1 Introduction

In this paper, we consider the following Lotka–Volterra competitive system with local vs. nonlocal diffusions

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}=D_1\triangle u +r_1u(1-u-b_{1}v),\\ v_{t}=D_2(J*v-v)+r_2 v(1-v-b_{2}u), \end{array}\right. } \end{aligned}$$
(1.1)

where \(D_i\), \(r_i\) and \(b_i\) are positive constants, \(i=1,2\), u, v are the population densities at location \(x\in {\mathbb {R}}\) and time \(t>0\), \(D_1\) and \(D_2\) are the diffusion coefficients of species, \(r_1\) and \(r_2\) are the growth rates of species, the parameters \(b_1\) and \(b_2\) mean the competition coefficients of species. \(J*v-v\) is nonlocal dispersal operator with

$$\begin{aligned} {{(J*v)(x,t)=\int _{{\mathbb {R}}}J(x-y)v(y,t)dy,}} \end{aligned}$$

which can model nonlocal dispersal processes of species, see [6, 9, 15]. The kernel \(J:{\mathbb {R}}\rightarrow {\mathbb {R}}^+\) satisfies

(J1):

\(J\in C({\mathbb {R}})\), \(J(x)=J(-x)\), \(x\in {\mathbb {R}}\), and \(\int _{{\mathbb {R}}}J(x)dx=1\).

(J2):

For every \(\lambda >0\), \(\int _{{\mathbb {R}}}J(x)e^{-\lambda x}dx<\infty \).

Mathematically, for simplification, letting \(D_1=1\), \(D_2=d\), \(r_{1}=1\), \(r_2=\alpha \), then (1.1) is reduced to the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}=\triangle u +u(1-u-b_{1}v),\\ v_{t}=d(J*v-v)+\alpha v(1-v-b_{2}u). \end{array}\right. } \end{aligned}$$
(1.2)

We make the following assumption on the coefficients of (1.2):

(H1):

\(0<b_{1}<1<b_{2}\),

which implies that the original species v is the weak competitor to the species u. In other words, the species v shall lose the competition during the evolution. System (1.2) has three non-negative equilibria

$$\begin{aligned} E^0:=(0,0),\ E^1:=(1,0),\ E^2:=(0,1). \end{aligned}$$

By (H1), we can obtain from its corresponding space-homogeneous ordinary differential system, i.e.,

$$\begin{aligned} {\left\{ \begin{array}{ll} u'=u(1-u-b_{1}v),\\ v'=\alpha v(1-v-b_{2}u), \end{array}\right. } \end{aligned}$$

that \(E^1\) is stable and \(E^2\) is unstable, see, e.g., [13, 20].

Wang et al. [28] have studied system (1.2) and established the existence and stability of traveling wave solutions of (1.2) connecting \(E^2\) and \(E^1\). To the best of our knowledge, the investigation of the evolution systems with local vs. nonlocal diffusions can be traced back to Kao et al. [17], where they proposed the following competition system

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\mu \Delta u+uf(u+v,x),\\ v_t=d(J*v-v)+vf(u+v,x), \end{array}\right. } \end{aligned}$$
(1.3)

and studied the dynamics of (1.3). In recent years, it has gotten researcher’s attention. In [31, Remark 1.2], Xu et al. considered a cooperative system with local vs. nonlocal diffusions, and showed the acceleration propagation of every species. Hao and Zhang [11] investigated the existence and stability of traveling wavefronts for the model (1.2) with nonlocal delay. More recently, the free boundary problems with local vs. nonlocal diffusions have been extensively studied, see, e.g., [19, 29, 30] and references therein. We should point out that it is a hot topic to study competitive systems with various types of diffusion. We refer readers to [2,3,4, 16, 23] for random (local) diffusion and [13, 20, 32, 33] for nonlocal dispersal.

By a change of variables \(u={\widetilde{u}}\), \(v=1-{\widetilde{v}}\), system (1.2) turns into a cooperative system as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} \widetilde{u_{t}}=\triangle {\widetilde{u}}+{\widetilde{u}}(1-{\widetilde{u}}-b_{1}+b_{1}{\widetilde{v}}),\\ \widetilde{v_{t}}=d(J*{\widetilde{v}}-{\widetilde{v}})+\alpha (1-{\widetilde{v}})(b_{2}{\widetilde{u}}-{\widetilde{v}}). \end{array}\right. } \end{aligned}$$
(1.4)

The equilibria \(E^2\), \(E^0\) and \(E^1\) become, respectively,

$$\begin{aligned}&E_0=(0,0), E_1=(0,1), E_2=(1,1). \end{aligned}$$

A traveling wave solution of (1.4) connecting the equilibria \(E_2\) and \(E_0\) takes the form

$$\begin{aligned} ({\widetilde{u}},{\widetilde{v}})(x,t) = (U,V)(z), \quad z=x-ct, \end{aligned}$$

where \(c> 0\) is the wave speed and (U,V) is called the wave profile. Substituting (U, V)(z) into (1.4) leads to the following wave profile problem

$$\begin{aligned} {\left\{ \begin{array}{ll} U''+cU'+U(1-U-b_{1}+b_{1}V)=0,\\ d(J*V-V)+cV'+\alpha (1-V)(b_{2}U-V)=0,\\ (U,V)(-\infty )= E_{2},\quad (U,V)(+\infty )= E_{0}. \end{array}\right. } \end{aligned}$$
(1.5)

This is equivalent to studying traveling wave solutions for the original competition system (1.2) that connect two boundary equilibria (0, 1) and (1, 0). Motivated by [28, Theorem 2.1], where the monotone semiflow method was used, we can also obtain that there exists \(c_{\min }\ge 0\) (called the minimal wave speed for this system) so that a monotone non-increasing traveling wave solution to (1.5) exists if and only if \(c \ge c_{\min }\). The upper-lower solutions method can also be applied to get the same conclusion (see, e.g., [7, 10]). Note that

$$\begin{aligned} c_{\min } \ge c_{0} = 2\sqrt{1-b_{1}}, \end{aligned}$$

where \(c_{0}\) is obtained by linearizing the system (1.5) at the null fixed point (for a detailed analysis, the reader is referred to Sect. 2). It is well-known that the minimal wave speed is a very important number. For example, in ecology, it may be the spreading speed of species invasion onto an unstable state, see, e.g., [18, 22]. We should point out that the formula of the minimal wave speed is usually not easy to determine unless it is equal to \(c_0\). When \(c_{\min }= c_{0}\), we say the minimal wave speed is linearly selected. On the other hand, when \(c_{\min }> c_{0}\), we say the minimal wave speed is nonlinearly selected, see the details in [16].

The speed selection mechanism has been studied widely in literature, see [2,3,4, 14, 16, 23, 26] for random diffusion systems and [24, 25, 27, 34] for lattice differential systems. For the nonlocal dispersal monostable systems, we only see the work of [12], where the authors studied the linear selection of the minimal speed of traveling wavefronts for a three-component competition system. The purpose of this work is to study the linear/nonlinear selection of the minimal wave speed for system (1.5). Note that system (1.5) has both local and nonlocal diffusions. Motivated by the above researches, the upper-lower solution method will be applied. We should point out that, for the linear selection, we concentrate on finding upper solutions only, and do not care about the existence of lower solutions, and for nonlinear selection, we only focus on the construction of a lower solution with fast decay rate.

The rest of the paper is organized as follows. In Sect. 2, we study the behavior of solutions of (1.5) near the equilibrium point \(E_{0}\) which plays a main role in constructing the required upper or lower solutions. In Sect. 3, by constructing suitable upper or lower solutions to system (1.5), we derive a general criterion for the linear and nonlinear speed selection. In Sect. 4, we further find explicit conditions for the speed selection.

2 Preliminaries

To investigate the speed selection, we need to analyze the local asymptotic behaviour for the positive wave profile of (1.5) near the equilibrium point \(E_{0}\). By linearizing the system around \(E_{0}\), we obtain the following constant coefficient system

$$\begin{aligned} {\left\{ \begin{array}{ll} U''+cU'+U(1-b_{1})=0,\\ d(J*V-V)+cV'+\alpha (b_{2}U-V)=0. \end{array}\right. } \end{aligned}$$
(2.1)

Letting \((U,V)(z)=(\xi _{1},\xi _{2})e^{-\mu z}\) for some positive constants \(\xi _{1}\), \(\xi _{2}\) and \(\mu \), and substituting it into (2.1), then we have

$$\begin{aligned} { \left( \begin{array}{ccc} T_1(\mu )&{} 0\\ \alpha b_{2}&{} T_2(\mu )\\ \end{array} \right) } { \left( \begin{array}{ccc} \xi _1\\ \xi _2\\ \end{array} \right) } = { \left( \begin{array}{ccc} 0\\ 0\\ \end{array} \right) } , \end{aligned}$$

where

$$\begin{aligned}&T_1(\mu ):=\mu ^{2}-c\mu +1-b_{1},\\&T_2(\mu ):=d\left( \int _{{\mathbb {R}}}J(y)e^{-\mu y}dy-1\right) -c\mu -\alpha . \end{aligned}$$

Note that the first equation of (2.1) is decoupled. If \(T_1(\mu )=0\), then \(\mu \) equals one of the following values

$$\begin{aligned} \mu _{1}(c)= \frac{c-\sqrt{c^{2}-4(1-b_{1})}}{2},\quad \mu _{2}(c)= \frac{c+\sqrt{c^{2}-4(1-b_{1})}}{2}. \end{aligned}$$
(2.2)

To make \(\mu _{1}(c)\) and \(\mu _{2}(c)\) real so that the solution U is positive, the speed c has to satisfy

$$\begin{aligned} c\ge c_{0}= 2\sqrt{1-b_{1}}, \end{aligned}$$
(2.3)

where \(c_{0}\) is called the linear speed of system (1.5). It is easy to see that \(\mu _1(c)\) is a decreasing function and \(\mu _2(c)\) is an increasing function with respect to c, satisfying \(\mu _1(c_0)=\mu _2(c_0)=\sqrt{1-b_{1}}\). It is easy to see that \(T_2(\mu )\) is a convex function of \(\mu \), and hence, \(T_2(\mu )=0\) has a unique positive root \(\mu _3(c)\).

By a similar argument as that in [1, Section 2], we see that for any \(c>c_{0}\), the behavior of solution U of (1.5) can be given by

$$\begin{aligned} U(z) \sim C_{1}e^{-\mu _{1}(c)z}+C_{2}e^{-\mu _{2}(c)z}, \end{aligned}$$

as \(z\rightarrow +\infty \), for constant \(C_{1}>0\), or \(C_{1}=0\) with \(C_{2}>0\), and the behavior of solution V is given by

$$\begin{aligned} V(z) \sim C_1 \frac{-\alpha b_{2}}{ T_2(\mu _{1}(c))}e^{-\mu _{1}(c)z} +C_2\frac{-\alpha b_{2}}{ T_2(\mu _{2}(c))}e^{-\mu _{2}(c)z} +C_3e^{-\mu _{3}(c)z}, \end{aligned}$$

as \(z\rightarrow +\infty \), for constants \(C_{1}>0\), \(C_3>0\), or \(C_{1}=0\) with \(C_{2}>0\), \(C_{3}>0\). The more accurate asymptotic behavior of (U, V) at positive infinity can be obtained by using the modified version of Ikehara’s theorem. We refer readers to [32] for a three species competition system with nonlocal dispersal.

3 The Speed Selection Mechanism

In this section, the speed selection mechanism of system (1.5) is studied by using the upper-lower solution method. We first give the definition of upper and lower solutions. For simplicity of notations, we denote

$$\begin{aligned} \Gamma _{u}[U,V]= & {} U''+cU'+U(1-U-b_{1}+b_{1}V),\end{aligned}$$
(3.1)
$$\begin{aligned} \Gamma _{v}[U,V]= & {} d(J*V-V)+cV'+\alpha (1-V)(b_{2}U-V). \end{aligned}$$
(3.2)

Definition 3.1

If \(({\overline{U}},{\overline{V}})(z)\) is continuous on \((-\infty ,\infty )\) and differentiable except at n finite number of points \(z_{q}\), \(q=1,2,...,n\), satisfying \({\overline{U}}'(z_{q}^{-})\ge {\overline{U}}'(z_{q}^{+})\) and \(\Gamma _{u}[{\overline{U}},{\overline{V}}](z)\le 0\), \(\Gamma _{v}[{\overline{U}},{\overline{V}}](z)\le 0\), for \(z\ne z_{q}\), then \(({\overline{U}},{\overline{V}})(z)\) is an upper solution to system (1.5).

The lower solution can be defined similarly by reversing the inequalities in the above definition of the upper solution. We first give the following two lemmas.

Lemma 3.2

For a given continuous and non-increasing function U(z) satisfying \(U(-\infty )=1\) and \(U(+\infty )=0\), there exists a non-increasing function V(z) satisfying the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} d(J*V-V)+cV'+\alpha (1-V)(b_{2}U-V)=0,\\ V(-\infty )= 1,\quad V(+\infty )=0. \end{array}\right. } \end{aligned}$$
(3.3)

Proof

We let \(W(z)=1-V(z)\). Then (3.3) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -c W'(z)=d(J*W(z)-W(z))+\alpha W(z)[r(z)-W(z)],\\ W(-\infty )=0,\quad W(+\infty )=1, \end{array}\right. } \end{aligned}$$
(3.4)

where \(r(z)=1-b_2U(z)\). It is clear that \(r(-\infty )=1-b_2U(-\infty )=1-b_2<0\) and \(r(+\infty )=1-b_2U(+\infty )=1\). It then follows from [21, Theorem 4.5] that for any given \(c>0\), (3.4) admits a non-decreasing positive solution W. Thus, (3.3) has a non-increasing positive solution V. The proof is complete. \(\square \)

Based on Lemma 3.2, we denote V as a function of U. The following monotonic property of V(U) with respect to U will play an important role in the study of nonlinear selection.

Lemma 3.3

The function V(U) is monotonically non-decreasing with respect to U in the sense that \(V (U_1)\ge V (U_2)\) if \(U_1\ge U_2\).

Proof

We rewrite (3.3) into the following equivalent form

$$\begin{aligned} -cV'=-kV+{\mathcal {H}}(U,V), \end{aligned}$$
(3.5)

where

$$\begin{aligned} {\mathcal {H}}(U,V)=kV+d(J*V-V)+\alpha (1-V)(b_{2}U-V), \end{aligned}$$
(3.6)

and k is chosen sufficiently large such that \({\mathcal {H}}(U,V)\) are monotone in V. By using the variation of parameters on system (3.5), we obtain

$$\begin{aligned} V(z)={\mathcal {G}}(U,V)(z), \end{aligned}$$

where

$$\begin{aligned} {\mathcal {G}}(U,V)(z):=-\frac{1}{c} e^{-\frac{k}{c}z}\int _{-\infty }^{z} e^{\frac{k}{c} s}{\mathcal {H}}(U,V)(s)ds. \end{aligned}$$

We define an iteration sequence as follows

$$\begin{aligned} {{V_{n+1}(z)={\mathcal {G}}(U,V_{n})(z),\quad n=0,1,\ldots ,}} \end{aligned}$$
(3.7)

where \(V_0\) is some function to be determined later. The upper-lower solution method is applied to show that this iteration sequence \(\{V_{n}(U)\}\) is convergent. It is easy to see that \({\underline{V}}=0\) is a lower solution of (3.3). As for the upper solution, we define a function as follows

$$\begin{aligned} {\overline{V}}(z)= \min \{1,1-\varphi (-z)\}, \end{aligned}$$

where \(\varphi (t)\) is the bistable traveling wavefronts of

$$\begin{aligned} d[(J*\varphi )(t)-\varphi (t)]+c_{\epsilon }\varphi '(t)+f(\varphi (t))=0, \end{aligned}$$
(3.8)

connecting \(1-\epsilon \) to \(-\epsilon \). Here,

$$\begin{aligned} f(\varphi )= {\left\{ \begin{array}{ll} \alpha \varphi (1-\epsilon -\varphi ), \quad &{}\varphi \ge 0,\\ \alpha \varphi (\epsilon +\varphi ), \quad &{}\varphi <0, \end{array}\right. } \end{aligned}$$

and \(0<\epsilon \ll 1\) is a small number. For the existence of solution \(\varphi \) of (3.8), we refer to [5, Theorem 3.1] and [8, Theorem 5.3]. The verification of \({\overline{V}}(z)\) to be an upper solution can be obtained directly. By choosing \(V_0={\underline{V}}=0\) in (3.7), we can obtain a sequence \(\{V_{n}(U)\}_{n=0}^\infty \) with \(V_{n}(U)\) being nonincreasing in \({\mathbb {R}}\) and nondecreasing in n, that is, \(0={\underline{V}}= V_0\le V_1\le V_2\le \cdots \le {\overline{V}}\le 1\). With the help of Helly’s lemma, we see that this sequence \(\{V_{n}(U)\}\) converges to a non-increasing function V(U) pointwise. The existence of the limits \(V(\pm \infty )\) follows from the monotonicity of V(z) as well as the inequalities \(0\le V(z)\le {\overline{V}}(z)\le 1\), \(z\in {\mathbb {R}}\). In the view of (3.5) and (3.6), we obtain

$$\begin{aligned} (1-V(+\infty ))V(+\infty )=0,\quad (1-V(-\infty ))(b_{2}-V(-\infty ))=0. \end{aligned}$$
(3.9)

Note that \({\overline{V}}(+\infty )=\epsilon \). It then follows from the first equation of (3.9) that \(V(+\infty )=0\). Since \(b_{2}>1\), we can obtain from the second equation of (3.9) that \(V(-\infty )=1\).

Now we are ready to show the monotonicity of V(U). Assume \(U_{1}(z)\ge U_{2}(z)\) for all \(z\in {\mathbb {R}}\). Then by (3.7), we can get two sequences \(\{V_{n}(U_{1})\}_{n=0}^{\infty }\) and \(\{V_{n}(U_{2})\}_{n=0}^{\infty }\). Note that \({\mathcal {H}}(U,V)\) is monotonically increasing in U. Then we obtain \(V_{n}(U_{1})\ge V_{n}(U_{2})\), \(\forall n\ge 1\). Taking the limit \(n\rightarrow \infty \) yields \(V(U_{1})=\lim \limits _{n\rightarrow \infty } V_{n}(U_{1}) \ge \lim \limits _{n\rightarrow \infty } V_{n}(U_{2})=V(U_{2})\). The proof is complete. \(\square \)

Next, we shall show that if (1.5) has a suitable upper solution, then the minimal wave speed \(c_{\min }\) is linearly selected.

Lemma 3.4

If there exists a monotone positive upper solution \(({\overline{U}}, {\overline{V}})(z)\), with \(c = c_0\), to system (1.5) so that \(0<{\overline{U}}(-\infty )\le 1\) and \({\overline{U}}(+\infty ) = 0\), then the minimal wave speed is linearly selected.

Proof

The existence of a monotone positive upper solution \(({\overline{U}}, {\overline{V}})(z)\) implies the existence of traveling wave solution (U, V)(z) with speed \(c=c_0\), see, e.g., [10, Lemma 2.5] and [7, Proposition 2]. Thus, we have \(c_{\min }\le c_0\). Together with the fact that \(c_{\min }\ge c_0\), we obtain that \(c_{\min }= c_0\), i.e., the minimal wave speed is linearly selected. The proof is complete. \(\square \)

We try to construct a suitable upper solution to the U-equation of (1.5). Define a continuous monotonic non-increasing function

$$\begin{aligned} {\overline{U}}(z)=\frac{1}{1+e^{\mu _{1}z}}, \quad \forall z\in {\mathbb {R}}, \end{aligned}$$
(3.10)

where \(\mu _1:=\mu _{1}(c_0)\) is defined in (2.2). It is easy to compute that

$$\begin{aligned} {\overline{U}}'&=-\mu _{1}{\overline{U}}(1-{\overline{U}}),\\ {\overline{U}}''&=\mu _{1}^{2}{\overline{U}}(1-{\overline{U}})(1-2{\overline{U}}). \end{aligned}$$

In view of Lemma 3.2, we let \({\overline{V}}(z)\) be the solution of V-equation of (1.5) with \(U(z)={\overline{U}}(z)\). Substituting \(({\overline{U}},{\overline{V}})(z)\) into (3.1), we obtain

$$\begin{aligned} \Gamma _{u}[{\overline{U}},{\overline{V}}] =&\, {\overline{U}}(1-{\overline{U}})\left\{ \mu _{1}^{2}(1-2{\overline{U}})-c\mu _{1}+1-\frac{ b_{1}(1-{\overline{V}})}{1-{\overline{U}}}\right\} \\ =&\,{\overline{U}}(1-{\overline{U}})\left\{ \mu _{1}^{2}-c\mu _{1}+1-b_1+{\overline{U}}\left( -2\mu _{1}^{2}+\frac{ b_{1}({\overline{V}}-{\overline{U}})}{{\overline{U}}(1-{\overline{U}})}\right) \right\} \\ =&\,{\overline{U}}^2(1-{\overline{U}})\left( -2\mu _{1}^{2}+\frac{ b_{1}({\overline{V}}-{\overline{U}})}{{\overline{U}}(1-{\overline{U}})}\right) . \end{aligned}$$

Note that \(\mu _{1}(c_{0})=\sqrt{1-b_{1}}\). We can see that \(({\overline{U}},{\overline{V}})(z)\) is an upper solution to system (1.5) if

$$\begin{aligned} -2(1-b_1)+G_1(z)<0, \end{aligned}$$
(3.11)

where \(G_{1}(z)=\frac{ b_{1}({\overline{V}}-{\overline{U}})}{{\overline{U}}(1-{\overline{U}})}\).

Now, we summarize the above discussion into the following theorem.

Theorem 3.5

The speed selection of the system (1.5) is linearly realized when (3.11) is satisfied for \(c=c_{0}\), with the choice of \({\overline{U}}(z)\) being given by (3.10).

We then turn to study the nonlinear selection through the upper and lower solutions method. The key observation is that, when a lower solution has an asymptotic behavior \(e^{-\mu _2 z}\) (i.e., the faster decay rate) as \(z\rightarrow +\infty \), the nonlinear selection will be realized. We give the following theorem as a justification.

Theorem 3.6

For \(c_{1}>c_{0}\), assume that \(({\underline{U}},{\underline{V}})(x-c_{1}t)\ge 0\) is a lower solution to the partial differential system

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}=u_{xx}+u(1-u-b_{1}{{+b_{1}v}}),\\ v_{t}=d{{(\int _{{\mathbb {R}}}J(y)v(x-y,t)dy-v)}}+\alpha (1-v)(b_{2}u-v). \end{array}\right. } \end{aligned}$$
(3.12)

In addition, also assume that \({\underline{U}}(z_{1})\), \((z_{1}=x-c_{1}t)\), is monotonic and satisfies

$$\begin{aligned} \limsup _{z_{1}\rightarrow -\infty }{\underline{U}}(z_{1})<1, \end{aligned}$$

and \({\underline{U}}(z_{1})\sim e^{-\mu _{2}(c_1)z_{1}}\) as \(z_{1}\rightarrow +\infty \), where \(\mu _{2}\) is defined in (2.2). Then the traveling wave solutions of (1.5) do not exist for \(c\in [c_{0},c_{1})\).

Proof

By contradiction, we assume that system (3.12) admits a monotone traveling wave solution \((U,V)(x-ct)\) subject to the initial data

$$\begin{aligned} u(x,0)=U(x),\quad v(x,0)=V(x), \end{aligned}$$

for some \(c\in (c_{0},c_{1})\). By the decay behaviors of the initial data, we can see that \({\underline{U}}(x)\le U(x)\), by shifting if necessary. It then follows from Lemma 3.3 that \(({\underline{U}},{\underline{V}})(x)\le (U,V)(x)\). Since \(({\underline{U}},{\underline{V}})(x-c_{1}t)\) is a lower solution to system (3.12) with the initial data \(({\underline{U}},{\underline{V}})(x)\), by the comparison principle in [28, Proposition 2.1], we have

$$\begin{aligned} {{{\underline{U}}(x-c_{1}t)\le U(x-ct),\quad {\underline{V}}(x-c_{1}t)\le V(x-ct),}} \end{aligned}$$
(3.13)

for all \((x,t)\in {\mathbb {R}}\times (0,\infty )\). We choose \(z_{1}=x-c_{1}t\) so that \({\underline{U}}(z_{1})> 0\). Thus, we infer from the first inequality of (3.13) that

$$\begin{aligned} 0<{\underline{U}}(z_{1})={\underline{U}}(x-c_{1}t)\le U(z_{1}+(c_{1}-c)t)\rightarrow 0,\ \text{ as }\ t\rightarrow +\infty , \end{aligned}$$

which leads to a contradiction. If there exists a traveling wave solution with speed \(c=c_{0}\), then there exists a traveling wave solution for \(c\in (c_0,c_1)\), since \(c_{0}\) becomes the minimal wave speed. Thus, the above analysis is still valid. The proof is complete. \(\square \)

Due to the above theorem, for the nonlinear selection, we only need to find a lower solution that has an asymptotic behavior \(e^{-\mu _{2}(c)z}\) as \(z\rightarrow +\infty \) for some \(c>c_0\). Define

$$\begin{aligned} {\underline{U}}(z)=\frac{k}{1+e^{\mu _{2}z}}, \quad \forall z\in {\mathbb {R}}, \end{aligned}$$
(3.14)

where \(\mu _{2}:=\mu _{2}(c)\) and \(0<k<1\). It is easy to see that

$$\begin{aligned} {\underline{U}}'=&-\mu _{2}{\underline{U}}\left( 1-\frac{{\underline{U}}}{k}\right) ,\\ {\underline{U}}''=&\mu _{2}^{2}{\underline{U}}\left( 1-\frac{{\underline{U}}}{k}\right) \left( 1-\frac{2{\underline{U}}}{k}\right) . \end{aligned}$$

Let \({\underline{V}}\) be the corresponding solution of V-equation of (1.5) with \(U(z)={\underline{U}}(z)\). Substituting \(({\underline{U}},{\underline{V}})(z)\) into the U-equation in (3.1), we have

$$\begin{aligned} \Gamma _{u}[{\underline{U}},{\underline{V}}]&\ge {\underline{U}}\left( 1-\frac{{\underline{U}}}{k}\right) \\&\qquad \left\{ \mu _{2}^{2}-c\mu _{2}+1-b_{1}+ \frac{{\underline{U}}}{k}\left( -2\mu _{2}^{2}+\frac{b_{1}{\underline{V}}-{\underline{U}}(1+\frac{b_{1}-1}{k})}{(1-\frac{{\underline{U}}}{k})\frac{{\underline{U}}}{k}}\right) \right\} \\&=\frac{{\underline{U}}^{2}}{k}\left( 1-\frac{{\underline{U}}}{k}\right) \left\{ -2\mu _{2}^{2}+\frac{b_{1}{\underline{V}}-{\underline{U}}(1+\frac{b_{1}-1}{k})}{(1-\frac{{\underline{U}}}{k})\frac{{\underline{U}}}{k}}\right\} . \end{aligned}$$

Thus, \(({\underline{U}},{\underline{V}})(z)\) is a lower solution to system (1.5) if

$$\begin{aligned} -2\mu _{2}^{2}+G_{2}(z)> 0, \end{aligned}$$
(3.15)

where

$$\begin{aligned} G_{2}(z)=\frac{b_{1}{\underline{V}}-{\underline{U}}(1+\frac{b_{1}-1}{k})}{(1-\frac{{\underline{U}}}{k})\frac{{\underline{U}}}{k}}. \end{aligned}$$

As such, we have the following theorem.

Theorem 3.7

The minimal wave speed of the system (1.5) is nonlinearly realized if (3.15) is satisfied for some \(c>c_{0}\) with the choice of \({\underline{U}}(z)\) being given by (3.14).

4 Explicit Conditions for the Speed Selection

In this section, we give some specific conditions for the linear/nonlinear speed selection. We start with the linear speed selection.

Theorem 4.1

Assume that (J1)–(J2) and (H1) hold. Then the minimal speed of system (1.5) is linearly selected if the following condition

$$\begin{aligned} \alpha < \frac{c_0\mu _{1}-d\int _{\mathbb {R}}J(y)(e^{-\mu _{1}y}-1)dy}{b_{2}-1} \end{aligned}$$
(4.1)

is satisfied, where \(\mu _{1}:=\mu _1(c_0)\).

Proof

Let \({\overline{U}}(z)\) be defined in (3.10). Define

$$\begin{aligned} {\overline{V}}(z)={\overline{U}}(z)=\frac{1}{1+e^{\mu _{1}z}},\quad \forall z\in {\mathbb {R}}. \end{aligned}$$

We shall prove that \(({\overline{U}}(z),{\overline{V}}(z))\) is an upper solution for system (1.5). Since \(b_1<1\) by (H1), we have

$$\begin{aligned} -2(1-b_1)+G_1(z)=-2(1-b_1)+\frac{ b_{1}({\overline{V}}-{\overline{U}})}{{\overline{U}}(1-{\overline{U}})}=-2(1-b_1)<0, \end{aligned}$$

which means (3.11) holds. Thus, \(\Gamma _{u}[{\overline{U}},{\overline{V}}]<0\). For the V-equation, we have

$$\begin{aligned} \Gamma _{v}[{\overline{U}},{\overline{V}}]\le&\, {\overline{U}}(1-{\overline{U}})d\int _{{\mathbb {R}}}J(y)\frac{{\overline{U}}(z-y)- {\overline{U}}(z)}{{\overline{U}}(1-{\overline{U}})}dy-c_0\mu _{1}{\overline{U}}(1-{\overline{U}})\\&+\alpha (b_2-1){\overline{U}}(1-{\overline{U}})\\ \le&\, {\overline{U}}(1-{\overline{U}})\left\{ d\int _{{\mathbb {R}}}J(y)\frac{(1+e^{\mu _1z})(1-e^{-\mu _1 y})}{1+e^{\mu _1(z-y)}}dy-c_0\mu _{1}+\alpha (b_2-1)\right\} \\ \le&\, {\overline{U}}(1-{\overline{U}})\Big \{d\int _{{\mathbb {R}}}J(y)(e^{\mu _{1}y}-1)dy+d\int _{{\mathbb {R}}}J(y)\frac{-e^{\mu _1 y}-e^{-\mu _1 y}+2}{1+e^{\mu _1(z-y)}}dy\\&-c_0\mu _{1}+\alpha (b_2-1)\Big \}. \end{aligned}$$

Note that \(-e^{\mu _1 y}-e^{-\mu _1 y}+2\le 0\) for any \(y\in {\mathbb {R}}\). Then we obtain

$$\begin{aligned} \Gamma _{v}[{\overline{U}},{\overline{V}}]\le {\overline{U}}(1-{\overline{U}})\left\{ d\int _{{\mathbb {R}}}J(y)(e^{-\mu _{1}y}-1)dy-c_0\mu _{1}+\alpha (b_{2}-1)\right\} \le 0, \end{aligned}$$

provided that (4.1) holds. Therefore, \(({\overline{U}},{\overline{V}})(z)\) is an upper solution to system (1.5). The proof is complete. \(\square \)

Remark 4.2

Note that \(\alpha >0\) and \(b_2-1>0\). Thus, in order to ensure that (4.1) holds, \(c_0\mu _{1}-d\int _{\mathbb {R}}J(y)(e^{-\mu _{1}y}-1)dy\) must be greater than 0. We should point out that it can be ensured if we choose some suitable parameters. For example, let \(J(x)=\frac{1}{\sqrt{\pi }}e^{-x^2}\), \(b_1=\frac{3}{5}\) and \(d=1\). Note that \(c_{0}= 2\sqrt{1-b_{1}}=\frac{2\sqrt{10}}{5}\), \(\mu _1:=\mu _1(c_0)= \sqrt{1-b_{1}}=\frac{\sqrt{10}}{5}\). Then we can compute that

$$\begin{aligned}&d\int _{{\mathbb {R}}}J(y)(e^{-\mu _{1}y}-1)dy-c_0\mu _{1}\\&=\left( \frac{1}{\sqrt{\pi }}\int _{{\mathbb {R}}}e^{-y^2}e^{-\mu _{1}y}dy-1\right) -\frac{4}{5}\\&=\left( \frac{1}{\sqrt{\pi }}e^{\frac{\mu _1^2}{4}}\int _{{\mathbb {R}}}e^{-(y+\frac{\mu _1}{2})^2}dy-1\right) -\frac{4}{5}\\&=\left( e^{\frac{\mu _1^2}{4}}-1\right) -\frac{4}{5}=\left( e^{\frac{1}{10}}-1\right) -\frac{4}{5}<0. \end{aligned}$$

Next, we derive some explicit conditions for the linear speed selection that are independent of the growth rate \(\alpha \).

Theorem 4.3

Assume that (J1)–(J2) and (H1) hold. Then the minimal speed of system (1.5) is linearly selected if the following conditions

$$\begin{aligned}{} & {} b_{1}< \frac{2}{3}, \end{aligned}$$
(4.2)
$$\begin{aligned}{} & {} \quad b_{2}<\frac{2(1-b_{1})}{b_{1}}, \end{aligned}$$
(4.3)
$$\begin{aligned}{} & {} \quad d\int _{{\mathbb {R}}}J(y)(e^{-\mu _{1}y}-1)dy-c_0\mu _{1}<0 \end{aligned}$$
(4.4)

are satisfied, where \(\mu _{1}:=\mu _1(c_0)\).

Proof

Let \({\overline{U}}(z)\) be defined in (3.10), and define

$$\begin{aligned} {\overline{V}}(z)= \min \{1,b_{2}{\overline{U}}(z)\}= {\left\{ \begin{array}{ll} \quad 1, &{} z\le z_{1},\\ b_{2}{\overline{U}}(z), &{}z> z_{1}. \end{array}\right. } \end{aligned}$$

We shall verify that \(({\overline{U}},{\overline{V}})(z)\) is an upper solution to system (1.5) under the conditions (4.2)–(4.4). Here we use a method different from that in Theorem 4.1 to verify \(\Gamma _{u}[{\overline{U}},{\overline{V}}]\le 0\). The argument will split into the following two cases.

Case 1: \(z\le z_{1}\). In this case, \({\overline{V}}(z)=1\). For the U-equation, it is easy to see that

$$\begin{aligned} \Gamma _{u}[{\overline{U}},{\overline{V}}] =&\, {\overline{U}}(1-{\overline{U}})\left( \mu _{1}^{2}(1-2{\overline{U}})-c_0\mu _{1}+1\right) \\ =&\, {\overline{U}}(1-{\overline{U}})f({\overline{U}}), \end{aligned}$$

where \(f({\overline{U}})\), for \({\overline{U}}\in \bigg [\frac{1}{b_{2}},1\bigg ]\) as \(z\le z_{1}\), is defined by

$$\begin{aligned} f({\overline{U}})=b_1-2(1-b_{1}){\overline{U}}. \end{aligned}$$

By (4.2) and (4.3), it is easy to see that

$$\begin{aligned} \max _{{\overline{U}}\in [\frac{1}{b_{2}},1]} f({\overline{U}})&=f\bigg (\frac{1}{b_2}\bigg )=\frac{b_1b_2-2(1-b_1)}{b_2}<0, \end{aligned}$$

which implies that \(\Gamma _{u}[{\overline{U}},{\overline{V}}](z)\le 0\) for all \(z\le z_{1}\).

For the V-equation, substituting \({\overline{V}}(z)=1\) into the V-equation and considering that \({\overline{V}}(z)\le 1\) for all \(z\in {\mathbb {R}}\), we derive

$$\begin{aligned} \Gamma _{v}[{\overline{U}},{\overline{V}}]&=d(J*{\overline{V}}-{\overline{V}})+c_0{\overline{V}}'+\alpha (1-{\overline{V}})(b_{2}{\overline{U}}-{\overline{V}})\le 0. \end{aligned}$$

Case 2: \(z > z_{1}\). In this case, \({\overline{V}}(z)=b_{2}{\overline{U}}(z)\). Then

$$\begin{aligned} \Gamma _{u}[{\overline{U}},{\overline{V}}]&={\overline{U}}(1-{\overline{U}})\left( \mu _{1}^{2}(1-2{\overline{U}})-c_0\mu _{1}+1-\frac{b_{1}(1-b_{2}{\overline{U}})}{1-{\overline{U}}}\right) \\&={\overline{U}}(1-{\overline{U}})g({\overline{U}}), \end{aligned}$$

where \(g({\overline{U}})\), for \({\overline{U}}\in [0,\frac{1}{b_{2}}]\) as \(z> z_{1}\), is defined by

$$\begin{aligned} g({\overline{U}})&=-2(1-b_{1}){\overline{U}}+b_{1}-\frac{b_{1}(1-b_{2}{\overline{U}})}{1-{\overline{U}}}. \end{aligned}$$

It is easy to obtain that

$$\begin{aligned} g'({\overline{U}})=-2(1-b_{1})+\frac{b_{1}(b_{2}-1)}{(1-{\overline{U}})^{2}},\\ g''({\overline{U}})=\frac{2b_{1}(b_{2}-1)}{(1-{\overline{U}})^{3}}>0. \end{aligned}$$

Hence,

$$\begin{aligned} \max _{{\overline{U}}\in [0,\frac{1}{b_{2}}]} g({\overline{U}})=\max \left\{ g(0),g(\frac{1}{b_{2}})\right\} =g(0)=0, \end{aligned}$$

since \(g(\frac{1}{b_{2}})=\frac{-2(1-b_1)+b_1b_2}{b_2}<0\). Thus, \(\Gamma _{u}[{\overline{U}},{\overline{V}}](z)\le 0\) for all \(z>z_{1}\).

For the V-equation, substituting \({\overline{V}}(z)=b_{2}{\overline{U}}(z)\) into the V-equation, we compute

$$\begin{aligned} \Gamma _{v}[{\overline{U}},{\overline{V}}]&\le b_{2}{\overline{U}}(1-{\overline{U}})d\int _{{\mathbb {R}}}J(y)\frac{{\overline{U}}(z-y)- {\overline{U}}(z)}{{\overline{U}}(1-{\overline{U}})}dy-b_2 c_0\mu _{1}{\overline{U}}(1-{\overline{U}})\\&\le b_{2}{\overline{U}}(1-{\overline{U}})\left\{ d\int _{{\mathbb {R}}}J(y)\frac{(1+e^{\mu _1z})(1-e^{-\mu _1 y})}{1+e^{\mu _1(z-y)}}dy-c_0\mu _{1}\right\} \\&\le b_{2}{\overline{U}}(1-{\overline{U}})\\&\left\{ d\int _{{\mathbb {R}}}J(y)(e^{\mu _{1}y}-1)dy+d\int _{{\mathbb {R}}}J(y)\frac{-e^{\mu _1 y}-e^{-\mu _1 y}+2}{1+e^{\mu _1(z-y)}}dy-c_0\mu _{1}\right\} \\&\le b_{2}{\overline{U}}(1-{\overline{U}})\left\{ d\int _{{\mathbb {R}}}J(y)(e^{-\mu _{1}y}-1)dy-c_0\mu _{1}\right\} \le 0 \end{aligned}$$

for all \(z>z_{1}\), due to (4.4). Hence, \(({\overline{U}},{\overline{V}})(z)\) is an upper solution to system (1.5). The proof is complete. \(\square \)

Finally, we are going to construct a typical lower solution to study the nonlinear selection.

Theorem 4.4

Assume that (J1)–(J2) and (H1) hold. Then the minimal speed of system (1.5) is nonlinearly selected if the following condition

$$\begin{aligned} \frac{d\int _0^{+\infty }J(y)(e^{-\mu _1y}-1)dy+c_0\mu _1+\alpha }{\alpha b_{2}}<1-2(1-b_{1}) \end{aligned}$$
(4.5)

is satisfied, where \(\mu _1:=\mu _1(c_{0})\).

Proof

Let \({\underline{U}}\) be defined in (3.14), and define

$$\begin{aligned} {\underline{V}}(z)=\frac{{\underline{U}}(z)}{k}=\frac{1}{1+e^{\mu _{2}z}}, \quad \forall z\in {\mathbb {R}}, \end{aligned}$$

where \(0<k<1\) is some constant specified later, and \(\mu _2:=\mu _{2}(c)\) with \(c=c_{0}+\epsilon \), \(0<\epsilon \ll 1\). It is easy to see that

$$\begin{aligned} -2\mu _{2}^{2}+G_{2}(z)&=\,-2\mu _{2}^{2} +\frac{b_{1}{\underline{V}}-{\underline{U}}(1+\frac{b_{1}-1}{k})}{(1-\frac{{\underline{U}}}{k})\frac{{\underline{U}}}{k}}\nonumber \\&=-2\mu _{2}^{2}+\frac{1-k}{1-\frac{{\underline{U}}}{k}}. \end{aligned}$$
(4.6)

It is clear that

$$\begin{aligned} \frac{1-k}{1-\frac{{\underline{U}}}{k}}>1-k. \end{aligned}$$

Thus, (4.6) becomes

$$\begin{aligned} -2\mu _{2}^{2}+G_{2}(z)\ge -2\mu _{2}^{2}+1-k\ge 0, \end{aligned}$$

provided that k satisfies

$$\begin{aligned} k<1-2(1-b_{1}), \end{aligned}$$

where we have made use of the fact that \(\mu _2(c)\sim \mu _1=\sqrt{1-b_1}\) for sufficiently small positive \(\epsilon \) in \(c=c_{0}+\epsilon \). Thus, (3.15) holds. Hence, \(\Gamma _{u}[{\underline{U}},{\underline{V}}]\ge 0\).

For the V-equation, we derive

$$\begin{aligned} \Gamma _{v}[{\underline{U}},{\underline{V}}]&={\underline{V}}(1-{\underline{V}})\left\{ d\int _{{\mathbb {R}}}J(y)\frac{{\underline{V}}(z-y)-{\underline{V}}(z)}{{\underline{V}}(1-{\underline{V}})}dy -c\mu _{2}+\frac{\alpha (b_{2}{\underline{U}}-{\underline{V}})}{{\underline{V}}}\right\} \\&{{={\underline{V}}(1-{\underline{V}})\left\{ d\int _{{\mathbb {R}}}J(y)\frac{(1-e^{-\mu _2y})(1+e^{\mu _2 z})}{1+e^{\mu _2(z-y)}}dy -c\mu _{2}+\frac{\alpha (b_{2}{\underline{U}}-{\underline{V}})}{{\underline{V}}}\right\} }}\\&\ge {\underline{V}}(1-{\underline{V}})\left\{ d\int _{-\infty }^0J(y)(1-e^{-\mu _{2}y})dy-c\mu _{2}+\frac{\alpha (b_{2}{\underline{U}}-{\underline{V}})}{{\underline{V}}}\right\} \\&=\frac{{\underline{U}}}{k}(1-\frac{{\underline{U}}}{k})\left\{ d\int _0^{+\infty }J(y)(1-e^{\mu _{2}y})dy-c\mu _{2}-\alpha +\alpha b_{2}k\right\} \\&\rightarrow \frac{{\underline{U}}}{k}(1-\frac{{\underline{U}}}{k})\left\{ d\int _0^{+\infty }J(y)(1-e^{\mu _1y})dy-c_0\mu _1-\alpha +\alpha b_{2}k\right\} , \end{aligned}$$

as \(\epsilon \rightarrow 0\). Thus, if

$$\begin{aligned} k>\frac{d\int _0^{+\infty }J(y)(e^{-\mu _1 y}-1)dy+c_0\mu _1+\alpha }{\alpha b_{2}}, \end{aligned}$$

then we see that \(\Gamma _{v}[{\underline{U}},{\underline{V}}]\ge 0\).

Therefore, by (4.5), we obtain that \(({\underline{U}}, {\underline{V}})\) is a lower solution to system (1.5). The proof is complete. \(\square \)

Remark 4.5

We should point out that the condition (4.5) can be ensured if we choose some suitable parameters. For example, let \(J(x)=\frac{1}{\sqrt{\pi }}e^{-x^2}\), \(b_1=\frac{3}{5}\), \(b_2=10\), \(\alpha =1\) and \(d=1\). Note that \(c_{0}= 2\sqrt{1-b_{1}}=\frac{2\sqrt{10}}{5}\), \(\mu _1:=\mu _1(c_0)= \sqrt{1-b_{1}}=\frac{\sqrt{10}}{5}\). Then we can compute that

$$\begin{aligned}&d\int _0^{+\infty }J(y)(e^{-\mu _{1}y}-1)dy+c_0\mu _{1}+\alpha \\&=\left( \frac{1}{\sqrt{\pi }}\int _0^{+\infty }e^{-y^2}e^{-\mu _{1}y}dy-\frac{1}{2}\right) +\frac{4}{5}+1\\&{{=\left( \frac{1}{\sqrt{\pi }}e^{\frac{\mu _1^2}{4}}\int _0^{+\infty } e^{-(y+\frac{\mu _1}{2})^2}dy-\frac{1}{2}\right) +\frac{4}{5}+1}}\\&{{\le \left( \frac{1}{\sqrt{\pi }}e^{\frac{\mu _1^2}{4}}\int _0^{+\infty } e^{-s^2}ds-\frac{1}{2}\right) +\frac{4}{5}+1}}\\&{ {= \frac{1}{2}\left( e^{\frac{\mu _1^2}{4}}-1\right) +\frac{4}{5}+1=\frac{1}{2}\left( e^{\frac{1}{10}}-1\right) +\frac{4}{5}+1,}} \end{aligned}$$

and hence,

$$\begin{aligned} \frac{d\int _0^{+\infty }J(y)(e^{-\mu _1y}-1)dy+c_0\mu _1+\alpha }{\alpha b_{2}}\le 0.186. \end{aligned}$$

Note that \(1-2(1-b_{1})=\frac{1}{5}\). Therefore, (4.5) holds.