left\{ \begin{aligned}&\overline{u}(x,t)= {P}(\phi _{11}(x+{\bar{c}}t-p_1(t)),\phi _{21}(x+{\bar{c}}t+p_1(t)),\phi _{31}(x+{\bar{c}}t+p_2(t))),\\&\overline{v}(x,t)= {Q}(\phi _{12}(x+{\bar{c}}t-p_1(t)),\phi _{22}(x+{\bar{c}}t+p_1(t)),\phi _{32}(x+{\bar{c}}t+p_2(t))) \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned}\left\{ \begin{aligned}&{\underline{u}}(x,t)= {P}(\phi _{11}(x+{\bar{c}}t-r_1(t)),\phi _{21}(x+{\bar{c}}t+r_1(t)),\phi _{31}(x+{\bar{c}}t+r_2(t))),\\&{\underline{v}}(x,t)= {Q}(\phi _{12}(x+{\bar{c}}t-r_1(t)),\phi _{22}(x+{\bar{c}}t+r_1(t)),\phi _{32}(x+{\bar{c}}t+r_2(t))) \end{aligned}\right. \end{aligned}$$

are a pair of super- and sub-solutions of (1.1) for $(x,t)\in {\mathbb {R}}\times (-\infty ,t_2]$ with some $t_2<0$. Moreover, (3.21) and (3.22) hold for $(\overline{u}(x,t),\overline{v}(x,t))$ and $({\underline{u}}(x,t),{\underline{v}}(x,t))$.

3.5 IV-Type Entire Solution

The fourth type of entire solution originates from two monostable and one bistable traveling wave solutions, namely $({\hat{c}}_2,{\hat{\Phi }}_2)$, $({\hat{c}}_1,{\hat{\Phi }}_1)$, and $(c_0,\Phi )$ with ${\hat{c}}_2<{\hat{c}}_1<c_0$. We see Fig. 1d for the profile of this entire solution as $t\rightarrow -\infty$.

Theorem 3.20

Assume (J1)–(J4) hold and $c_0>{\hat{c}}_1>{\hat{c}}_2$. Then (1.1) admits an entire solution ${\mathcal {W}}_4(x,t):{\mathbb {R}}^2\rightarrow [0,u_2^*]\times [0,v_2^*]$ satisfying

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }&\left\{ \sup \limits _{x\le \rho _1(t)}\Vert {\mathcal {W}}_4(x,t)-{\hat{\Phi }}_2(x+{\hat{c}}_2t-\vartheta _{5})\Vert \right. \nonumber \\&\quad +\sup \limits _{\rho _1(t)\le x\le \rho _2(t)}\Vert {\mathcal {W}}_4(x,t)-{\hat{\Phi }}_1(x+{\hat{c}}_1t+\vartheta _{5})\Vert \nonumber \\&\left. \quad +\sup \limits _{\rho _2(t)\le x}\Vert {\mathcal {W}}_4(x,t)-\Phi (x+c_0t+\vartheta _{5})\Vert \right\} =0, \end{aligned}$$

where $\vartheta _5$ is a constant satisfying

$$\begin{aligned} \vartheta _{5}:=-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}-\frac{L}{\nu _1}\right] , \end{aligned}$$

and $\rho _1,\rho _2:{\mathbb {R}}\mapsto {\mathbb {R}}$ are defined by

$$\begin{aligned} \rho _1(t):=\frac{-({\hat{c}}_2+{\hat{c}}_1)t}{2},\rho _2(t):=\frac{-({\hat{c}}_1+c_0)t}{2}. \end{aligned}$$

Moreover, when (J5) holds, there are two positive constants $D_1$ and $D_2$ such that for any $(x,t)\in {\mathbb {R}}^2$ and $\eta >0$,

$$\begin{aligned} \Vert {\mathcal {W}}_4(x+\eta ,t)-{\mathcal {W}}_4(x,t)\Vert \le D_1\eta ,\left\| \frac{\partial {\mathcal {W}}_4}{\partial t}(x+\eta ,t)-\frac{\partial {\mathcal {W}}_4}{\partial t}\right\| \le D_2\eta . \end{aligned}$$

Now, we give the construction of auxiliary functions and super- and sub-solutions for ${\mathcal {W}}_4$, which is the main difficulty, and the other details in the proof of Theorem 3.20 are omitted. Throughout this subsection, we choose the values of $s_1$, $s_2$, and $s_3$ as follows:

$$\begin{aligned} s_1:={\hat{c}}_2\le {\hat{c}}_2^*<0,~s_2:={\hat{c}}_1\ge {\hat{c}}_1^*>0,~s_3:=c_0. \end{aligned}$$

Denote

$$\begin{aligned} \varvec{\phi }_1(\xi ):={\hat{\Phi }}_2(\xi ),~\varvec{\phi }_2(\xi ):={\hat{\Phi }}_1(\xi ),~\varvec{\phi }_3(\xi ):=\Phi (\xi ) \end{aligned}$$

and assume that

$$\begin{aligned} \phi _{11}(0)= & {} \frac{u_1^*}{2},~(\phi _{21}(0),\phi _{22}(0))\ge \left( \frac{u_1^*+u_2^*}{2},\frac{v_1^*+v_2^*}{2}\right) ,~\text {and}~(\phi _{31}(0),\phi _{32}(0))\\\ge & {} \left( \frac{u_1^*+u_2^*}{2},\frac{v_1^*+v_2^*}{2}\right) . \end{aligned}$$

The two auxiliary functions are defined as follows:

$$\begin{aligned} {P}(x,y,z)=y+(u_2^*-y)\frac{(u_2^*-x)(u_1^*-y)z+(x-u_1^*)y(u_2^*-z)}{(u_2^*-x)(u_1^*-y)u_2^*+(u_2^*-u_1^*)y(u_2^*-z)},(x,y,z)\in {D}_1 \end{aligned}$$

and

$$\begin{aligned} {Q}(x,y,z)=y+(v_2^*-y)\frac{(v_2^*-x)(v_1^*-y)z+(x-v_1^*)y(v_2^*-z)}{(v_2^*-x)(v_1^*-y)v_2^*+(v_2^*-v_1^*)y(v_2^*-z)},(x,y,z)\in {D}_2, \end{aligned}$$

where

$$\begin{aligned} {D}_1:=\left\{ (x,y,z)\in [u_1^*,u_2^*]\times [0,u_1^*]\times [0,u_2^*]\big |x(u_1^*-y)u_2^*+u_1^*y(u_2^*-z)>0\right\} \end{aligned}$$

and

$$\begin{aligned} {D}_2:=\left\{ (x,y,z)\in [v_1^*,v_2^*]\times [0,v_1^*]\times [0,v_2^*]\big |x(v_1^*-y)v_2^*+v_1^*y(v_2^*-z)>0\right\} . \end{aligned}$$

The super- and sub-solutions are given by the following lemma.

Lemma 3.21

Assume (J1)–(J4) hold and $c_0>{\hat{c}}_1>{\hat{c}}_2$. Let $({\tilde{p}}_1(t),{\tilde{p}}_2(t),{\tilde{r}}_1(t),$ ${\tilde{r}}_2(t))$ be the solution of (3.3). Then, the functions defined by

$$\begin{aligned}\left\{ \begin{aligned}&\overline{u}(x,t)= {P}(\phi _{11}(x+{\bar{c}}t-{\tilde{p}}_1(t)),\phi _{21}(x+{\bar{c}}t+{\tilde{r}}_1(t)),\phi _{31}(x+{\bar{c}}t+{\tilde{p}}_2(t))),\\&\overline{v}(x,t)= {Q}(\phi _{12}(x+{\bar{c}}t-{\tilde{p}}_1(t)),\phi _{22}(x+{\bar{c}}t+{\tilde{r}}_1(t)),\phi _{32}(x+{\bar{c}}t+{\tilde{p}}_2(t))) \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned}\left\{ \begin{aligned}&{\underline{u}}(x,t)= {P}(\phi _{11}(x+{\bar{c}}t-{\tilde{r}}_1(t)),\phi _{21}(x+{\bar{c}}t+{\tilde{p}}_1(t)),\phi _{31}(x+{\bar{c}}t+{\tilde{r}}_2(t))),\\&{\underline{v}}(x,t)= {Q}(\phi _{12}(x+{\bar{c}}t-{\tilde{r}}_1(t)),\phi _{22}(x+{\bar{c}}t+{\tilde{p}}_1(t)),\phi _{32}(x+{\bar{c}}t+{\tilde{r}}_2(t))) \end{aligned}\right. \end{aligned}$$

are a pair of super- and sub-solutions of (1.1) for $(x,t)\in {\mathbb {R}}\times (-\infty ,t_3]$ with some $t_3<0$. Moreover, (3.21) and (3.22) hold for $(\overline{u}(x,t),\overline{v}(x,t))$ and $({\underline{u}}(x,t),{\underline{v}}(x,t))$.

3.6 V-Type Entire Solution

The fifth type of entire solution originates from two monostable and one bistable traveling wave solutions, namely $(c_1,\Phi _1)$, $(c_2,\Phi _2)$, and $({\hat{c}}_0,{\hat{\Phi }})$ with $c_1<c_2<{\hat{c}}_0$. We see Fig. 1e for the profile of this entire solution as $t\rightarrow -\infty$.

Theorem 3.22

Assume (J1)–(J4) hold and ${\hat{c}}_0>c_2>c_1$. Then, (1.1) admits an entire solution ${\mathcal {W}}_5(x,t):{\mathbb {R}}^2\rightarrow [0,u_2^*]\times [0,v_2^*]$ satisfying

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }&\left\{ \sup \limits _{x\le \rho _1(t)}\Vert {\mathcal {W}}_5(x,t)-\Phi _1(x+c_1t-\vartheta _{6})\Vert \right. \nonumber \\&\quad +\sup \limits _{\rho _1(t)\le x\le \rho _2(t)}\Vert {\mathcal {W}}_5(x,t)-\Phi _2(x+c_2t+\vartheta _{5})\Vert \nonumber \\&\left. \quad +\sup \limits _{\rho _2(t)\le x}\Vert {\mathcal {W}}_5(x,t)-{\hat{\Phi }}(x+{\hat{c}}_0t+\vartheta _{6})\Vert \right\} =0, \end{aligned}$$

where $\vartheta _5$ and $\vartheta _5$ are some constants satisfying

$$\begin{aligned} \vartheta _{5}:=-\frac{1}{\kappa }\ln \left[ e^{-\kappa r_0}-\frac{L}{\nu _1}\right] ,\quad \vartheta _{6}:=-\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}+\frac{L}{\nu _1}\right] , \end{aligned}$$

and $\rho _1,\rho _2:{\mathbb {R}}\mapsto {\mathbb {R}}$ are defined by

$$\begin{aligned} \rho _1(t):=\frac{-(c_1+c_2)t}{2},\rho _2(t):=\frac{-(c_2+{\hat{c}}_0)t}{2}. \end{aligned}$$

Moreover, when (J5) holds, there are two positive constants $D_1$ and $D_2$ such that for any $(x,t)\in {\mathbb {R}}^2$ and $\eta >0$,

$$\begin{aligned} \Vert {\mathcal {W}}_5(x+\eta ,t)-{\mathcal {W}}_5(x,t)\Vert \le D_1\eta ,\left\| \frac{\partial {\mathcal {W}}_5}{\partial t}(x+\eta ,t)-\frac{\partial {\mathcal {W}}_5}{\partial t}\right\| \le D_2\eta . \end{aligned}$$

Similar to other types of entire solutions, we focus only on the construction of auxiliary functions and super- and sub-solutions for ${\mathcal {W}}_5$ and omit the other details in the proof of Theorem 3.22. Throughout this subsection, we choose the values of $s_1$, $s_2$, and $s_3$ as follows:

$$\begin{aligned} s_1:=c_1,~s_2:=c_2,~s_3:={\hat{c}}_0. \end{aligned}$$

Denote

$$\begin{aligned} \varvec{\phi }_1(\xi ):=\Phi _1(\xi ),~\varvec{\phi }_2(\xi ):=\Phi _2(\xi ),~\varvec{\phi }_3(\xi ):={\hat{\Phi }}(\xi ) \end{aligned}$$

and assume that

$$\begin{aligned} \phi _{11}(0)= & {} \frac{u_1^*}{2},~\phi _{21}(0)=\frac{u_1^*+u_2^*}{2},~\text {and}~(\phi _{31}(0),\phi _{32}(0))\ge (u_1^*,v_1^*). \end{aligned}$$

The two auxiliary functions are defined as follows:

$$\begin{aligned} {P}(x,y,z)=y+(-y)\frac{x(y-u_1^*)(u_2^*-z)+(u_1^*-x)(u_2^*-y)z}{x(y-u_1^*)u_2^*+u_1^*(u_2^*-y)z},(x,y,z)\in {D}_1 \end{aligned}$$

and

$$\begin{aligned} {Q}(x,y,z)=y+(-y)\frac{x(y-v_1^*)(v_2^*-z)+(v_1^*-x)(v_2^*-y)z}{x(y-v_1^*)v_2^*+v_1^*(v_2^*-y)z},(x,y,z)\in {D}_2, \end{aligned}$$

where

$$\begin{aligned} {D}_1:=\left\{ (x,y,z)\in [0,u_1^*]\times [u_1^*,u_2^*]\times [0,u_2^*]\big |x(y-u_1^*)u_2^*+u_1^*(u_2^*-y)z>0\right\} \end{aligned}$$

and

$$\begin{aligned} {D}_2:=\left\{ (x,y,z)\in [0,v_1^*]\times [v_1^*,v_2^*]\times [0,v_2^*]\big |x(y-v_1^*)v_2^*+v_1^*(v_2^*-y)z>0\right\} . \end{aligned}$$

The super- and sub-solutions are given by the following lemma.

Lemma 3.23

Assume (J1)–(J4) hold and ${\hat{c}}_0>c_2>c_1$. Let $({\tilde{p}}_1(t),{\tilde{p}}_2(t),{\tilde{r}}_1(t),$ ${\tilde{r}}_2(t))$ be the solution of (3.3), and ${{\bar{c}}}:=(c_1+c_2)/2$. Then, the functions defined by

$$\begin{aligned}\left\{ \begin{aligned}&\overline{u}(x,t)= {P}(\phi _{11}(x+{\bar{c}}t-{\tilde{r}}_1(t)),\phi _{21}(x+{\bar{c}}t+{\tilde{p}}_1(t)),\phi _{31}(x+{\bar{c}}t+{\tilde{r}}_2(t))),\\&\overline{v}(x,t)= {Q}(\phi _{12}(x+{\bar{c}}t-{\tilde{r}}_1(t)),\phi _{22}(x+{\bar{c}}t+{\tilde{p}}_1(t)),\phi _{32}(x+{\bar{c}}t+{\tilde{r}}_2(t))) \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned}\left\{ \begin{aligned}&{\underline{u}}(x,t)= {P}(\phi _{11}(x+{\bar{c}}t-{\tilde{p}}_1(t)),\phi _{21}(x+{\bar{c}}t+{\tilde{r}}_1(t)),\phi _{31}(x+{\bar{c}}t+{\tilde{p}}_2(t))),\\&{\underline{v}}(x,t)= {Q}(\phi _{12}(x+{\bar{c}}t-{\tilde{p}}_1(t)),\phi _{22}(x+{\bar{c}}t+{\tilde{r}}_1(t)),\phi _{32}(x+{\bar{c}}t+{\tilde{p}}_2(t))) \end{aligned}\right. \end{aligned}$$

are a pair of super- and sub-solutions of (1.1) for $(x,t)\in {\mathbb {R}}\times (-\infty ,t_4]$ with some $t_4<0$. Moreover, (3.21) and (3.22) hold for $(\overline{u}(x,t),\overline{v}(x,t))$ and $({\underline{u}}(x,t),{\underline{v}}(x,t))$.

3.7 VI-Type Entire Solution

The sixth type of entire solution originates from two monostable and one bistable traveling wave solutions, namely $({\hat{c}}_2,{\hat{\Phi }}_2)$, $(c_2,\Phi _2)$, and $({\hat{c}}_0,{\hat{\Phi }})$ with ${\hat{c}}_2<c_2<{\hat{c}}_0$. We see Fig. 1f for the profile of this entire solution as $t\rightarrow -\infty$.

Theorem 3.24

Assume (J1)–(J4) hold and ${\hat{c}}_0>c_2>{\hat{c}}_2$. Then, (1.1) admits an entire solution ${\mathcal {W}}_6(x,t):{\mathbb {R}}^2\rightarrow [0,u_2^*]\times [0,v_2^*]$ satisfying

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }&\left\{ \sup \limits _{x\le \rho _1(t)}\Vert {\mathcal {W}}_{6}(x,t)-{\hat{\Phi }}_2(x+{\hat{c}}_2t-\vartheta _{3})\Vert \right. \nonumber \\&\quad +\sup \limits _{\rho _1(t)\le x\le \rho _2(t)}\Vert {\mathcal {W}}_{6}(x,t)-\Phi _2(x+c_2t+\vartheta _{3})\Vert \nonumber \\&\left. \quad +\sup \limits _{\rho _2(t)\le x}\Vert {\mathcal {W}}_{6}(x,t)-{\hat{\Phi }}(x+{\hat{c}}_0t+\vartheta _{4})\Vert \right\} =0, \end{aligned}$$

where $\vartheta _3$ and $\vartheta _4$ are some constants satisfying

$$\begin{aligned} \vartheta _3:=-\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}-\frac{L}{\nu _1}\right] ,\quad \vartheta _4:=\frac{1}{\kappa }\ln \left[ e^{-\kappa p_0}-\frac{L}{\nu _1}\right] +p_0+r_0, \end{aligned}$$

and $\rho _1,\rho _2:{\mathbb {R}}\mapsto {\mathbb {R}}$ are defined by

$$\begin{aligned} \rho _1(t):=\frac{-({\hat{c}}_2+c_2)t}{2},\rho _2(t):=\frac{-(c_2+{\hat{c}}_0)t}{2}. \end{aligned}$$

Moreover, when (J5) holds, there are two positive constants $D_1$ and $D_2$ such that for any $(x,t)\in {\mathbb {R}}^2$ and $\eta >0$,

$$\begin{aligned} \Vert {\mathcal {W}}_6(x+\eta ,t)-{\mathcal {W}}_6(x,t)\Vert \le D_1\eta ,\left\| \frac{\partial {\mathcal {W}}_6}{\partial t}(x+\eta ,t)-\frac{\partial {\mathcal {W}}_6}{\partial t}\right\| \le D_2\eta . \end{aligned}$$

Now, the construction of auxiliary functions and super- and sub-solutions for ${\mathcal {W}}_6$ is provided, and the other details in the proof of Theorem 3.24, which are similar to other types of entire solutions, are omitted. Throughout this subsection, we choose the values of $s_1$, $s_2$, and $s_3$ as follows:

$$\begin{aligned} s_1:={\hat{c}}_2,~s_2:=c_2,~s_3:={\hat{c}}_0. \end{aligned}$$

Denote

$$\begin{aligned} \varvec{\phi }_1(\xi ):={\hat{\Phi }}_2(\xi ),~\varvec{\phi }_2(\xi ):=\Phi _2(\xi ),~\varvec{\phi }_3(\xi ):={\hat{\Phi }}(\xi ) \end{aligned}$$

and assume that

$$\begin{aligned} (\phi _{11}(0),\phi _{12}(0))< & {} \left( \frac{u_1^*+u_2^*}{2},\frac{v_1^*+v_2^*}{2}\right) ,~\phi _{21}(0)=\frac{u_1^*+u_2^*}{2},~(\phi _{31}(0),\phi _{32}(0))\\> & {} \left( \frac{u_1^*+u_2^*}{2},\frac{v_1^*+v_2^*}{2}\right) . \end{aligned}$$

The two auxiliary functions are defined as follows:

$$\begin{aligned}&{P}(x,y,z)=y\\&+\frac{(u_2^*-x)(y-u_1^*)(u_2^*-z)(-y)+(x-u_1^*)(u_2^*-y)z(u_2^*-y)}{(u_2^*-x)(y-u_1^*)u_2^*+(u_2^*-u_1^*)(u_2^*-y)z},~~(x,y,z)\in {D}_1 \end{aligned}$$

and

$$\begin{aligned}&{Q}(x,y,z)=y\\&+\frac{(v_2^*-x)(y-v_1^*)(v_2^*-z)(-y)+(x-v_1^*)(v_2^*-y)z(v_2^*-y)}{(v_2^*-x)(y-v_1^*)v_2^*+(v_2^*-v_1^*)(v_2^*-y)z},~~(x,y,z)\in {D}_2, \end{aligned}$$

where

$$\begin{aligned} {D}_1:= & {} \left\{ (x,y,z)\in [u_1^*,u_2^*]\times [u_1^*,u_2^*]\times [0,u_2^*]\big |(u_2^*-x)(y-u_1^*)u_2^*\right. \\&\quad \left. +(u_2^*-u_1^*)(u_2^*-y)z>0\right\} \end{aligned}$$

and

$$\begin{aligned} {D}_2:= & {} \left\{ (x,y,z)\in [v_1^*,v_2^*]\times [v_1^*,v_2^*]\times [0,v_2^*]\big |(v_2^*-x)(y-v_1^*)v_2^*\right. \\&\quad \left. +(v_2^*-v_1^*)(v_2^*-y)z>0\right\} . \end{aligned}$$

The super- and sub-solutions are given by the following lemma.

Lemma 3.25

Assume (J1)–(J4) hold and ${\hat{c}}_0>c_2>{\hat{c}}_2$. Let $(p_1(t),p_2(t),r_1(t),$ $r_2(t))$ be the solution of (3.2), and ${{\bar{c}}}:=({\hat{c}}_2+c_2)/2$. Then, the functions defined by

$$\begin{aligned}\left\{ \begin{aligned}&\overline{u}(x,t)= {P}\left( \phi _{11}(x+{\bar{c}}t-r_1(t)),\phi _{21}(x+{\bar{c}}t+r_1(t)),\phi _{31}(x+{\bar{c}}t+r_2(t))\right) ,\\&\overline{v}(x,t)= {Q}\left( \phi _{12}(x+{\bar{c}}t-r_1(t)),\phi _{22}(x+{\bar{c}}t+r_1(t)),\phi _{32}(x+{\bar{c}}t+r_2(t))\right) \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned}\left\{ \begin{aligned}&{\underline{u}}(x,t)= {P}\left( \phi _{11}(x+{\bar{c}}t-p_1(t)),\phi _{21}(x+{\bar{c}}t+p_1(t)),\phi _{31}(x+{\bar{c}}t+p_2(t))\right) ,\\&{\underline{v}}(x,t)= {Q}\left( \phi _{12}(x+{\bar{c}}t-p_1(t)),\phi _{22}(x+{\bar{c}}t+p_1(t)),\phi _{32}(x+{\bar{c}}t+p_2(t))\right) \end{aligned}\right. \end{aligned}$$

are a pair of super- and sub-solutions of (1.1) for $(x,t)\in {\mathbb {R}}\times (-\infty ,t_5]$ with some $t_5<0$. Moreover, (3.21) and (3.22) hold for $(\overline{u}(x,t),\overline{v}(x,t))$ and $({\underline{u}}(x,t),{\underline{v}}(x,t))$.

3.8 Nonexistence of N-Wave Entire Solutions for $N\ge 5$

In this subsection, we show that when $c_0\ne {{\hat{c}}}_0$, there is no entire solution originating from more than four traveling wave solutions. When we say $c_0= {{\hat{c}}}_0$, it means that the nondecreasing bistable traveling wave solution $\Phi (x+c_0t)$ and the nonincreasing bistable traveling wave solution ${\hat{\Phi }}(x+{\hat{c}}_0t)$ have the same propagating direction and the same speed. For the special case where the nondecreasing bistable traveling wave solution and the nonincreasing bistable traveling wave solution are symmetric, we write not $c_0= {{\hat{c}}}_0$, but $c_0=-{{\hat{c}}}_0$. Obviously, the assumption $c_0\ne {{\hat{c}}}_0$ covers most cases of bistable spreading speeds. However, when $c_0= {{\hat{c}}}_0$, we cannot prove whether or not there exists N-wave entire solution with $N\ge 5$.

We first introduce some definitions. Let $(c_j, \varvec{\phi }_j)$ with $j=1,2,\ldots , N$ be N traveling wave solutions of (1.1). We say that $S_N=\{(c_1, \varvec{\phi }_1), (c_2, \varvec{\phi }_2),\ldots ,(c_N, \varvec{\phi }_N)\}$ is a generative sequence, if there is an entire solution (u, v) of (1.1) originating from $S_N$, in the sense that, (u, v) and $S_N$ satisfy that

$$\begin{aligned} c_1\leqslant c_2\leqslant \ldots \leqslant c_N, \end{aligned}$$

(3.27)

and there exist constants $\vartheta _j$ with $j=1,2,\ldots ,N$ such that (1.4) holds. For example, the sequences corresponding to ${\mathcal {W}}_1$–${\mathcal {W}}_6$ in Sects. 3.2–3.7 are generative, which are given by

$$\begin{aligned} \begin{aligned}&\{(c_2,\Phi _2),({\hat{c}}_0,{\hat{\Phi }}),(c_0,\Phi )\},&\{({\hat{c}}_1,{\hat{\Phi }}_1),(c_0,\Phi ),({\hat{c}}_0,{\hat{\Phi }})\},&\{(c_1,\Phi _1),({\hat{c}}_1,{\hat{\Phi }}_1),(c_0,\Phi )\},\\&\{({\hat{c}}_2,{\hat{\Phi }}_2),({\hat{c}}_1,{\hat{\Phi }}_1),(c_0,\Phi )\},&\{(c_1,\Phi _1),(c_2,\Phi _2),({\hat{c}}_0,{\hat{\Phi }})\},&\{({\hat{c}}_2,{\hat{\Phi }}_2),(c_2,\Phi _2),({\hat{c}}_0,{\hat{\Phi }})\}. \end{aligned} \end{aligned}$$

For any $j=1,2,\ldots , N-1$, when $x=\rho _{j+1}(t)$, it follows from (1.4) that

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }(u,v)(\rho _{j+1}(t),t)= & {} \lim \limits _{t\rightarrow -\infty }\varvec{\phi }_j(\rho _{j+1}(t)+c_j t+\vartheta _j)\\= & {} \lim \limits _{t\rightarrow -\infty }\varvec{\phi }_{j+1}(\rho _{j+1}(t)+c_{j+1} t+\vartheta _{j+1}), \end{aligned}$$

namely,

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }\varvec{\phi }_j\left( \frac{(c_j-c_{j+1})t}{2} +\vartheta _j\right) =\lim \limits _{t\rightarrow -\infty }\varvec{\phi }_{j+1}\left( \frac{(c_{j+1}-c_{j})t}{2} +\vartheta _{j+1}\right) . \end{aligned}$$

By (3.27), we have that

$$\begin{aligned} \varvec{\phi }_j(+\infty )=\varvec{\phi }_{j+1}(-\infty )~\text {for any}~j=1,2,\ldots ,N-1. \end{aligned}$$

(3.28)

Therefore, (3.27) and (3.28) are necessary for that $S_N$ is a generative sequence. By (3.28), when $S_N$ is a generative sequence, we can denote a sequence

$$\begin{aligned} P_{N}=\{p_0,p_1,\ldots ,p_N\}, \end{aligned}$$

where

$$\begin{aligned} p_0\triangleq \varvec{\phi }_1(-\infty ),~ p_j\triangleq \varvec{\phi }_j(+\infty )=\varvec{\phi }_{j+1}(-\infty ),~j\in \{1,\ldots ,N-1\},~\text {and}~p_N\triangleq \varvec{\phi }_N(+\infty ). \end{aligned}$$

We say that $S_N=\{(c_1, \varvec{\phi }_1), (c_2, \varvec{\phi }_2),\ldots ,(c_N, \varvec{\phi }_N)\}$ is a non-generative sequence, if there is no entire solution originating from $S_N$. Obviously, $S_N$ is non-generative if $\varvec{\phi }_j(+\infty )\ne \varvec{\phi }_{j+1}(-\infty )$ for some $j\in \{1,2,\ldots ,N-1\}$ or there exist $i,j\in \{1,2,\ldots ,N\}$ such that $c_i> c_j$ and $i<j$.

Lemma 3.26

In the following two cases, the sequence $S_N=\{(c_1, \varvec{\phi }_1), (c_2, \varvec{\phi }_2),\ldots ,(c_N, \varvec{\phi }_N)\}$ is non-generative.

(i):

When $c_0\ne {{\hat{c}}}_0$, there exist $i,j\in \{1,2,\ldots ,N\}$ with $i<j$ such that

$$\begin{aligned} (c_i,\varvec{\phi }_i)=(c_j,\varvec{\phi }_j)=(c_0,\Phi )\quad \text {or}\quad (c_i,\varvec{\phi }_i)=(c_j,\varvec{\phi }_j)=({\hat{c}}_0,{\hat{\Phi }}). \end{aligned}$$

(ii):

There exist $i,j\in \{1,2,\ldots ,N\}$ with $i<j$ such that

$$\begin{aligned} \varvec{\phi }_i(-\infty )=\varvec{\phi }_j(+\infty )=E_1. \end{aligned}$$

Proof

(i) We only consider the case where there exist $i,j\in \{1,2,\ldots ,N\}$ with $i<j$ such that $(c_i,\varvec{\phi }_i)=(c_j,\varvec{\phi }_j)=(c_0,\Phi )$, and the proof for the case $(c_i,\varvec{\phi }_i)=(c_j,\varvec{\phi }_j)=({\hat{c}}_0,{\hat{\Phi }})$ is similar. Suppose, by contradiction, that $S_N$ is generative. Obviously, it holds that $i<j-1$; otherwise, $\varvec{\phi }_i(+\infty )=E_2\ne \varvec{\phi }_j(-\infty )=E_0$ with $i=j-1$. Note that $p_i=\varvec{\phi }_i(+\infty )=E_2$ and $p_{j-1}=\varvec{\phi }_j(-\infty )=E_0$. Consider the sequence $\{p_i,\ldots ,p_{j-1}\}$, and choose $i_0$ with $i\leqslant i_0< j$ satisfying that $p_{i_0}$ is the last $E_2$ in $\{p_i,\ldots ,p_{j-1}\}$. Next, consider the sequence $\{p_{i_0},\ldots ,p_{j-1}\}$ and choose $j_0$ with $i_0< j_0\leqslant j$ satisfying that $p_{j_0-1}$ is the first $E_0$ in $\{p_{i_0},\ldots ,p_{j-1}\}$. Let $S=\{p_{i_0},\ldots ,p_{j_0-1}\}$. Then, we have either

$$\begin{aligned} S=\{p_{i_0},p_{j_0-1}\}=\{E_2,E_0\}~\text {with}~i_0=j_0-2. \end{aligned}$$

(3.29)

or

$$\begin{aligned} S=\{p_{i_0},p_{i_0+1}=p_{j_0-2},p_{j_0-1}\}=\{E_2, E_1,E_0\}~\text {with}~ i_0=j_0-3. \end{aligned}$$

(3.30)

When (3.29) holds, we have that $\varvec{\phi }_{i_0+1}={{\hat{\Phi }}}$ and $c_{i_0+1}={{\hat{c}}}_0$. Then, (3.27) and $c_i=c_j=c_0$ imply that $c_{i_0+1}=c_0={{\hat{c}}}_0$. It is a contradiction with $c_0\ne {{\hat{c}}}_0$. When (3.30) holds, we have that $\varvec{\phi }_{i_0+1}={{\hat{\Phi }}}_2$ and $\varvec{\phi }_{i_0+2}={{\hat{\Phi }}}_1$ with $c_{i_0+1}={{\hat{c}}}_2$ and $c_{i_0+2}={{\hat{c}}}_1$. From (3.27) and $c_i=c_j=c_0$, it follows that ${{\hat{c}}}_2={{\hat{c}}}_1=c_0$, which is a contradiction with ${{\hat{c}}}_2\le c_R<c_L\le {{\hat{c}}}_1$ in (2.3).

(ii) By Theorems 2.1 and 2.2 , $\varvec{\phi }_i$ must be ${\hat{\Phi }}_1$ or ${\Phi }_2$ (which implies that $c_i\ge c_L$), and $\varvec{\phi }_j$ must be ${\Phi }_1$ or ${\hat{\Phi }}_2$ (which implies that $c_j\le c_R$). From (2.3), it follows that $c_i\ge c_L>c_R\ge c_j$. Hence, $S_N$ is non-generative. $\square$

Theorem 3.27

When $c_0\ne {{\hat{c}}}_0$, any sequence $S_N=\{(c_1, \varvec{\phi }_1), (c_2, \varvec{\phi }_2),\ldots ,(c_N, \varvec{\phi }_N)\}$ with $N\ge 5$ is non-generative.

Proof

Let $S_N$ be a generative sequence. Obviously, there are at most two bistable traveling wave solutions in $S_N$ (otherwise, Lemma 3.26 (i) happens). Now, we prove that there are at most two monostable traveling wave solutions in $S_N$. We claim that $E_1$ appears at most one time in $P_N$. Indeed suppose, by contradiction, that $E_1$ appears at least two times in $P_N$. Then, any two $E_1$ in $P_N$ must be nonadjacent since there is no traveling wave solution $(c, \varvec{\phi }_c)$ satisfying that $\varvec{\phi }_c(\pm \infty )=E_1$. Let $i,j\in \{0,1,\ldots ,N\}$ satisfy $p_i=p_j=E_1$ and $i+1<j$. Then, we have that

$$\begin{aligned} \varvec{\phi }_{i+1}(-\infty )=p_i=E_1,\quad \varvec{\phi }_j(+\infty )= p_j=E_1. \end{aligned}$$

From Lemma 3.26 (ii), we get that $S_N$ is non-generative, which is a contradiction. Therefore, $E_1$ appears at most one time in $P_N$, which implies that there are at most two monostable traveling wave solutions in $S_N$. Finally, we get that if $S_N$ be a generative sequence, then $N\le 4$, namely, there exist at most two monostable and two bistable traveling wave solutions in $S_N$. $\square$

Discussions. These twelve types of three-wave entire solutions in Figs. 1 and 2 contain all possibilities of entire solutions originating from three traveling wave solutions $(c_j, \varvec{\phi }_j)_{j=1,2, 3}$. By (3.28) in Sect. 3.8, the profiles must satisfy $\varvec{\phi }_j(+\infty )=\varvec{\phi }_{j+1}(-\infty )$ for $j=1,2$. The twelve types in Figs. 1 and 2 show all possibilities of $(c_j, \varvec{\phi }_j)_{j=1,2, 3}$ satisfying this condition. Moreover, the wave speeds $\{c_j\}_{j=1,2,3}$ also satisfy $c_1\le c_2\le c_3$, and otherwise, the two adjacent traveling wave solutions with $c_j>c_{j+1}$ for some $j\in \{1,2\}$ must intersect at some negative time. When the speeds of bistable and monostable traveling wave solutions are nonzero, the existence results in Sect. 3 show all possibilities of three-wave entire solutions satisfying $c_1<c_2<c_3$. But for the case $c_1=c_2$ or $c_2=c_3$, it remains open whether or not the entire solution of (1.1) originating from $(c_j, \varvec{\phi }_j)_{j=1,2, 3}$ exists.

Note that four-wave entire solutions have not been considered in this paper, and it is interesting to show whether or not there exist four-wave entire solutions of (1.1). The main difficulty in the proof of the existence is how to construct some appropriate auxiliary functions. We will consider this problem in future research.

Appendix A. Proofs of Some Lemmas in Section 3.2

1.1 A.1 Proof of Lemma 3.8

We denote

$$\begin{aligned}&\epsilon _1:=\min \Big \{\frac{\phi _{11}(0)}{u_1^*},\frac{\phi _{12}(0)}{v_1^*}\Big \},\quad \epsilon _2:=\max \Big \{\frac{\phi _{11}(0)}{u_2^*},\frac{\phi _{12}(0)}{v_2^*}\Big \},\\&\epsilon _3:=\min \Big \{\frac{\phi _{21}(0)}{u_1^*},\frac{\phi _{22}(0)}{v_1^*}\Big \},\quad \epsilon _4:=\max \Big \{\frac{\phi _{21}(0)}{u_2^*},\frac{\phi _{22}(0)}{v_2^*}\Big \},\\&\epsilon _5:=\min \Big \{\frac{\phi _{31}(0)}{u_1^*},\frac{\phi _{32}(0)}{v_1^*}\Big \},\quad \epsilon _6:=\max \Big \{\frac{\phi _{31}(0)}{u_2^*},\frac{\phi _{32}(0)}{v_2^*}\Big \}. \end{aligned}$$

It follows that $\epsilon _1$, $\epsilon _3>1$, $\epsilon _2$, $\epsilon _4$, $\epsilon _5$, $\epsilon _6\in (0,1)$ and

$$\begin{aligned} E_1\ll \epsilon _1 E_1\le \varvec{\phi }_1(0)\le \epsilon _2 E_2\ll E_2,\\ E_0\ll \epsilon _3 E_1\le \varvec{\phi }_2(0)\le \epsilon _4 E_2\ll E_2,\\ E_0\ll \epsilon _5 E_1\le \varvec{\phi }_3(0)\le \epsilon _6 E_2\ll E_2. \end{aligned}$$

In what follows, we give the upper and lower bounds of $\varvec{\phi }_1(\xi -q_1)$, $\varvec{\phi }_2(\xi +q_2)$, and $\varvec{\phi }_3(\xi +q_2)$ in four cases. The calculations are not complicated and we omit them. First, when $\xi \le q_1(t)$, it holds that

$$\begin{aligned} E_1\le \varvec{\phi }_1(\xi -q_1)\le \epsilon _2 E_2, ~\epsilon _3 E_1\le \varvec{\phi }_2(\xi +q_2)\le E_2, ~E_0 \le \varvec{\phi }_3(\xi +q_2)\le \epsilon _6 E_2. \end{aligned}$$

Second, when $q_1(t)\le \xi \le -q_1(t)$, we have

$$\begin{aligned} \epsilon _1 E_1\le \varvec{\phi }_1(\xi -q_1)\le E_2, ~\epsilon _3 E_1\le \varvec{\phi }_2(\xi +q_2)\le E_2, ~E_0 \le \varvec{\phi }_3(\xi +q_2)\le \epsilon _6 E_2. \end{aligned}$$

Third, when $-q_1(t)\le \xi \le -q_2(t)$, it follows that

$$\begin{aligned} \epsilon _1 E_1\le \varvec{\phi }_1(\xi -q_1)\le E_2,~ E_0\le \varvec{\phi }_2(\xi +q_2)\le \epsilon _4 E_2,~ E_0 \le \varvec{\phi }_3(\xi +q_2)\le \epsilon _6 E_2. \end{aligned}$$

Fourth, when $\xi \ge -q_2(t)$, we can get

$$\begin{aligned} \epsilon _1 E_1\le \varvec{\phi }_1(\xi -q_1)\le E_2, ~E_0\le \varvec{\phi }_2(\xi +q_2)\le \epsilon _4 E_2, ~\epsilon _5 E_1\le \varvec{\phi }_3(\xi +q_2)\le E_2. \end{aligned}$$

Some calculations imply that

$$\begin{aligned} \begin{aligned} 0&<u_1^*u_2^*\min \{\epsilon _3(1-\epsilon _6)(u_2^*-u_1^*),(\epsilon _1-1)(1-\epsilon _4)u_2^*\}\\&\le u_2^*[\phi _{11}(\xi -q_1)-u_1^*][u_2^*-\phi _{21}(\xi +q_1)]\\&\quad +(u_2^*-u_1^*)\phi _{21}(\xi +q_1)[u_2^*-\phi _{31}(\xi +q_2)]\\&\le 2{u_2^*}^2(u_2^*-u_1^*) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} 0&<v_1^*v_2^*\min \{\epsilon _3(1-\epsilon _6)(v_2^*-v_1^*),(\epsilon _1-1)(1-\epsilon _4)v_2^*\}\\&\le v_2^*\left[ \phi _{12}(\xi -q_1)-v_1^*\right] [v_2^*-\phi _{22}(\xi +q_1)]\\&\quad +(v_2^*-v_1^*)\phi _{22}(\xi +q_1)[v_2^*-\phi _{32}(\xi +q_2)]\\&\le 2{v_2^*}^2(v_2^*-v_1^*) \end{aligned} \end{aligned}$$

for any $\xi \in {\mathbb {R}}$ and $q_2$, $q_1$ satisfying (3.17). Then, the second and third inequalities in Lemma 3.8 can be easily obtained. For the first inequality, we get from (3.12) and $\varvec{\phi }_2'<0$ that

$$\begin{aligned} (\phi _{21}(\xi +q_1),\phi _{22}(\xi +q_1))\ge (\phi _{21}(0),\phi _{22}(0))\ge \left( \frac{u_1^*+u_2^*}{2},\frac{v_1^*+v_2^*}{2}\right) ~~\text {when}~\xi \le q_1(t). \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned}&u_2^*[\phi _{31}(\xi +q_2)-u_1^*][u_2^*-\phi _{21}(\xi +q_1)]+\phi _{21}(\xi +q_1)[u_2^*-\phi _{31}(\xi +q_2)](u_2^*-u_1^*)\\&\quad \ge u_2^*[\phi _{31}(\xi +q_2)-u_1^*]\left( u_2^*-\frac{u_1^*+u_2^*}{2}\right) +\frac{u_1^*+u_2^*}{2}[u_2^*-\phi _{31}(\xi +q_2)](u_2^*-u_1^*)\\&\quad \ge \frac{u_2^*}{2}(u_2^*-u_1^*)^2, \end{aligned} \end{aligned}$$

and then

$$\begin{aligned} \begin{aligned} P^{x}(\xi ,t)&= \phi _{21}(u_2^*-\phi _{31})\frac{(u_2^*-u_1^*)[u_2^*(\phi _{31}-u_1^*)(u_2^*-\phi _{21})+\phi _{21}(u_2^*-\phi _{31})(u_2^*-u_1^*)]}{[u_2^*(\phi _{11}-u_1^*)(u_2^*-\phi _{21})+(u_2^*-u_1^*)\phi _{21}(u_2^*-\phi _{31})]^2}\\&\ge \frac{\epsilon _3(1-\epsilon _6)u_1^*u_2^*(u_2^*-u_1^*)\frac{u_2^*}{2}(u_2^*-u_1^*)^2}{[2{u_2^*}^2(u_2^*-u_1^*)]^2}=\frac{\epsilon _3(1-\epsilon _6)u_1^*(u_2^*-u_1^*)}{8{u_2^*}^2}. \end{aligned} \end{aligned}$$

It completes the proof.

1.2 A.2 Proof of Lemma 3.9

When $\xi \le -q_1(t)$, we have that

$$\begin{aligned} \phi _{31}(\xi +q_2)\le \phi _{31}(0)\le u_1^*\le \phi _{11}(\xi -q_1),\quad ~\text {and}~P^{y}(\xi ,t)\ge 0. \end{aligned}$$

By $P^{z},\phi _{11}',\phi _{31}'\ge 0$ and $\phi _{21}'\le 0$, we have that when $\xi \le q_1(t)$,

$$\begin{aligned} A_1(\xi ,t)=P^{x}(\xi ,t)\phi _{11}'-P^{y}(\xi ,t)\phi _{21}'+P^{z}(\xi ,t)\phi _{31}'\ge P^x(\xi ,t)\phi _{11}'\ge \frac{1}{2}P^x(\xi ,t)|\phi _{11}'|, \end{aligned}$$

and when $\xi \le -q_1(t)$,

$$\begin{aligned} A_1(\xi ,t)\ge \frac{1}{2}\left[ P^{x}(\xi ,t)|\phi _{11}'|+P^{y}(\xi ,t)|\phi _{21}'|\right] . \end{aligned}$$

When $-q_1(t)\le \xi \le -q_2(t)$, we have that $P^{y}$, $P^{z}\ge 0$. Then, we compute that

$$\begin{aligned}&A_1(\xi ,t)-\frac{1}{2}\left[ P^{y}(\xi ,t)|\phi _{21}'(\xi +q_1)|+P^{z}(\xi ,t)|\phi _{31}'(\xi +q_2)|\right] \\&\quad =P^x(\xi ,t)\phi _{11}'+\frac{1}{2}P^y(\xi ,t)|\phi _{21}'|+\frac{1}{2}P^z(\xi ,t)\phi _{31}'\ge P^x(\xi ,t)\phi _{11}'+\frac{1}{2}P^y(\xi ,t)|\phi _{21}'|\\&\quad \ge \frac{\phi _{21}(u_2^*-\phi _{31})(u_2^*-u_1^*)[u_2^*(\phi _{31}-u_1^*)(u_2^*-\phi _{21})+\phi _{21}(u_2^*-\phi _{31})(u_2^*-u_1^*)]}{[u_2^*(\phi _{11}-u_1^*)(u_2^*-\phi _{21})+(u_2^*-u_1^*)\phi _{21}(u_2^*-\phi _{31})]^2}\phi _{11}'\\&\qquad +\frac{\rho \phi _{21}(\phi _{11}-u_1^*)(u_2^*-\phi _{31}){u_2^*}^2(u_2^*-u_1^*)(\phi _{11}-\phi _{31})}{2[u_2^*(\phi _{11}-u_1^*)(u_2^*-\phi _{21})+(u_2^*-u_1^*)\phi _{21}(u_2^*-\phi _{31})]^2}\\&\quad \ge \frac{\phi _{21}(u_2^*-u_1^*)u_1^*{u_2^*}^3}{[2u_2^*(u_2^*-u_1^*)]^2}\left[ -C_0e^{-2\eta _2\delta }+\frac{\rho }{2}(\varepsilon _1-1)(1-\varepsilon _6)(\phi _{11}(0)-\phi _{31}(0))\right] \ge 0 \end{aligned}$$

for $\delta$ sufficiently large. When $\xi \ge -q_2(t)$, we have that $P^z\ge 0$. Similarly, we obtain that

$$\begin{aligned}\begin{aligned}&A_1(\xi ,t)-\frac{1}{2}P^z(\xi ,t)|\phi _{31}'(\xi +q_2)|=P^x(\xi ,t)\phi _{11}'+P^y(\xi ,t)|\phi _{21}'|\\&\quad +\frac{1}{2}P^z(\xi ,t)\phi _{31}' \ge \frac{u_2^*-\phi _{31}}{[2u_2^*(u_2^*-u_1^*)]^2}\left\{ -(u_2^*-u_1^*)u_1^*{u_2^*}^3C_0e^{-2\eta _2\delta }\right. \\&\left. \quad -(u_2^*-u_1^*)^3{u_2^*}^2C_0e^{-\eta _2\delta }+\frac{\rho }{2}[(\varepsilon _1-1)u_1^*(1-\varepsilon _4){u_2^*}^2]^2\right\} \ge 0 \end{aligned}\end{aligned}$$

for $\delta$ sufficiently large. It completes the proof.

1.3 A.3 Proof of Lemma 3.10

For given $(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-$ and $s\in {\mathbb {R}}$, when there is no confusion, we simply write

$$\begin{aligned} {\hat{\phi }}_{11}(\theta ):= & {} \phi _{11}(\xi -q_1(t)-\theta s),~{\hat{\phi }}_{21}(\theta ) :=\phi _{21}(\xi +q_1(t)-\theta s),~\text {and}~{\hat{\phi }}_{31}(\theta )\\:= & {} \phi _{31}(\xi +q_2(t)-\theta s), \end{aligned}$$

where $\theta \in [0,1]$. Then, $H_1(\xi ,t)$ can be represented as follows:

$$\begin{aligned} \begin{aligned} H_1(\xi ,t)=&\int _{{\mathbb {R}}}J_1(s)[P({\hat{\phi }}_{11}(1),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))-P({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))]{\text {d}}s\\&\quad -P_{x}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))\int _{{\mathbb {R}}}J_1(s)[{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]{\text {d}}s\\&\quad -P_{y}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))\int _{{\mathbb {R}}}J_1(s)[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]{\text {d}}s\\&\quad -P_{z}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))\int _{{\mathbb {R}}}J_1(s)[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]{\text {d}}s. \end{aligned}\end{aligned}$$

We denote

$$\begin{aligned} \begin{aligned}&{\mathcal {I}}_1=P_{x}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0)) [{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)],\\&{\mathcal {I}}_2=P_{y}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)],\\&{\mathcal {I}}_3=P_{z}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]. \end{aligned} \end{aligned}$$

By mean value theorem, there exist $\theta _1$, $\theta _2$, and $\theta _3$ in (0, 1) such that

$$\begin{aligned} P({\hat{\phi }}_{11}(1),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))-P({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(0)) ={\mathcal {I}}_4+{\mathcal {I}}_5+{\mathcal {I}}_6, \end{aligned}$$

where

$$\begin{aligned}\begin{aligned}&{\mathcal {I}}_4=P_{x}(\theta _1{\hat{\phi }}_{11}(1)+(1-\theta _1){\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)],\\&{\mathcal {I}}_5=P_{y}({\hat{\phi }}_{11}(0),\theta _2{\hat{\phi }}_{21}(1)+(1-\theta _2){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)],\\&{\mathcal {I}}_6=P_{z}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _3{\hat{\phi }}_{31}(1)+(1-\theta _3){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]. \end{aligned}\end{aligned}$$

Then, we can get that

$$\begin{aligned} H_1(\xi ,t)=\int _{{\mathbb {R}}}J_1(s)[{\mathcal {I}}_4+{\mathcal {I}}_5+{\mathcal {I}}_6-{\mathcal {I}}_1-{\mathcal {I}}_2-{\mathcal {I}}_3]{\text {d}}s. \end{aligned}$$

Also by mean value theorem, there exist $\theta _4$, $\theta _5$, and $\theta _6$ in (0, 1) such that

$$\begin{aligned}\begin{aligned}&{\mathcal {I}}_4-{\mathcal {I}}_1=P_{xx}(\theta _4{\hat{\phi }}_{11}(1)+(1-\theta _4){\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))\theta _1[{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]^2\\&\quad +P_{xy}({\hat{\phi }}_{11}(0),\theta _5{\hat{\phi }}_{21}(1)+(1-\theta _5){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)][{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]\\&\quad +P_{xz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _6{\hat{\phi }}_{31}(1)+(1-\theta _6){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)][{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]\\&\quad =\theta _1G_1(\xi ,t,s;\theta _4)+G_2(\xi ,t,s;\theta _5)+G_3(\xi ,t,s;\theta _6), \end{aligned}\end{aligned}$$

where

$$\begin{aligned}\begin{aligned} G_1(\xi ,t,s;\theta _4)&= P_{xx}(\theta _4{\hat{\phi }}_{11}(1)+(1-\theta _4){\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(1),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]^2,\\ G_2(\xi ,t,s;\theta _5)&= P_{xy}({\hat{\phi }}_{11}(0),\theta _5{\hat{\phi }}_{21}(1)+(1-\theta _5){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))\\&\quad [{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)][{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)],\\ G_3(\xi ,t,s;\theta _6)&= P_{xz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _6{\hat{\phi }}_{31}(1)+(1-\theta _6){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]\\&\quad [{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)]. \end{aligned}\end{aligned}$$

There exist $\theta _7$ and $\theta _8$ in (0, 1) such that

$$\begin{aligned}\begin{aligned} {\mathcal {I}}_5-{\mathcal {I}}_2&=P_{yy}({\hat{\phi }}_{11}(0),\theta _7{\hat{\phi }}_{21}(1)+(1-\theta _7){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))\theta _2[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]^2\\&\quad -P_{yz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _8{\hat{\phi }}_{31}(1)+(1-\theta _8){\hat{\phi }}_{31}(0))\\&\quad [{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)][{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]\\&=\theta _2G_4(\xi ,t,s;\theta _7)+G_5(\xi ,t,s;\theta _8), \end{aligned}\end{aligned}$$

where

$$\begin{aligned}\begin{aligned} G_4(\xi ,t,s;\theta _7)&= P_{yy}({\hat{\phi }}_{11}(0),\theta _7{\hat{\phi }}_{21}(1)+(1-\theta _7){\hat{\phi }}_{21}(0),{\hat{\phi }}_{31}(1))[{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]^2,\\ G_5(\xi ,t,s;\theta _8)&= P_{yz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _6{\hat{\phi }}_{31}(1)+(1-\theta _6){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]\\&\quad [{\hat{\phi }}_{21}(1)-{\hat{\phi }}_{21}(0)]. \end{aligned}\end{aligned}$$

There is a constant $\theta _9\in (0,1)$ such that

$$\begin{aligned} \begin{aligned} {\mathcal {I}}_6-{\mathcal {I}}_3&=P_{zz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _9{\hat{\phi }}_{31}(1)+(1-\theta _9){\hat{\phi }}_{31}(0))\theta _3[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]^2\\&=\theta _3G_6(\xi ,t,s;\theta _9), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} G_6(\xi ,t,s;\theta _9):=P_{zz}({\hat{\phi }}_{11}(0),{\hat{\phi }}_{21}(0),\theta _9{\hat{\phi }}_{31}(1)+(1-\theta _9){\hat{\phi }}_{31}(0))[{\hat{\phi }}_{31}(1)-{\hat{\phi }}_{31}(0)]^2. \end{aligned}$$

Based on the above formulas, we can get that

$$\begin{aligned} \begin{aligned} H_1(\xi ,t)&=\int _{{\mathbb {R}}}J_1(s)[\theta _1G_1(\xi ,t,s;\theta _4)+G_2(\xi ,t,s;\theta _5)+G_3(\xi ,t,s;\theta _6)\\&\quad +\theta _2G_4(\xi ,t,s;\theta _7)+G_5(\xi ,t,s;\theta _8)+\theta _3G_6(\xi ,t,s;\theta _9)]{\text {d}}s. \end{aligned} \end{aligned}$$

(3.31)

In what follows, we provide the detailed estimations only for $|G_1(\xi ,t,s;\theta _4)/ A_1(\xi ,t)|$, and for $|G_i(\xi ,t,s;\theta _{i+3})/ A_1(\xi ,t)|$ with $i=2,\ldots ,6$, the methods are similar. We get from the mean value theorem that

$$\begin{aligned} |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|= & {} |{\phi }_{11}(\xi -q_1-s)-{\phi }_{11}(\xi -q_1)| =|{\phi }_{11}'(\xi -q_1-\theta _{10}s)s|\\= & {} |{\hat{\phi }}_{11}'(\theta _{10})s|, \end{aligned}$$

where $\theta _{10}$ is some constant in [0, 1]. In addition, for the $C_3$ given by Lemma 3.6, there is a sufficiently large constant $C_4$ such that

$$\begin{aligned} C_3|{\phi }_{11}(\xi -q_1-s)-{\phi }_{11}(\xi -q_1)|<C_4\quad \text {for any}~(\xi ,t)\in {\mathbb {R}}\times {\mathbb {R}}^-,s\in {\mathbb {R}}. \end{aligned}$$

Now, we consider the following six cases. Recall that the constant m, which appears below, is defined in assumption (J4).

Case 1: When $\xi \le q_1(t)$, based on Lemmas 3.6, 3.7, 3.8, and 3.9 (i), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_1$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }} _{11}(1)-{\hat{\phi }}_{11}(0)|}{P^{x}(\xi ,t)|\phi _{11}'(\xi -q_1)|}\\\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _1|\phi _{11}'(\xi -q_1)|} \le \frac{2C_4m}{\mu _1}|\phi _{21}'(\xi +q_1-s)|\\&\cdot \frac{|u_2^*-\phi _{21}(\xi +q_1-s)|}{|\phi _{21}'(\xi +q_1-s)|}\cdot \frac{|\phi _{11}'(\xi -q_1-\theta _{10}s)|}{|\phi _{11}(\xi -q_1-\theta _{10}s)-u_1^*|}\\&\cdot \frac{|\phi _{11}(\xi -q_1-\theta _{10}s)-u_1^*|}{|\phi _{11}(\xi -q_1)-u_1^*|}\cdot \frac{|\phi _{11}(\xi -q_1)-u_1^*|}{|\phi _{11}'(\xi -q_1)|}\\\le & {} \frac{2C_4m}{\mu _1} C_0e^{\eta _1(\xi +q_1-s)}\frac{1}{C_1} C_2C\frac{1}{C_1}\le {\tilde{M}}_1e^{\eta _1q_1}. \end{aligned}$$

Case 2: When $q_1(t)\le \xi \le 0$, based on Lemmas 3.6, 3.7, 3.8, and 3.9 (ii), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_2$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^x(\xi ,t)|\phi _{11}'(\xi -q_1)|+P^y(\xi ,t)|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _1|\phi _{11}'(\xi -q_1)|}\le \frac{2C_4m}{\mu _1}|\phi _{21}'(\xi +q_1-s)|\\&\cdot \frac{|u_2^*-\phi _{21}(\xi +q_1-s)|}{|\phi _{21}'(\xi +q_1-s)|}\cdot \frac{|\phi _{11}'(\xi -q_1-\theta _{10}s)|}{|u_2^*-\phi _{11}(\xi -q_1-\theta _{10}s)|}\\&\cdot \frac{|u_2^*-\phi _{11}(\xi -q_1-\theta _{10}s)|}{|u_2^*-\phi _{11} (\xi -q_1)|}\cdot \frac{|u_2^*-\phi _{11}(\xi -q_1)|}{|\phi _{11}'(\xi -q_1)|}\\\le & {} \frac{2C_4m}{\mu _1}C_0e^{\eta _1(\xi +q_1-s)}\frac{1}{C_1} C_2C\frac{1}{C_1}\le {\tilde{M}}_2e^{\eta _1q_1}. \end{aligned}$$

Case 3: When $0\le \xi \le -q_1(t)$, based on Lemmas 3.6, 3.7,  3.8, and 3.9 (ii), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_3$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^x(\xi ,t)|\phi _{11}'(\xi -q_1)|+P^y(\xi ,t)|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _2|\phi _{21}'(\xi +q_1)|}\le \frac{2C_4m}{\mu _2}|\phi _{11}'(\xi -q_1-\theta _{10}s)|\\&\cdot \frac{|u_2^*-\phi _{21}(\xi +q_1-s)|}{|u_2^*-\phi _{21}(\xi +q_1)|}\cdot \frac{|u_2^*-\phi _{21}(\xi +q_1)|}{|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4m}{\mu _2}C_0e^{-\eta _2(\xi -q_1-\theta _{10}s)}C\frac{1}{C_1}\le {\tilde{M}}_3e^{\eta _2q_1}. \end{aligned}$$

Case 4: When $-q_1(t)v\le \xi \le (-q_1(t)-q_2(t))/2$, based on Lemma 3.6, 3.7,  3.8, and 3.9(iii), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_4$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^y(\xi ,t)|\phi _{21}'(\xi +q_1)|+P^z(\xi ,t)|\phi _{31}'(\xi +q_2)|}\\\le & {} \frac{2C_4|{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _2|\phi _{21}'(\xi +q_1)|} \le \frac{2C_4m}{\mu _2}|\phi _{11}'(\xi -q_1-\theta _{10}s)|\\&\cdot \frac{|\phi _{21}(\xi +q_1-s)|}{|\phi _{21}(\xi +q_1)|}\cdot \frac{|\phi _{21}(\xi +q_1)|}{|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4m}{\mu _2}C_0e^{-\eta _2(\xi -q_1-\theta _{10}s)}C\frac{1}{C_1}\le {\tilde{M}}_4e^{\eta _2q_1}. \end{aligned}$$

Case 5: When $(-q_1(t)-q_2(t))/2\le \xi \le -q_2(t)$, based on Lemmas 3.6, 3.7,  3.8, and 3.9 (iii), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_5$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^y(\xi ,t)|\phi _{21}'(\xi +q_1)|+P^z(\xi ,t)|\phi _{31}'(\xi +q_2)|}\\\le & {} \frac{2C_4|{\hat{\phi }}_{21}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _2|\phi _{21}'(\xi +q_1)|}\le \frac{2C_4m}{\mu _2}|\phi _{11}'(\xi -q_1-\theta _{10}s)|\\&\cdot \frac{|\phi _{21}(\xi +q_1-s)|}{|\phi _{21}(\xi +q_1)|}\cdot \frac{|\phi _{21}(\xi +q_1)|}{|\phi _{21}'(\xi +q_1)|}\\\le & {} \frac{2C_4m}{\mu _2}C_0e^{-\eta _2(\xi -q_1-\theta _{10}s)}C\frac{1}{C_1}\le {\tilde{M}}_5e^{\eta _2q_1}. \end{aligned}$$

Case 6: When $\xi \ge -q_2(t)$, based on Lemmas 3.6, 3.7,  3.8, and 3.9 (iv), we have that for $s\in \text {supp}(J_1)$, there exists a positive constant ${\tilde{M}}_6$ such that

$$\begin{aligned} \bigg |\frac{G_1(\xi ,t,s;\theta _4)}{A_1(\xi ,t)}\bigg |\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{31}(1)|\cdot |{\hat{\phi }}_{11}(1)-{\hat{\phi }}_{11}(0)|}{P^z(\xi ,t)|\phi _{31}'(\xi +q_2)|}\\\le & {} \frac{2C_4|u_2^*-{\hat{\phi }}_{31}(1)|\cdot |{\hat{\phi }}_{11}'(\theta _{10})s|}{\mu _3|\phi _{31}'(\xi +q_2)|}\le \frac{2C_4m}{\mu _3}|\phi _{11}'(\xi -q_1-\theta _{10}s)|\\&\cdot \frac{|u_2^*-\phi _{31}(\xi +q_2-s)|}{|u_2^*-\phi _{31}(\xi +q_2)|}\cdot \frac{|u_2^*-\phi _{31}(\xi +q_2)|}{|\phi _{31}'(\xi +q_2)|}\\\le & {} \frac{2C_4m}{\mu _3}C_0e^{-\eta _2(\xi -q_1-\theta _{10}s)}C\frac{1}{C_1}\le {\tilde{M}}_6e^{\eta _2q_1}. \end{aligned}$$

With the above estimates, we can get (3.19) from (3.31) and $\int _{{\mathbb {R}}}J_1(x) dx=1$, immediately. It completes the proof.