Traveling Wavefronts to a Model of Precursor and Differentiated Cells: Impacting Parameter-Structure Transition from Monostable to Bistable, and from Monotone to Non-monotone

Abstract

This paper provides a novel analysis of the rich and complex propagation dynamics to a model of precursor and differentiated cells, with the appearance of non-isolated equilibria on a line in the phase space. We established the existence of traveling waves in the monostable monotone case by means of continuation argument via perturbation in a weighted functional space, by applying the abstract implicit function theorem. We proved necessary and sufficient conditions of the minimal wave speed selection and showed the existence of the transition (turning point) $k^*$ for the minimal wave speed when the parameters $\lambda$ and $\gamma$ are fixed. Two explicit estimates about $k^*$ were given by the easy-to-apply theorem we derived. We investigated the decay rate of the minimal traveling wave as $z\rightarrow \infty$ in terms of the value of k. We further proved the existence of non-negative wavefronts in the monostable non-monotone case and found that the minimal wave speed is always linearly selected. Finally in the bistable monotone case, the existence and uniqueness of bistable traveling waves were proved via constructing an auxiliary parabolic non-local equation.

1 Introduction

Mathematical models have been of great significance in studying biological systems of migrating and proliferating cells (see the references in [17, 38, 40, 42]). Here, we consider a population system of migrating, proliferating and differentiating cells that invade a certain region, motivated by experimental studies of developing enteric nervous system:

$$\begin{aligned} \left\{ \begin{aligned}&c_t=dc_{xx}+\sigma c\left( 1-\frac{c+\gamma n}{\rho _1}\right) -\vartheta c\left( 1-\frac{n}{\rho _2}\right) , \\&n_t=\vartheta c\left( 1-\frac{n}{\rho _2}\right) .\\ \end{aligned}\right. \end{aligned}$$

(1.1)

Here, c(t, x) and n(t, x) are the population of precursor and differentiated cells at time t and location x, respectively. The parameter d is the diffusion coefficient of precursor cells. By experiment, the differentiated cells have negligible migratory capability compared with the precursor cells so that its diffusion rate can be viewed as zero in the model. Proliferation of the precursor cells is modeled by the logistic growth term with a proliferation rate $\sigma$ and a carrying capacity $\rho _1$. The parameter $\gamma$ is the relative contribution that the differentiated cell population makes to the carrying capacity. Differentiation is limited by a carrying capacity $\rho _2$ for the differentiated cells and has a maximum rate $\vartheta$. For more information on the biological background of this model, see [38, 40, 42] and references therein.

For the developing enteric nervous system, the colonization process is time-sensitive (Farlie et al. [13]), as the failure of neural crest cells in colonizing the required length of the gut finally results in a developmental disorder known as Hirschsprung’s disease (Newgreen and Young[38]). The development of this process seems propagative or invasive, linking mathematically to an important nonlinear phenomenon such as traveling wavefronts to the precursor and differentiated cells. The results we obtain may assist experimentalists in identifying the resource that limits the proliferation of precursor cells so as to find effective treatments.

Scale (1.1) by

$$\begin{aligned} c'=\frac{c}{\rho _1}, \quad n'=\frac{n}{\rho _1}, \quad t'=\sigma t, \quad x'=\sqrt{\frac{\sigma }{d}}x, \end{aligned}$$

and drop the prime notation for convenience. It yields

$$\begin{aligned} \left\{ \begin{aligned}&c_t=c_{xx}+c(1-c-\gamma n)-\lambda c(1-kn), \\&n_t=\lambda c(1-kn), \\ \end{aligned}\right. \end{aligned}$$

(1.2)

where $\lambda =\frac{\vartheta }{\sigma }$, and $k=\frac{\rho _1}{\rho _2}$. The parameter $\lambda$ is the ratio of the differentiation and proliferation rates of the precursor cells, and k is the ratio of the carrying capacities of the precursor and differentiated cell populations. All parameters are non-negative, and we always assume $\frac{\gamma }{k} <1$ so that a positive equilibrium exists. System (1.2) has infinite number of constant equilibria

$$\begin{aligned} {\varvec{e}}_\beta =\left( 1-\frac{\gamma }{k},\frac{1}{k}\right) , \quad e_a=(0,a), \end{aligned}$$

(1.3)

where a is an arbitrary non-negative constant. If $a=0$, then $e_0=(0,0)$. The kinetic system of model (1.2) is

$$\begin{aligned} \left\{ \begin{aligned}&\frac{dc}{dt}=c(1-c-\gamma n)-\lambda c(1-kn), \\&\frac{dn}{dt}=\lambda c(1-kn). \\ \end{aligned}\right. \end{aligned}$$

(1.4)

The particular importance of parameter $\lambda$ is that the value of this parameter determines whether the system (1.2) is monostable or bistable. Indeed, stability analysis of (1.4) around the equilibria indicates that $e_\beta$ is stable, and $e_0$ is unstable for $\lambda < 1$; $e_\beta$ is stable, and $e_0$ is non-asymptotically stable for $\lambda >1$. That is, $\lambda =1$ is the transition point so that $e_\beta$ and $e_0$ of (1.2) have the typical monostability when $\lambda \in (0,1)$, and the bistability properties when $\lambda >1$. Furthermore, system (1.2) transits from monotonic to non-monotonic when $\lambda k-\gamma$ changes from positive to negative. Observe that system (1.2) becomes the well-known Fisher-KPP equation when $\lambda k-\gamma =0$, and the related results can be seen in [24]. In this paper, we consider the following three cases of system (1.2):

• MM (monostable, monotonic): $\lambda k-\gamma >0,\quad \lambda <1;$

• MN (monostable, non-monotonic): $\lambda k-\gamma<0,\quad \lambda <1;$

• BM (bistable, monotonic): $\lambda k-\gamma>0,\quad \lambda >1.$

Since we require that $e_{\beta }$ is positive, the case $\lambda k-\gamma <0$ with $\lambda \ge 1$ does not hold in our model. However, for $\lambda =1$ and $\lambda k-\gamma >0$, the system becomes exceptional with algebraically decay traveling wave solutions. We will study it separately in the future.

To find a traveling wave solution of system (1.2) connecting two equilibria $e_\beta$ to $e_0$, we let

$$\begin{aligned} (c,n)(t,x)=(C,N)(z), \quad z=x-st. \end{aligned}$$

Then the traveling wave speed $s\in \mathbb {R}$ and (C, N)(z) satisfy

$$\begin{aligned} \left\{ \begin{aligned}&C''+sC'+C[1-\lambda -C+(\lambda k-\gamma )N]=0, \\&sN'+\lambda C(1-kN)=0, \\ \end{aligned}\right. \end{aligned}$$

(1.5)

with boundary conditions

$$\begin{aligned} (C,N)(-\infty )=e_\beta ,\quad (C,N)(\infty )=e_0. \end{aligned}$$

(1.6)

In the MM and MN cases when $\lambda \le 1$, the solution to (1.5)–(1.6) is usually not unique, as can be seen in the classical KPP-Fisher model [15]. Denote $s_{\min }$ as

$$\begin{aligned} s_{\min }=\inf \{s: (1.5)-(1.6) \ \text {has a non-negative solution}\}. \end{aligned}$$

Standard linearization near the equilibrium point $e_0$ shows $s_{\min }\ge s_0=2\sqrt{1-\lambda }$. For convenience, we give the following definition on the minimal wave speed selection in the monostable case.

Definition 1.1

If $s_{\min }=s_0$, then we say the minimal wave speed is linearly selected; If $s_{\min }>s_0$, then the minimal wave speed is said to be nonlinearly selected.

We should mention that in the critical case when $\lambda k-\gamma =0$, the first equation in (1.5) is independent of N and it becomes the classical Fisher-Kpp equation that is bounded by its linearized system at zero. Thus its minimal speed is always linearly selected, see [24, 25]. Furthermore in the BM case when $\lambda >1$, the situation is totally different and we will show that there exists a unique (up to translation) traveling wave of (1.5)–(1.6).

While degenerate reaction-diffusion models have attracted extensive attention in the research field (see e.g., [8, 11, 17]), system (1.2) is a complex one exhibiting the so-called sophistic transition dynamics(transition from monostable to bistable, and from monotone to non-monotone, when parameters vary), and due to the appearance of clustering-and-non-isolated equilibria on a line in the phase space (the equilibrium $e_0$ is not isolated), it becomes much trickier to study model (1.2). Such a characteristics of $e_0$ prevents us from applying the recent powerful Liang-Zhao general theory [25] or Fang and Zhao’s result [11, 12] of abstract monostable systems to our system. We should mention that via a shooting method, Gibbs[17] proved the existence of traveling waves of the Belousov–Zhabotinskii reaction system similar to (1.2). However, while this approach has geometrical insight, it is not suitable for high-dimensional phase space systems or for nonlocal systems, and the idea in [17] cannot be directly extended to work on our non-monotonic case. Here, in order to overcome the difficulty caused by $e_0$ non-isolated, we develop new methods to prove the existence of traveling wave solutions for the system with either monostable or bistable nonlinearity. Our method relies on a combination of advanced analysis and dynamical technique, and can be applied to nonlocal problem and high-dimensional phase system. Furthermore, the minimal wave speed selection of (1.5)–(1.6) is another crucial and challenging issue that still needs to be solved. One part of the goals of our paper is to provide a solution to this problem.

For the monostable and monotonic case MM, we prove the existence of traveling waves of system (1.5)–(1.6) by using continuation via perturbation in a weighted functional space and the abstract implicit function theorem. Moreover, a necessary and sufficient conditions to distinguish between linear selection and nonlinear selection are given. We find that the minimal wave speed is non-decreasing with respect to k, when $\gamma$ and $\lambda$ are fixed, and there exists a transition (turning) point $k^*$ for the wave speed selection, at which the wavefront decays like a pair of exponential functions. We also derive an easy-to-apply theorem for speed selection, which can be used to get a series of explict conditions on speed selection by constructing various upper and lower solutions. By applying this theorem, two explicit estimates about $k^*$ are given. We also give details of the decay rate of the minimal traveling wave as k varies, which implies the transition wave behavies completely different than other pulled waves for $k<k^*$. For the monostable and non-monotonic case MN, the non-monotonicity of the system invalidates the monotonic dynamical theory and the comparison principle. By constructing a suitable invariant region, we can utilize the Schauder fixed point theorem to prove the existence of non-negative solutions of (1.5)–(1.6). We show that the minimal wave speed is always linearly selected without other restrictions on the system parameters. Last, for the bistable and monotone case BM, we have developed a new method to find the bistable wavefront. Solving the second equation in (1.5) transforms the system into a nonlocal equation with wave speed s also in the reaction term. Consider the wave speed s in the reaction term as a parameter. We then establish the existence of bistable traveling wave to an auxiliary parabolic equation, by constructing appropriate upper and lower solutions and using monotonic dynamic theory in a weighted space. Returing to the original system, we show that there exists at most one (up to translation) bistable traveling wave solution. The novelty of our results in comparison to previous works is in the discussion section.

Finally, we would like to provide readers with a few more references related to our study. The problem of the existence of traveling waves and their speed sign for monotonic systems have been studied in many literatures. See eg., [2, 4, 5, 9, 11, 12, 14, 15, 17, 24, 25, 30, 32, 34, 35, 41, 42, 45, 47,48,49]. For non-monotone systems, the methods used include phase plane analysis, shooting method, Conley index and fixed point theorem, and we refer to [10, 16, 19, 26, 27]. The problem of the minimal speed selection was widely discussed in various models such as the Lotka-Volterra model, Stream-Population model, population genetics, etc. See, e.g., [1, 3, 4, 20, 21, 28, 29, 31, 33, 39, 46, 50, 51].

The remainder of this paper is organized as follows. In Sect. 2, we establish the existence of the monotone traveling wave of (1.5)–(1.6) for the case MM. Then we show results about the minimal wave speed selection and the decay rate of the minimal traveling wave as $z\rightarrow \infty$, when k is in different value regions. The existence of non-negative solution of (1.5)–(1.6) for the case MN is established in Sect. 3. In Sect. 4, we prove the existence and uniqueness of the monotone bistable traveling wave of (1.5)–(1.6) for the case BM. Numerical results will be presented in Sect. 5. Section 6 is about conclusion and discussion.

2 Traveling Waves and Speed Selection in the Case MM

In this section, we consider system (1.2) in the case MM. Note that $e_\beta$ is stable, and $e_0$ is unstable. Linearizing (1.5) at $e_0$, we obtain the characteristic equation as

$$\begin{aligned} \mu ^2-s\mu +(1-\lambda )=0, \end{aligned}$$

(2.1)

which has two roots as follows

$$\begin{aligned} \mu _1=\mu _1(s)=\frac{s-\sqrt{s^2-4(1-\lambda )}}{2},\quad \mu _2=\mu _2(s)=\frac{s+\sqrt{s^2-4(1-\lambda )}}{2}. \end{aligned}$$

(2.2)

Denote $s_0=2\sqrt{1-\lambda }$ as the linear speed so that $\mu _1$ and $\mu _2$ are real when $s\ge s_0$. In terms of the second equation of system (1.5) and $(C, N)(\infty )=e_0$, N(z) can be expressed as

$$\begin{aligned} N(z)=\frac{1}{k}\bigg [ 1-e^{-k\int _{z}^{\infty }\frac{\lambda C(t)}{s}dt}\bigg ] \end{aligned}$$

(2.3)

which is non-decreasing in C(t). After a substitution, system (1.5) becomes a nonlocal equation

$$\begin{aligned} C''+sC'+C[1-\lambda -C+(\lambda k-\gamma )N(z)]=0, \end{aligned}$$

(2.4)

with the boundary condition

$$\begin{aligned} C(-\infty )=1-\frac{\gamma }{k}, \quad C(\infty )=0, \end{aligned}$$

(2.5)

where N(z) is defined in equation (2.3).

The following theorem gives the existence of the traveling wave of system (1.5)–(1.6) in the case MM.

Theorem 2.1

Assume that MM holds. There is a finite $s_{\min }\ge s_0$ such that monotone and positive traveling wave solutions of (1.5)–(1.6) exist if and only if $s\ge s_{\min }$.

Before proving Theorem 2.1, we introduce some important lemmas that play a great role in the proof of the above theorem.

Lemma 2.2

For $s<s_0$, there exists no non-negative solution of system (1.5)–(1.6).

Proof

Since $\mu _1$ and $\mu _2$ are complex for $s<s_0$, we get that $e_1$ is a spiral point. Therefore there does not exist a non-negative solution of system (1.5)–(1.6). $\square$

Lemma 2.3

Assume that MM holds. We have the following two results:

• (i) If (1.5)–(1.6) has a non-increasing traveling wave solution (C(z), N(z)) at $s=s_2>s_0,$ and $C(z)\sim A_1e^{-\mu _1(s_2) z}$ as $z\rightarrow \infty$ with $A_1>0$, then there exists $\delta >0$ such that (1.5)–(1.6) has a non-increasing traveling wave solution for $s=s_2-\delta$;

• (ii) If (1.5)–(1.6) has a non-increasing traveling wave solution (C(z), N(z)) at $s=s_2>s_0$, and $C(z)\sim A_2e^{-\mu _2(s_2) z}$ as $z\rightarrow \infty$ with $A_2>0$, then there exist no non-negative traveling wave solution of (1.5)–(1.6) for $s<s_2$.

Proof

We first give the proof of result (i). Let $\alpha$ be large enough so that

$$\begin{aligned} F(C):=\alpha C+C[1-\lambda -C+(\lambda k-\gamma )N] \end{aligned}$$

is non-decreasing in C. Equation (2.4) can be written as

$$\begin{aligned} C''+sC'-\alpha C=-F(C). \end{aligned}$$

(2.6)

Define $\beta _1$ and $\beta _2$ as

$$\begin{aligned} \beta _1=\frac{-s-\sqrt{s^2+4\alpha }}{2}<0, \text {and}\quad \beta _2=\frac{-s+\sqrt{s^2+4\alpha }}{2}>0. \end{aligned}$$

(2.7)

In view of variation of parameters, the integral form of (2.6) is given by

$$\begin{aligned} C(z)=\frac{1}{\beta _2-\beta _1}\bigg \{\int _{-\infty }^{z}e^{\beta _1(z-t)}F(C)(t)dt +\int _{z}^{\infty }e^{\beta _2(z-t)}F(C)(t)dt\bigg \} =:T_1(F)(z). \end{aligned}$$

(2.8)

For $s=s_2>s_0$, suppose there exists a non-increasing traveling wave solution $C_2(z)$ of (2.4)–(2.5) with $C_2(z) \sim A_1e^{-\mu _1(s_2) z}$ as $z\rightarrow \infty$. Then $C_2(z)$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&C_2''+s_2C_2'+C_2[1-\lambda -C_2+(\lambda k-\gamma )N[C_2]]=0, \\&C_2(-\infty )=1-\frac{\gamma }{k},\quad C_2(\infty )=0.\\ \end{aligned}\right. \end{aligned}$$

(2.9)

Consider $s=s_\delta =s_2-\delta <s_2$, and $\delta$ is sufficiently small to be determined. Let $\bar{C}(z)=C_2(z) \omega (z)$, where

$$\begin{aligned} \omega =\frac{1}{1+\delta \exp \{(\mu _1(s_\delta )-\mu _1(s_2))z\}}. \end{aligned}$$

By way of asymptotic analysis, we have $\bar{C}(z)\sim \frac{A_1}{\delta }e^{-\mu _1(s_\delta )z}$ as $z\rightarrow \infty$.

To find the solution of (2.6), say $C_\delta (z)$ at $s=s_{\delta }$, let $C_\delta (z)=\bar{C}(z)+W_1(z)$, where $W_1(z) \in B_0$ is to be determined, and here $B_0$ is defined as

$$\begin{aligned} B_0=\{u\in C(-\infty , \infty ):u(\pm \infty )=0\}. \end{aligned}$$

If we can prove the existence of $W_1(z) \in B_0$ such that $C_\delta (z)$ is a solution of (2.4) or (2.6) with speed $s_\delta ,$ then result (i) is proved. Substituting $C_\delta (z)$ into (2.4), and by (2.9), we find that $W_1$ satisfies

$$\begin{aligned} W_1''+s_2W_1'-\alpha W_1=-(F_0+F_{\delta }+F_h), \end{aligned}$$

(2.10)

where

$$\begin{aligned} \begin{aligned}&F_0=\left\{ \alpha +1-\lambda -2C_2+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda C_2(\tau )}{s_2}d\tau }\right] \right\} W_1\\&\qquad +C_2(\lambda k-\gamma )e^{-k\int _{z}^{\infty }\frac{\lambda C_2(\tau )}{s_2}d\tau }\int _{z}^{\infty }\frac{\lambda W_1(\tau )}{s_2}d\tau ,\\&F_\delta =G(s_\delta , C_2\omega +W_1)-G(s_2, C_2+W_1)+G(s_2, C_2)-\omega G(s_2, C_2)+\\&\quad \quad \quad 2C_2'\omega '+C_2\omega ''+(s_2-\delta )C_2\omega '-\delta (W_1'+C_2'\omega ),\\&F_h=-W_1^2+\left( \lambda -\frac{\gamma }{k}\right) e^{-k\int _{z}^{\infty }\frac{\lambda C_2(\tau )}{s_2}d\tau }(C_2\varphi _1-W_1\varphi _2),\\&G(s, C)=C\left\{ 1-\lambda -C+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda C(\tau )}{s}d\tau }\right] \right\} , \\&\varphi _1=1-e^{-k\int _{z}^{\infty }\frac{\lambda W_1(\tau )}{s_2}d\tau }-k\int _{z}^{\infty }\frac{\lambda W_1(\tau )}{s_2}d\tau ,\\&\varphi _2=e^{-k\int _{z}^{\infty }\frac{\lambda W_1(\tau )}{s_2}d\tau }-1. \end{aligned} \end{aligned}$$

Therefore, we have

$$\begin{aligned} W_1=T_1(F_0)+T_1(F_\delta )+T_1(F_h) \end{aligned}$$

(2.11)

with $s=s_2$, where $T_1$ is as (2.8). Note that $\omega '=-(\mu _1(s_\delta )-\mu _1(s_2))\omega (1-\omega )$ and $\omega ''=(\mu _1(s_\delta )-\mu _1(s_2))^2\omega (1-\omega )(1-2\omega )$. Clearly $T_1(F_\delta )$ is of $O(\delta )$ when $\delta \rightarrow 0$. Define

$$\begin{aligned} T(W_1)=T_1(F_0(W_1)). \end{aligned}$$

The linear operator T is compact and strongly positive. It has a simple principle eigenvalue $\lambda =1$ with the associated positive eigenfunction $-C_2'$. In order to remove this eigenfunction from $B_0$, we define the following weighted functional space

$$\begin{aligned} \mathcal {H}=\{h(\xi )\in B_0: he^{\mu _1(s_\delta )\xi }=o(1) \quad \textrm{as} \quad \xi \rightarrow \infty \}. \end{aligned}$$

Since the eigenfunction $-C_2'$ is not in the space $\mathcal {H}$, this implies that T does not have eigenvalue $r=1$ in the functional space $\mathcal {H}$, that is, $I-T$ has a bounded inverse in $\mathcal {H}$, where I is the identity operator. It follows by abstract implicit function theorem that there exists $\delta _0$ so that a solution $W_1$ of the equation (2.11) exists for any $\delta \in [0, \delta _0)$. Hence, there exists a solution $C_\delta (z)$ of (2.4)–(2.5) with $s=s_\delta$.

We then give the proof of the result (ii). Assume that there exists $\hat{s}\in (s_0,s_2)$ such that system (1.2) has a monotone traveling wave solution $(\bar{C},\bar{N})(x-\hat{s}t)$ with initial conditions

$$\begin{aligned} c(0,x)=\bar{C}(x) \quad \text {and} \quad n(0,x)=\bar{N}(x). \end{aligned}$$

Clearly, $(\bar{C},\bar{N})(z)$ satisfies system (1.5) with $s=\hat{s}$. Since $\mu _2(s)$ is non-decreasing with respect to s, we get $\mu _1(\hat{s})\le \mu _2(\hat{s})\le \mu _2(s_2)$. Recall that $C(z)\sim A_2 e^{-\mu _2(s_2) z}$ as $z\rightarrow \infty$. It follows that $C(z)\le \bar{C}(z)$ as $z\rightarrow \infty$. Note that $C(z)\sim (1-\frac{\gamma }{k})-A_3e^{\tau z}$ as $z\rightarrow -\infty$ for positive $A_3$ and $\tau$, and $\tau$ is non-increasing with respect to s. Therefore, $C(z)\le \bar{C}(z)$ as $z\rightarrow -\infty$. Making a translation if necessary, we can always have $C(z)\le \bar{C}(z)$ for all z. According to the second equation of (1.2), we then have $(C,N)(x)\le (\bar{C},\bar{N})(x)$ for all $x\in \mathbb {R}$ (by shift if necessary). By comparison, it gives that

$$\begin{aligned} C(x-s_2t)\le \bar{C}(x-\hat{s}t). \end{aligned}$$

(2.12)

Fix $\hat{z}=x-s_2t$ such that $C(\hat{z})>0$. Observe that

$$\begin{aligned} \bar{C}(x-\hat{s}t)=\bar{C}(\hat{z}+(s_2-\hat{s})t))\rightarrow 0, \quad \text {as}\quad t \rightarrow \infty . \end{aligned}$$

Thus, we obtain

$$\begin{aligned} C(\hat{z})\le \bar{C}(x-\hat{s}t)\rightarrow 0, \quad \text{ as }\quad t \rightarrow \infty , \end{aligned}$$

which contradicts the fact $C(\hat{z})>0$. Therefore, there is no traveling wave solution of system (1.5)–(1.6) for any $\hat{s}\in [s_0,s_2)$. $\square$

We now prove that for large wave speed s, system (1.5)–(1.6) has traveling wave solutions via upper/lower solution method. Indeed, for any $s>s_0$, define a continuous function

$$\begin{aligned} \underline{C}_0(z)=\left\{ \begin{aligned}&e^{-\mu _1(s) z} (1-Me^{-\delta _1 z}),\quad&z\ge z_1, \\&0,\ {}&z<z_1,\\ \end{aligned} \right. \end{aligned}$$

(2.13)

where $0<\delta _1\ll 1$, M is a positive constant to be determined, and $z_1=\frac{\log M}{\delta _1}$.

Lemma 2.4

Assume MM holds. If $s=s_0+\delta _2$ and $\delta _2>0$, then $(\underline{C}_0, \underline{N}_0)(z)$ is a lower solution of system (1.5)–(1.6), where $\underline{C}_0(z)$ is defined as (2.13), and $\underline{N}_0(z)$ is the solution of N-equation of the system with $C(z)=\underline{C}_0(z)$.

Proof

Substituting $\underline{C}_0(z)$ into C-equation of (1.5), we obtain

$$\begin{aligned} \begin{aligned}&\underline{C}_0''+s\underline{C}_0'+\underline{C}_0[1-\lambda -\underline{C}_0+(\lambda k-\gamma )\underline{N}_0]\\&\quad =e^{-\mu _1 z}[\mu _1^2-s\mu _1+(1-\lambda )]-Me^{-(\mu _1+\delta _1)z}[(\mu _1+\delta _1)^2-s(\mu _1+\delta _1)+1-\lambda ]\\&\qquad -e^{-2\mu _1 z}(1-Me^{-\delta _1 z})^2+(\lambda k-\gamma )\underline{N}_0e^{-\mu _1 z}(1-Me^{-\delta _1 z}). \end{aligned} \end{aligned}$$

Note that the second term is positive. Let M be large enough such that the second term plus the third one is positive. The first term is 0, and the last term is positive. Therefore, the proof is complete. $\square$

Lemma 2.5

Assume that MM holds. For $s\ge 2\sqrt{1-\frac{\gamma }{k}}$, there exists a non-increasing and non-negative traveling wave solution for system (1.5)–(1.6).

Proof

We shall use the upper-lower solution method to prove it. Note that $s\ge 2\sqrt{1-\frac{\gamma }{k}}>s_0$ as $\lambda k-\gamma >0$. A lower solution is defined as (2.13). We only need to construct a suitable upper solution. Take $\hat{\mu }=\frac{s-\sqrt{s^2-4(1-\frac{\gamma }{k})}}{2}$ satisfying the equation $\hat{\mu }^2-s\hat{\mu }+1-\frac{\gamma }{k}=0$. We define $\bar{C}_0$ as

$$\begin{aligned} \bar{C}_0(z)=\left\{ \begin{aligned}&1-\frac{\gamma }{k},\quad&z\le z_0, \\&e^{-\hat{\mu } z},\ {}&z>z_0,\\ \end{aligned} \right. \end{aligned}$$

(2.14)

where $e^{-\hat{\mu } z_0}=1-\frac{\gamma }{k}$. Let $\bar{N}_0(z)$ satisfy the second equation of (1.5), i.e., (2.3) with $C(z)=\bar{C}_0(z)$. For $z\le z_0$, it follows that

$$\begin{aligned} \begin{aligned}&\bar{C}_0''+s\bar{C}_0'+\bar{C}_0\left\{ 1-\lambda -\bar{C}_0+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda \bar{C}_0(\tau )}{s}d\tau }\right] \right\} \\&\quad \le \left( 1-\frac{\gamma }{k}\right) \left[ 1-\lambda -\left( 1-\frac{\gamma }{k}\right) +\left( \lambda -\frac{\gamma }{k}\right) \right] =0. \end{aligned} \end{aligned}$$

(2.15)

For $z>z_0$, we get

$$\begin{aligned} \begin{aligned}&\bar{C}_0''+s\bar{C}_0'+\bar{C}_0\left\{ 1-\lambda -\bar{C}_0+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda \bar{C}_0(\tau )}{s}d\tau }\right] \right\} \\&\quad \le \hat{\mu }^2e^{-\hat{\mu } z}-s\hat{\mu }e^{-\hat{\mu } z}+e^{-\hat{\mu } z}\left( 1-\frac{\gamma }{k}\right) =0. \end{aligned} \end{aligned}$$

(2.16)

Thus, $(\bar{C}_0,\bar{N}_0)(z)$ is the upper solution of (1.5)–(1.6). By Lemma 2.4 and the upper-lower solution method, there exists a monotone traveling wave of (1.5)–(1.6) for speed $s\ge 2\sqrt{1-\frac{\gamma }{k}}$. Therefore, the proof is complete. $\square$

The following lemma provides a limiting argument.

Lemma 2.6

Assume that MM holds. If for all $s=s_k, k=1,2, \cdots$, where $\lim \limits _{k\rightarrow \infty }s_k=\bar{s}$, system (1.5)–(1.6) has a non-increasing traveling wave solution $(C_k, N_k)(z)$, where $C_k(z)\sim A_k e^{-\mu _1(s_k) z}$ as $z\rightarrow \infty$ with $A_k>0$, then there exists a non-increasing traveling wave solution of (1.5)–(1.6) for $s=\bar{s}$.

Proof

We shall use a limiting argument to prove it, see [6, 41]. Using Lemmas 2.3 and 2.5, we may assume that the sequence $\{s_k\}$ is decreasing such that system (1.5)–(1.6) has a non-increasing traveling wave solution $(C_k, N_k)(z)$ with $s=s_k$. Via shift, fix $C_k(0)=\frac{1}{2}(1-\frac{\gamma }{k})$ for each k. Note that $C_k(z)$ also satisfies the integral equation (2.8), and $N_k(z)$ is defined as in (2.3) with $C(z)=C_k(z)$. Then it follows

$$\begin{aligned} \begin{aligned} \bigg |\frac{dC_k(z)}{dz}\bigg |&=\frac{1}{\beta _2-\beta _1}\bigg |\bigg \{\int _{-\infty }^{z}\beta _1e^{\beta _1(z-t)}+\int _{z}^{\infty }\beta _2e^{\beta _2(z-t)}\bigg \}[F(C_k, N_k)(t)]dt\bigg |\\&\le \frac{\beta _2}{\beta _2-\beta _1}\bigg |\bigg \{\int _{-\infty }^{z}e^{\beta _1(z-t)}+\int _{z}^{\infty }e^{\beta _2(z-t)}\bigg \}[F(C_k, N_k)(t)]dt\bigg |\\&\le \beta _2< \infty . \end{aligned} \end{aligned}$$

This means that $C_k(z)$ is equicontinuous and uniformly bounded. By the Ascoli-Arzela theorem, there is a subsequence $\{s_{k_j}\}$ such that $C_{k_j}(z)$ converges uniformly on any compact interval and pointwise on $\mathbb {R}$ to a limit, say $C_{\bar{s}}(z)$. Let $j \rightarrow \infty$. In terms of the dominated convergence theorem, $C_{\bar{s}}(z)$ satisfies the integral equation (2.8), and $N_{\bar{s}}(z)$ satisfies (2.3) with $C(z)=C_{\bar{s}}(z)$ for $s=\bar{s}$. Obviously, $(C_{\bar{s}}(z),N_{\bar{s}}(z))$ is a non-increasing solution of (1.5) with $s=\bar{s}$ and $C_{\bar{s}}(0)=\frac{1}{2}(1-\frac{\gamma }{k})$. Letting $z\rightarrow \pm \infty$, and by the monotone convergence theorem, $C_{\bar{s}}(\pm \infty )$ satisfy the integral equation (2.8). Due to the fact that $1-\frac{\gamma }{k}\ge C_{\bar{s}}(-\infty )\ge C_{\bar{s}}(0)\ge C_{\bar{s}}(\infty ) \ge 0$, it follows that $C_{\bar{s}}(-\infty )=1-\frac{\gamma }{k}$ and $C_{\bar{s}}(\infty )=0$. Since $N_{\bar{s}}(z)$ satisfies (2.3) with $C(z)=C_{\bar{s}}(z)$, we get $N_{\bar{s}}(-\infty )=\frac{1}{k}$ and $N_{\bar{s}}(\infty )=0$. Therefore, the proof is complete. $\square$

Now we work on the proof of Theorem 2.1. By Lemma 2.5, monotone traveling wave of (1.5)–(1.6) exists when $s=s_1$, where $s_1\ge 2\sqrt{1-\frac{\gamma }{k}}>s_0$. If the traveling wave solution C(z) decays in $\mu _2$ as $z\rightarrow \infty$ when $s=s_1$, then $s_{\min }=s_1$ by result (ii) in Lemma 2.3. If the traveling wave solution C(z) decays in $\mu _1$ when $s=s_1$ as $z\rightarrow \infty$, then for some small enough $\delta$, monotone traveling wave of (1.5)–(1.6) exists for wave speed $s=s_2=s_1-\delta$, by result (i) in Lemma 2.3. And so on, we can get a decreasing sequence $\{s_k\}_{k=1}^\infty$ such that system (1.5)–(1.6) has the traveling wave solution $(C_k, N_k)$, where $C_k=A_ke^{-\mu _1 z}$ as $z\rightarrow \infty$. By Lemma 2.2 and the fact $s_k\ge s_0$ for each k, the limit of this decreasing sequence exists. Let $\bar{s}=\lim \limits _{k\rightarrow \infty }s_k$. In terms of Lemma 2.6, monotone traveling wave of system (1.5)–(1.6) exists for $s=\bar{s}$. It follows that $s_{\min }=\bar{s}\ge s_0$. Therefore, the proof of Theorem 2.1 is complete.

The following theorem gives a necessary and sufficient condition for the nonlinear selection of the minimal wave speed of system (1.5)–(1.6), which is a development of the abstract results in [31] to the system (1.2) with infinite equilibria.

Theorem 2.7

Assume that MM holds. The minimal wave speed $s_{\min }$ of (1.5)–(1.6) is nonlinearly selected if and only if there exists $s_2>s_0$ such that

$$\begin{aligned} C(z)\sim A_2e^{-\mu _2(s_2) z},\ \text {as} \ z\rightarrow \infty ,\ z=x-s_2t, \end{aligned}$$

with $A_2>0$. Furthermore, $s_2=s_{\min }$.

Proof

This result can be easily proved by Lemma 2.3. $\square$

The above result has an assumption on the wave solution which is unknown. To overcome this difficult, we shall give an easy-to-apply theorem for speed selection. We can use this general result to derive a series of explict conditions on speed selection by constructing various upper and lower solutions.

Theorem 2.8

(Linear, nonlinear selection and estimate of $s_{\min }$) Assume that MM holds.

(i). For $s_1>s_0$, if there exists a monotonic lower solution $(\underline{C},\underline{N})(x-s_1t)$ of system (1.5)–(1.6) such that $\underline{C}(z)\sim \underline{A} e^{-\mu _2 z}$ as $z\rightarrow \infty$ with $\underline{A}>0$, and $\limsup _{z\rightarrow -\infty }\underline{C}(z)<1-\frac{\gamma }{k}$, then no traveling wave solution exists for $s\in [s_0, s_1)$. Thus, we have $s_{\min }>s_0$.

(ii). For $s=s_0+\varepsilon$ where $\varepsilon$ is any small positive number, if there exists a nonnegative and monotonic upper solution $(\overline{C},\overline{N})(x-st)$ of system (1.5)–(1.6) such that $\overline{C}(z)\sim \overline{A} e^{-\mu _1 z}$ as $z\rightarrow \infty$ with $\overline{A}>0$, and $\liminf _{z\rightarrow -\infty }\overline{C}(z)\ge 1-\frac{\gamma }{k}$, then we have $s_{\min }=s_0$.

Proof

For part (i), we prove it by contradiction. For $s\in (s_0, s_1)$, we assume there exists a monotone traveling wave solution $(C,N)(x-st)$ of (1.2) with the initial conditions

$$\begin{aligned} c(0,x)=C(x),\quad n(0,x)=N(x). \end{aligned}$$

Note that $\underline{C}(x)\le C(x)$ for $x\in \mathbb {R}$ by shifting if necessary. According to equation (2.3), we have $(\underline{C},\underline{N})(x)\le (C,N)(x)$ for $x\in \mathbb {R}$. By the comparison theorem, we get

$$\begin{aligned}{} & {} \underline{C}(x-s_1t)\le C(x-st), \\{} & {} \underline{N}(x-s_1t)\le N(x-st). \end{aligned}$$

Fix $z^*=x-s_1t$ such that $\underline{C}(z^*)>0$. Since

$$\begin{aligned} C(x-st)=C(z^*+(s_1-s)t)\rightarrow 0, \ \text {as} \ t\rightarrow \infty , \end{aligned}$$

it follows that $\underline{C}(z^*)\le 0$, which is a contradiction.

For part (ii), we know (2.13) is a lower solution when $s=s_0+\varepsilon$ by Lemma 2.4. Let $\varepsilon \rightarrow 0$. Our result holds by the upper-lower solution method. Thus, the proof is complete. $\square$

The following theorem gives a necessary and sufficient criterion to distinguish between linear and nonlinear selection in terms of the value of k.

Theorem 2.9

Assume that MM holds. For fixed $\lambda$ and $\gamma$, the minimal speed $s_{\min }$ is non-decreasing with respect to k. Furthermore, there exists a finite value $k^*$ such that $s_{\min }=s_0$ for $k \in [\frac{\gamma }{\lambda }, k^*]$, and $s_{\min }>s_0$ for $k>k^*$.

Proof

Let $k_2>k_1\ge \frac{\gamma }{\lambda }$. By Theorem 2.1, then there exists a monotone traveling wave solution $C_{k_2}(z)$ of system (2.4)–(2.5) with speed $s=s_{\min }(k_2)$ for $k=k_2$. Note that $C_{k_2}(z)$ is an upper solution of the system for $k=k_1$ in terms of the comparison theorem. A lower solution is defined as (2.13). Using the upper-lower solution method, monotone traveling wave solution of the system exists for $k=k_1$ when $s=s_{\min }(k_2)$, and thus $s_{\min }(k_1)\le s_{\min }(k_2)$. Thus, $s_{\min }$ is non-decreasing with respect to k.

Now we prove that there is a finite value of k so that transition of the speed selection is realized.

Claim: If k is sufficiently large, then the minimal wave speed is nonlinearly selected, that is, $s_{\min }(k)>s_0$.

Define $\underline{C}_M(z)$ as

$$\begin{aligned} \underline{C}_M(z)=\frac{\epsilon }{1+e^{\mu _2 z}}, \end{aligned}$$

(2.17)

where $\epsilon <1-\frac{\gamma }{k}$ is small, and $\underline{N}_M(z)=N(z)$ is defined by (2.3) with $C(z)=\underline{C}_M(z)$. Substituting $\underline{C}_M(z)$ into equation (2.4) gives

$$\begin{aligned} \begin{aligned}&\underline{C}_M''+s\underline{C}_M'+\underline{C}_M[1-\lambda -\underline{C}_M+(\lambda k-\gamma )\underline{N}_M]\\&\quad =\underline{C}_M\left( 1-\frac{\underline{C}_M}{\epsilon }\right) \left\{ \mu _2^2-s\mu _2+1-\lambda +\frac{\underline{C}_M}{\epsilon }\left[ -2\mu _2^2+\frac{1-\lambda -\epsilon }{1-\frac{\underline{C}_M}{\epsilon }}+\frac{(\lambda k-\gamma )\underline{N}_M}{\left( 1-\frac{\underline{C}_M}{\epsilon }\right) \frac{\underline{C}_M}{\epsilon }}\right] \right\} \\&\quad =\frac{\underline{C}_M^2}{\epsilon }\left( 1-\frac{\underline{C}_M}{\epsilon }\right) \left\{ -2\mu _2^2+\frac{1-\lambda -\epsilon }{1-\frac{\underline{C}_M}{\epsilon }}+\frac{(\lambda -\frac{\gamma }{k})\left[ 1-\left( \frac{e^{\mu _2 z}}{e^{\mu _2 z}+1}\right) ^{\frac{\lambda k \epsilon }{s \mu _2}}\right] }{\left( 1-\frac{\underline{C}_M}{\epsilon }\right) \frac{\underline{C}_M}{\epsilon }}\right\} \\&\quad =\frac{\underline{C}_M^2}{\epsilon }\left( 1-\frac{\underline{C}_M}{\epsilon }\right) \left\{ -2\mu _2^2+\frac{1-\lambda -\epsilon }{1-y}+\frac{(\lambda -\frac{\gamma }{k})[1-(1-y)^{\hat{M}}]}{y(1-y)}\right\} \\&\quad =:\frac{\underline{C}_M^2}{\epsilon }\left( 1-\frac{\underline{C}_M}{\epsilon }\right) Y(y),\\ \end{aligned} \end{aligned}$$

(2.18)

where $y=\frac{1}{1+e^{\mu _2 z}}$ and $\hat{M}=\frac{\lambda k \epsilon }{s \mu _2}$. Since k is sufficiently large, it follows that $\hat{M}$ is large. Note that

$$\begin{aligned} \begin{aligned} Y'(y)&=\frac{(1-\lambda -\epsilon )y^2+(\lambda -\frac{\gamma }{k})\hat{M}y(1-y)^{\hat{M}}-(\lambda -\frac{\gamma }{k})(1-2y) [1-(1-y)^{\hat{M}}]}{y^2(1-y)^2}\\&=:\frac{Y_1(y)}{y^2(1-y)^2}. \end{aligned} \end{aligned}$$

By calculation, $Y_1'(y)\ge 0$ for $y\in [0,1]$, and thus we get $Y_1(y)\ge Y_1(0)=0$. It follows that $Y'(y)\ge 0$. For $y\rightarrow 0$, we have

$$\begin{aligned} \lim \limits _{y\rightarrow 0}Y(y)=-2\mu _2^2+1-\lambda -\epsilon +\hat{M}\left( \lambda -\frac{\gamma }{k}\right) \ge 0, \end{aligned}$$

due to the fact that $\hat{M}$ is sufficiently large, and $\mu _2\sim \sqrt{1-\lambda }$ when $s=s_0+\epsilon _1$ for small $\epsilon _1$. Clearly, $Y(y)\ge 0$. Therefore, $\underline{C}_M''+s\underline{C}_M'+\underline{C}_M[1-\lambda -\underline{C}_M+(\lambda k-\gamma )\underline{N}_M]\ge 0$ for all $z\in \mathbb {R}$. Thus, $\underline{C}_M(z)$ is a lower solution of (2.4) when k is sufficiently large. By Theorem 2.8, the above claim is true.

Note that C-equation of (2.4) becomes the Fisher-KPP equation when $k=\frac{\gamma }{\lambda }$. It gives $s_{\min }=s_0$, which can be seen in [24, 25]. By using the fact $s_{\min }$ is non-decreasing with respect to k and the above claim, then there exists finite $k^*$ such that $s_{\min }=s_0$ for $k \in [\frac{\gamma }{\lambda }, k^*]$, and $s_{\min }>s_0$ for $k>k^*$. Therefore, the proof is complete. $\square$

By an analogous argument as in the proof of Theorem 2.9, we shall derive the following theorem for the monotonicity of $s_{\min }$ with respect to $\gamma$.

Theorem 2.10

Assume that MM holds. For fixed $\lambda$ and k, $s_{\min }$ is non-increasing with respect to $\gamma$.

Proof

Let $\gamma _1<\gamma _2\le \lambda k$. By Theorem 2.1, there exists a monotone traveling wave solution $C_{\gamma _1}(x-s_{\min }(\gamma _1)t)$ satisfying system (2.4)–(2.5) with $\gamma =\gamma _1$. Note that $C_{\gamma _1}(x-s_{\min }(\gamma _1)t)$ is an upper solution of the system with $\gamma =\gamma _2$. A lower solution is defined as in (2.13). Then there exists a traveling wave solution of the system for $\gamma =\gamma _2$ when $s=s_{\min }(\gamma _1)$. It follows that $s_{\min }(\gamma _2)\le s_{\min }(\gamma _1)$. Therefore, $s_{\min }$ is non-increasing with respect to $\gamma$. $\square$

Next, we shall give some estimates about $k^*$ based on Theorem 2.8.

Theorem 2.11

Assume that MM holds. If parameters $\lambda$, $\gamma$ and k satisfy

$$\begin{aligned} \left\{ \begin{aligned}&k<\gamma +\frac{2(1-\lambda )}{\lambda },\\&(3\lambda -2)k<\gamma ,\\ \end{aligned}\right. \end{aligned}$$

(2.19)

then $s_{\min }=s_0$.

Proof

We can prove it by Theorem 2.8. Define a pair of continuous function $(\bar{C_1}, \bar{N_1})(z)$ as

$$\begin{aligned} \bar{C_1}(z)=\frac{1-\frac{\gamma }{k}}{1+e^{\mu _1 z}}, \ \bar{N_1}(z)= \left\{ \begin{aligned}&\frac{1}{k},\quad \quad z\le z_2,\\&\eta _1\bar{C}_1,\ \ z>z_2,\\ \end{aligned}\right. \end{aligned}$$

(2.20)

where $\eta _1=\frac{2(1-\lambda )}{(\lambda k-\gamma )(1-\frac{\gamma }{k})}>\frac{1}{k(1-\frac{\gamma }{k})}$ by (2.19), and $\eta _1\bar{C}_1(z_2)=\frac{1}{k}$. Let $s=s_0+\varepsilon _2$, where $\varepsilon _2$ is sufficiently small. We have $\mu _1\sim \sqrt{1-\lambda }$. It follows that

$$\begin{aligned}{} & {} \begin{aligned}&\bar{C}_1''+s\bar{C}_1'+\bar{C}_1[1-\lambda -\bar{C}_1+(\lambda k-\gamma )\bar{N}_1]\\&\quad =\bar{C}_1\left( 1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\right) \left\{ \mu _1^2-s\mu _1+1-\lambda +\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\left[ -2\mu _1^2+\frac{(\lambda k-\gamma ) \bar{N}_1-\left( \frac{\gamma }{k}-\lambda \right) \frac{\bar{C}_1}{1-\frac{\gamma }{k}}}{\left( 1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\right) \frac{\bar{C}_1}{1-\frac{\gamma }{k}}}\right] \right\} \\&\quad =\left\{ \begin{aligned}&\frac{\bar{C}_1^2}{1-\frac{\gamma }{k}}\left( 1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\right) \left[ -2\mu _1^2+\ \frac{\left( \lambda -\frac{\gamma }{k}\right) \left( 1-\frac{\gamma }{k}\right) }{\bar{C}_1}\right] \\&\frac{\bar{C}_1^2}{1-\frac{\gamma }{k}}\left( 1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\right) \left[ -2\mu _1^2+\frac{\frac{\gamma }{k}-\lambda +(\lambda k-\gamma )\left( 1-\frac{\gamma }{k}\right) \eta _1 }{\left( 1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\right) }\right] \end{aligned}\right. \\&\quad \le \left\{ \begin{aligned}&\frac{\bar{C}_1^2}{1-\frac{\gamma }{k}}\left( 1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\right) \left[ -2\mu _1^2+(\lambda k-\gamma )\left( 1-\frac{\gamma }{k}\right) \eta _1\right] \le 0,\ z\le z_2,\\&\frac{\bar{C}_1^2}{1-\frac{\gamma }{k}}\left( 1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\right) \left[ -2\mu _1^2+(\lambda k-\gamma )\left( 1-\frac{\gamma }{k}\right) \eta _1\right] \le 0,\ z>z_2,\\ \end{aligned}\right. \end{aligned} \\{} & {} s\bar{N}_1'+\lambda \bar{C}_1(1-k\bar{N}_1)=0,\ \ \ \text {for}\ z\le z_2, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&s\bar{N}_1'+\lambda \bar{C}_1(1-k\bar{N}_1)\\&\quad =\bar{C}_1\left( 1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\right) \left[ -s\eta _1\mu _1+\frac{\lambda (1-k \eta _1 \bar{C}_1)}{1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}}\right] \\&\quad \le \bar{C}_1\left( 1-\frac{\bar{C}_1}{1-\frac{\gamma }{k}}\right) [-s\eta _1\mu _1+\lambda ]\le 0, \ \ \text {for}\ z> z_2, \end{aligned} \end{aligned}$$

by using (2.19). Therefore, $(\bar{C_1}, \bar{N_1})(z)$ is an upper solution of (1.5)–(1.6). By Theorem 2.8, we have $s_{\min }=s_0$ when (2.19) holds. The proof is complete. $\square$

Theorem 2.12

Assume that MM holds. If $\lambda$, $\gamma$ and k satisfy

$$\begin{aligned} k>\frac{2(1-\lambda )+\lambda \gamma }{\lambda (2\lambda -1)}>0, \end{aligned}$$

(2.21)

then $s_{\min }>s_0$.

Proof

We shall use Theorem 2.8 to prove this theorem. The key point is to find a suitable lower solution satisfying all conditions in Theorem 2.8 (i). Define a pair of continuous function $(\underline{C}_1, \underline{N}_1)(z)$ as

$$\begin{aligned} \underline{C}_1(z)=\frac{\underline{k}_1}{1+e^{\mu _2 z}}, \quad \quad \underline{N}_1(z)=\frac{\underline{C}_1}{k\underline{k}_1}, \end{aligned}$$

(2.22)

where $0<\underline{k}_1<1-\frac{\gamma }{k}$. We need to prove $(\underline{C}_1, \underline{N}_1)(z)$ is the lower solution of system (1.5)–(1.6). According to (2.21), $\underline{k}_1$ can be chosen such that

$$\begin{aligned} \frac{2(1-\lambda )}{\lambda k}<\underline{k}_1<1-\frac{\gamma }{k}-2(1-\lambda ). \end{aligned}$$

(2.23)

We have

$$\begin{aligned} \begin{aligned}&\underline{C}_1''+s\underline{C}_1'+\underline{C}_1[1-\lambda -\underline{C}_1+(\lambda k-\gamma ) \underline{N}_1]\\&\quad =\underline{C}_1\left( 1-\frac{\underline{C}_1}{\underline{k}_1}\right) \left\{ \mu _2^2-s\mu _2+1-\lambda +\frac{\underline{C}_1}{\underline{k}_1}\left[ -2\mu _2^2+\frac{(1-\lambda )\frac{\underline{C}_1}{\underline{k}_1}-\underline{C}_1+(\lambda k-\gamma )\underline{N}_1}{\left( 1-\frac{\underline{C}_1}{\underline{k}_1}\right) \frac{\underline{C}_1}{\underline{k}_1}}\right] \right\} \\&\quad \ge \frac{\underline{C}_1^2}{\underline{k}_1}\left( 1-\frac{\underline{C}_1}{\underline{k}_1}\right) \left[ -2\mu _2^2+1-\frac{\gamma }{k}-\underline{k}_1\right] \ge 0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&s\underline{N}_1'+\lambda \underline{C}_1(1-k\underline{N}_1)\\&\quad =\underline{C}_1\left( 1-\frac{\underline{C}_1}{\underline{k}_1}\right) \left[ -\frac{s\mu _2}{k\underline{k}_1}+\lambda \right] \ge 0, \end{aligned} \end{aligned}$$

by using (2.23), and $\mu _2\sim \sqrt{1-\lambda }$ for $s=s_0+\varepsilon _3$, where $\varepsilon _3>0$ is small. Thus, $(\underline{C}_1, \underline{N}_1)(z)$ is a lower solution of (1.5)–(1.6). It follows that $s_{\min }>s_0$ by Theorem 2.8. Thus, the proof is complete. $\square$

The following theorem gives details of the decay rate of the minimal traveling wave as k varies. We will show that the decay rate of the minimal traveling wave goes from $ze^{-\mu _1(s_0) z}$ to $e^{-\mu _1(s_0) z}$ as $k\rightarrow k^*$ from below. Wu, Xiao and Zhou [51, Theorem 1.2 and Theorem 1.5] established a similar result for the scalar reaction-diffusion equations and the two-species Lotka-Volterra competition systems by constructing a very complex upper solution, which is relatively difficult to be applied to other models. For this model of precursor and differentiated cells, we develop a new technique to prove this decay behavior.

Theorem 2.13

Assume that MM holds. Let $k^*$ be the turning point for the minimal speed selection, that is, $s_{\min }=s_0$ for $k \in [\frac{\gamma }{\lambda }, k^*]$, and $s_{\min }>s_0$ for $k>k^*$. Then the minimal traveling wave solution C(z) of system (2.4)–(2.5) satisfies:

• (1) if $k\in [\frac{\gamma }{\lambda },k^*)$, then $C(z)\sim Aze^{-\mu _1(s_0) z}$ as $z\rightarrow \infty$, where $A>0$;

• (2) if $k=k^*$, then $C(z)\sim Be^{-\mu _1(s_0) z}$ as $z\rightarrow \infty$, where $B>0$;

• (3) if $k>k^*$, then $C(z)\sim Be^{-\mu _2(s_{\min }) z}$ as $z\rightarrow \infty$, where $B>0$.

Proof

For $k \in [\frac{\gamma }{\lambda }, k^*]$, we have $s_{\min }=s_0$ by Theorem 2.9, and then

$$\begin{aligned} C(z)\sim Aze^{-\mu _1(s_0) z}+Be^{-\mu _1(s_0) z}, \ \ \text {as} \ z\rightarrow \infty , \end{aligned}$$

(2.24)

where $A> 0$, or $B>0$ if $A=0$, see [4]. We first prove result (2) by using a similar method to Lemma 2.3 (i). Denote the minimal traveling wave solution C(z) by $C^*(z)$ for $k=k^*$. Assume $C^*(z)\sim Aze^{-\mu _1(s_0) z}$ as $z\rightarrow \infty$ with $A>0$. Clearly, $C^*(z)$ satisfies

$$\begin{aligned} {C^*}''+s_0{C^*}'+G_1(k^*, s_0, C^*)=0, \end{aligned}$$

(2.25)

where

$$\begin{aligned} G_1(k^*,s_0, C^*)=C^*\left\{ 1-\lambda -C^*+\left( \lambda -\frac{\gamma }{k^*}\right) \left[ 1-e^{-k^*\int _{z}^{\infty }\frac{\lambda C^*(\tau )}{s_0}d\tau }\right] \right\} . \end{aligned}$$

Let $k_\varepsilon =k^*+\varepsilon$, where $\varepsilon$ is sufficiently small to be determined. Let $C_\varepsilon (z)=C^*(z)+W_2(z)$, where $W_2(z)\in B_0=\{u\in C(-\infty , \infty ), u(\pm \infty )=0\}$. If we can prove the existence of $W_2$ such that $C_\varepsilon (z)$ is a solution of (2.4) with $k=k_\varepsilon$ for $s=s_0$, then it contradicts the definition of $k^*$, and then result (2) holds. Suppose that $C_\varepsilon (z)$ is a solution of (2.4) with $k=k_\varepsilon$ and $s=s_0$. It means

$$\begin{aligned} C_\varepsilon ''+s_0C_\varepsilon '+G_1(k_\varepsilon , s_0, C_\varepsilon )=0. \end{aligned}$$

(2.26)

By (2.25) and (2.26), $W_2$ should satisfy

$$\begin{aligned} W_2''+s_0W_2'-\alpha W_2=-(P_0+P_{\varepsilon }+P_h), \end{aligned}$$

(2.27)

where

$$\begin{aligned} \begin{aligned}&P_0=\left\{ \alpha +1-\lambda -2C^*+\left( \lambda -\frac{\gamma }{k^*}\right) \left[ 1-e^{-k^*\int _{z}^{\infty }\frac{\lambda C^*(\tau )}{s_0}d\tau }\right] \right\} W_2\\&\qquad +C^*(\lambda k^*-\gamma )e^{-k^*\int _{z}^{\infty }\frac{\lambda C^*(\tau )}{s_0}d\tau }\int _{z}^{\infty }\frac{\lambda W_2(\tau )}{s_0}d\tau ;\\&P_\varepsilon =G_1(k_\varepsilon , s_0, C_\varepsilon )-G_1(k^*, s_0, C_\varepsilon );\\&P_h=-W_2^2+\left( \lambda -\frac{\gamma }{k^*}\right) e^{-k^*\int _{z}^{\infty }\frac{\lambda C^*(\tau )}{s_0}d\tau }(C^*\phi _1-W_2\phi _2);\\&\phi _1=1-e^{-k^*\int _{z}^{\infty }\frac{\lambda W_2(\tau )}{s_0}d\tau }-k^*\int _{z}^{\infty }\frac{\lambda W_2(\tau )}{s_0}d\tau ;\\&\phi _2=e^{-k^*\int _{z}^{\infty }\frac{\lambda W_2(\tau )}{s_0}d\tau }-1. \end{aligned} \end{aligned}$$

Therefore, for $s=s_0$, we have

$$\begin{aligned} W_2=T_1(P_0)+T_1(P_\varepsilon )+T_1(P_h), \end{aligned}$$

(2.28)

where $T_1$ is defined in (2.8). Clearly, $T_1(P_\varepsilon )$ is of $O(\varepsilon )$ when $\varepsilon \rightarrow 0$. Define

$$\begin{aligned} T_2(W_2)=T_1(P_0(W_2)). \end{aligned}$$

The linear operator $T_2$ is compact and strongly positive, and it has a simple principle eigenvalue $r=1$ with the associated positive eigenfunction as $-{C^*}'$. In order to remove this eigenfunction from $B_0$, we define the following weighted functional space

$$\begin{aligned} \mathcal {H}_1=\{h(\xi )\in B_0: he^{\mu _1(s_0)\xi }=o(1) \quad \textrm{as} \quad \xi \rightarrow \infty \}. \end{aligned}$$

Since $-{C^*}'$ is not in the space $\mathcal {H}_1$, then $T_2$ does not have eigenvalue $\lambda =1$ for $W_2$ in $\mathcal {H}_1$, that is, $I-T_2$ has a bounded inverse in $\mathcal {H}_1$, where I is the identity operator. Then there exists a solution $W_2$ of equation (2.28) for any $\varepsilon \in [0, \varepsilon _0)$, where $\varepsilon _0$ is a small positive number. It follows that there exists a $\varepsilon _0$ so that the solution $C_\varepsilon$ of (2.4)–(2.5) exists with $s=s_0$ for $k=k_\varepsilon =k^*+\varepsilon <k^*+\varepsilon _0$, which contradicts the conclusion in Theorem 2.9. Therefore, result (2) is true.

Result (1) can be proved by a similar method as in [51, Theorem 1.2]. Assume that there exists $k\in [\frac{\gamma }{\lambda }, k^*)$ such that the minimal traveling wave satisfies

$$\begin{aligned} C_k(z)\sim B_1e^{-\mu _1(s_0)z},\ \text {as} \ z\rightarrow \infty ,\ B_1>0. \end{aligned}$$

(2.29)

By result (2), for $k=k^*$, the minimal traveling wave $C_{k^*}$ satisfies

$$\begin{aligned} C_{k^*}(z)\sim Be^{-\mu _1(s_0)z}, \ \text {as} \ z\rightarrow \infty ,\ B>0. \end{aligned}$$

(2.30)

When $z\rightarrow -\infty$, we get

$$\begin{aligned}{} & {} C_k\sim 1-\frac{\gamma }{k}-b_1e^{\mu (k) z},\ \text {as} \ z\rightarrow -\infty , b_1>0, \\{} & {} C_{k^*}\sim 1-\frac{\gamma }{k^*}-be^{\mu (k^*) z},\ \text {as} \ z\rightarrow -\infty , b>0, \end{aligned}$$

where $\mu (k^*)>\mu (k)>0$. Then there exists an $L>0$ such that $C_{k^*}(z-L)>C_k(z)$ for all $z\in \mathbb {R}$. Define

$$\begin{aligned} L^*:=\inf \{L\in \mathbb {R}| C_{k^*}(z-L)\ge C_k(z), \ \forall z\in \mathbb {R}\}. \end{aligned}$$

If there exists $z_1\in \mathbb {R}$ such that $C_{k^*}(z_1-L^*)=C_k(z_1)$, then by the strong maximum principle, $C_{k^*}(z-L^*)=C_k(z)$ for all $z\in \mathbb {R}$, which contradicts the fact that $C_{k^*}(z-L^*)$ and $C_k(z)$ satisfy different equations. Thus, $C_{k^*}(z-L^*)>C_k(z)$ for all $z\in \mathbb {R}$.

We then have the following claim.

Claim: $\lim \limits _{z\rightarrow \infty } \frac{C_{k^*}(z-L^*)}{C_k(z)}=1$.

Assume $\lim \limits _{z\rightarrow \infty } \frac{C_{k^*}(z-L^*)}{C_k(z)}>1$. By $C_{k^*}(z-L^*)>C_k(z)$ for all $z\in \mathbb {R}$, then there exists a sufficiently small $\varepsilon _1>0$ such that $C_{k^*}(z-(L^*-\varepsilon _1))\ge C_k(z)$ for $z\in \mathbb {R}$, which contradicts the definition of $L^*$. Therefore, the claim is true.

Thanks to the above claim, we have $Be^{\mu _1(s_0)L^*}=B_1$. Let $W(z)=C_{k^*}(z-L^*)-C_k(z)$. Then W(z) satisfies

$$\begin{aligned} W''+s_0W'+G_1(k^*, s_0, C_{k^*}(z-L^*))-G_1(k, s_0, C_k(z))=0. \end{aligned}$$

(2.31)

Note that

$$\begin{aligned} W(z)\sim (Dz+E)e^{-\mu _1(s_0) z}, \ \text {as} \ z\rightarrow \infty , \end{aligned}$$

where $D\ge 0$, or $E>0$ if $D=0$. By (2.29), (2.30), and $Be^{\mu _1(s_0)L^*}=B_1$, we know $E=0$ and $D=0$. This is a contradiction. Thus, result (1) holds.

Result (3) can be derived by Theorems 2.7 and 2.9. Therefore, the proof is complete. $\square$

3 Traveling Waves and Linear Selection in the Case MN

In this section, we consider system (1.2) in the case MN. Clearly, system (1.2) is non-monotonic, and the equilibrium $e_\beta$ is stable, with $e_0$ unstable.

Define $\Omega =\{u:u \text { is a bounded and uniformly continuous function from }\mathbb {R}\text { to }\mathbb {R}^2\}$. Let $\mu >0$ be a constant such that $2\mu <-\beta _1$, where $\beta _1$ is defined in (2.7). Further define a weighted functional space from $\Omega$ as $B_\mu$=$\{ u\in \Omega : \sup \limits _{z\in \mathbb {R}}|u(z)|e^{-\mu |z|}<\infty \}$, equipped with $|u|_\mu =\sup \limits _{z\in \mathbb {R}}|u(z)|e^{-\mu |z|}$. Then $B_\mu$ is a Banach space with the norm $|\cdot |_\mu$.

We shall apply the method of upper and lower solutions to prove the existence of non-negative solutions of (1.5)–(1.6) under MN. The upper and lower solutions are crossly-defined as follows.

Definition 3.1

Assume that MN holds. Two pair of continuous functions $(\bar{C},\bar{N})$ and $(\underline{C}, \underline{N})$ which are twice differentiable on $\mathbb {R}$ except at finite number of points $z_i, i=1,..., n,$ and satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\bar{C}''+s\bar{C}'+\bar{C}[1-\lambda -\bar{C}+(\lambda k-\gamma )\underline{N}]\le 0, \\&s\bar{N}'+\lambda \bar{C}(1-k\bar{N})\le 0, \\&\underline{C}''+s\underline{C}'+\underline{C}[1-\lambda -\underline{C}+(\lambda k-\gamma )\bar{N}]\ge 0,\\&s\underline{N}'+\lambda \underline{C}(1-k\underline{N})\ge 0 \end{aligned} \right. \end{aligned}$$

(3.1)

for $z\ne z_i$, with $\bar{C}'(z_i^+)\le \bar{C}'(z_i^-)$ and $\underline{C}'(z_i^-)\le \bar{C}'(z_i^+)$, are upper and lower solutions of (1.5)

For any $s>s_0$, we shall construct

$$\begin{aligned} \begin{aligned}&\bar{C}=\min \{1-\lambda , (1-\lambda )e^{-\mu _1 z}\}, \quad \quad \bar{N}=\left\{ \begin{aligned}&\frac{1}{k},\quad \quad z\le z_3,\\&Me^{-\mu _1 z},\ \ z>z_3,\\ \end{aligned}\right. \\&\underline{C}=\left\{ \begin{aligned}&\epsilon ,\quad \quad \quad \quad \quad \quad z\le z_4,\\&e^{-\mu _1 z}-qe^{-\eta \mu _1 z},\ \ z>z_4,\\ \end{aligned}\right. \quad \quad \underline{N}=0, \end{aligned} \end{aligned}$$

(3.2)

where parameters $M, q, \eta$ satisfy

$$\begin{aligned} \begin{aligned}&M\ge \max \{\lambda , \frac{1}{k}\},\\&q=\frac{1-(\lambda k-\gamma )M}{-(\eta ^2\mu _1^2-s\mu _1\eta +1-\lambda )}+1,\\&\mu _1<\eta \mu _1\le \min \{\mu _2, 2\mu _1, \left( \frac{\ln q}{\ln (Mk)}+1\right) \mu _1\}, \end{aligned} \end{aligned}$$

(3.3)

$\mu _1$, $\mu _2$ are defined in (2.2), $Me^{-\mu _1 z_3}=\frac{1}{k}$, $e^{-\mu _1 z_4}-qe^{-\eta \mu _1 z_4}=\epsilon$, and $\epsilon$ is sufficiently small. Obviously $\eta >1$.

We have the following lemma.

Lemma 3.2

Assume that MN holds. Then $(\bar{C},\bar{N})$ and $(\underline{C}, \underline{N})$, defined in (3.2), are upper and lower solutions of (1.5), respectively.

Proof

When $\bar{C}=1-\lambda$, it follows that $\bar{C}''+s\bar{C}'+\bar{C}[1-\lambda -\bar{C}+(\lambda k-\gamma )\underline{N}]=0$. When $\bar{C}=(1-\lambda )e^{-\mu _1 z}$, this yields $\bar{C}''+s\bar{C}'+\bar{C}[1-\lambda -\bar{C}+(\lambda k-\gamma )\underline{N}]=-\bar{C}^2\le 0$.

For $\bar{N}=\frac{1}{k}$, we get $s\bar{N}'+\lambda \bar{C}(1-k\bar{N})=0$. For $\bar{N}=Me^{-\mu _1 z}$, that is, $z\ge z_3\ge 0$ by (3.3), it implies $\bar{C}=(1-\lambda )e^{-\mu _1 z}$. Thus,

$$\begin{aligned} s\bar{N}'+\lambda \bar{C}(1-k\bar{N})=e^{-\mu _1 z}[-sM\mu _1+\lambda (1-\lambda )(1-k\bar{N})]\le e^{-\mu _1 z}[-M(1-\lambda )+\lambda (1-\lambda )] \le 0, \end{aligned}$$

thanks to the facts $\bar{N}\in [0,\frac{1}{k}]$ and (3.3).

For $\underline{C}=\epsilon$, then we have

$$\begin{aligned} \underline{C}''+s\underline{C}'+\underline{C}[1-\lambda -\underline{C}+(\lambda k-\gamma )\bar{N}]\ge \epsilon \left[ 1-\frac{\gamma }{k}-\epsilon \right] \ge 0. \end{aligned}$$

For $\underline{C}=e^{-\mu _1 z}-qe^{-\eta \mu _1 z}\ge \epsilon >0$, then $z> z_3\ge 0$ by (3.3). It follows that $\bar{N}=Me^{-\mu _1 z}$. Therefore, by (3.3), we have

$$\begin{aligned} \begin{aligned}&\underline{C}''+s\underline{C}'+\underline{C}[1-\lambda -\underline{C}+(\lambda k-\gamma )\bar{N}]\\&\quad =-qe^{-\eta \mu _1 z}[\eta ^2\mu _1^2-s\mu _1\eta +1-\lambda ]+(e^{-\mu _1 z}-qe^{-\eta \mu _1 z})[-e^{-\mu _1 z}+qe^{-\eta \mu _1 z}\\&\qquad +(\lambda k-\gamma )Me^{-u_{1} z}]\\&\quad \ge e^{-2\mu _1 z}\{-q[\eta ^2\mu _1^2-s\mu _1\eta +1-\lambda ]-1+(\lambda k-\gamma )M\}\ge 0. \end{aligned} \end{aligned}$$

Finally for $\underline{N}=0$, we have $s\underline{N}'+\lambda \underline{C}(1-k\underline{N})=\lambda \underline{C}\ge 0$.

Hence, $(\bar{C},\bar{N})$ and $(\underline{C}, \underline{N})$ are upper and lower solutions of (1.5) according to the Definition 3.1. $\square$

Lemma 3.3

Assume that MN holds. With the choice of upper and lower solutions $(\bar{C},\bar{N})$ and $(\underline{C}, \underline{N})$ as in (3.2) satisfying $(\bar{C}, \bar{N})\ge (\underline{C}, \underline{N})$,

it follows that (1.5) has a positive solution (C, N) such that

$$\begin{aligned} (\underline{C},\underline{N})\le (C, N)\le (\bar{C}, \bar{N}). \end{aligned}$$

(3.4)

Proof

Define $\Gamma =\{(C,N)\in \Omega : (\underline{C},\underline{N})\le (C, N)\le (\bar{C}, \bar{N})\}$. Note that $\Gamma$ is nonempty, convex, bounded and closed. Let $P_1(C, N)$ be defined as $T_1$ in (2.8), and $P_2(C)=\frac{1}{k}\big [1-e^{-k\int _{z}^{\infty }\frac{\lambda C(t)}{s}dt}\big ]$ as (2.3). We first want to prove $P=(P_1, P_2): \Gamma \rightarrow \Gamma$, that is,

$$\begin{aligned} \left\{ \begin{aligned}&\underline{C}(z)\le P_1(\underline{C},\bar{N})(z)\le P_1(\bar{C}, \underline{N})(z)\le \bar{C}(z), \ z\in \mathbb {R},\\&\underline{N}(z)\le P_2(\underline{C})(z)\le P_2(\bar{C})(z)\le \bar{N}(z), \ z\in \mathbb {R}.\\ \end{aligned} \right. \end{aligned}$$

(3.5)

Indeed, it follows first that

$$\begin{aligned} \begin{aligned} P_1(\underline{C}, \bar{N})(z)&=\frac{1}{\beta _2-\beta _1}\bigg \{\int _{-\infty }^{z}e^{\beta _1(z-t)}F(\underline{C}, \bar{N})(t)dt+\int _{z}^{\infty }e^{\beta _2(z-t)}F(\underline{C},\bar{N})(t)dt\bigg \}\\&\ge \frac{1}{\beta _2-\beta _1}\bigg \{\int _{-\infty }^{z}e^{\beta _1(z-t)}+\int _{z}^{\infty }e^{\beta _2(z-t)}\bigg \}(\alpha \underline{C}(t)-s\underline{C}'(t)-\underline{C}''(t))dt\\&=\underline{C}(z)+\frac{1}{\beta _2-\beta _1}\bigg [\sum _{j=1}^{m}e^{\lambda _1(z-z_j)}(\underline{C}'(z_j^+)-\underline{C}'(z_j^-))\\&+\sum _{j=m+1}^{n}e^{\lambda _2(z-z_j)}(\underline{C}'(z_j^+)-\underline{C}'(z_j^-))\bigg ]\\&\ge \underline{C}(z) \end{aligned} \end{aligned}$$

(3.6)

and

$$\begin{aligned} P_2(\bar{C})=\frac{1}{k}\left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda C(t)}{s}dt}\right] \le \frac{1}{k}\left[ 1-e^{-k\int _{z}^{\infty }\frac{-\bar{N}'(t)}{1-k\bar{N}(t)}dt}\right] =\bar{N}(z). \end{aligned}$$

(3.7)

Note that $P_2(C)$ is non-decreasing with respect to C, F(C, N) is non-decreasing with respect to C, and F(C, N) is non-increasing with respect to N. The remainder of (3.5) can be proved in a similar way.

For any $\Phi =(C_1, N_1)\in \Gamma$ and $\Psi =(C_2, N_2)\in \Gamma$, we get

$$\begin{aligned} \begin{aligned}&|P_1(C_1, N_1)(z)-P_1(C_2,N_2)(z)|e^{-\mu |z|}\\&\quad =\bigg |\frac{e^{-\mu |z|}}{\beta _2-\beta _1}\bigg \{\int _{-\infty }^{z}e^{\beta _1(z-t)}+\int _{z}^{\infty }e^{\beta _2(z-t)}\bigg \}[F(C_1, N_1)(t)-F(C_2, N_2)(t)]dt\bigg |\\&\quad \le \frac{Re^{-\mu |z|}}{\beta _2-\beta _1}\bigg \{\int _{-\infty }^{z}e^{\beta _1(z-t)}+\int _{z}^{\infty }e^{\beta _2(z-t)}\bigg \}[|C_1(t)-C_2(t)|+|N_1(t)-N_2(t)|]dt\\&\quad \le \frac{2R|\Phi -\Psi |_{\mu }}{\beta _2-\beta _1}\bigg \{\int _{-\infty }^{z}e^{\beta _1(z-t)}+\int _{z}^{\infty }e^{\beta _2(z-t)}\bigg \}e^{\mu |z-t|}dt\\&\quad =\frac{2R|\Phi -\Psi |_{\mu }}{\beta _2-\beta _1}\left[ \frac{1}{\beta _2-\mu }-\frac{1}{\beta _1+\mu }\right] , \end{aligned} \end{aligned}$$

(3.8)

where $R=\alpha +1-\lambda +(2+\gamma -\lambda k)\max \{\sup _{z\in \mathbb {R}}C_1(z), \sup _{z\in \mathbb {R}}C_2(z)\}$. It implies that

$$\begin{aligned} \sup _{z\in \mathbb {R}}|P_1(C_1, N_1)(z)-P_1(C_2,N_2)(z)|e^{-\mu |z|}\le \frac{2R|\Phi -\Psi |_{\mu }}{\beta _2-\beta _1}\left[ \frac{1}{\beta _2-\mu }-\frac{1}{\beta _1+\mu }\right] . \end{aligned}$$

Thus, $P_1(C,N)$ is continuous with respect to the decay norm $|\cdot |_\mu$. By the expression of $P_2(C)$, $P_2$ is also continuous. By (3.5), it follows

$$\begin{aligned} \begin{aligned} \bigg |\frac{d[P_1(C_1, N_1)(z)]}{dz}\bigg |&=\frac{1}{\beta _2-\beta _1}\bigg |\bigg \{\int _{-\infty }^{z}\beta _1e^{\beta _1(z-t)}+\int _{z}^{\infty }\beta _2e^{\beta _2(z-t)}\bigg \}[F(C_1, N_1)(t)]dt\bigg |\\&\le \frac{\beta _2}{\beta _2-\beta _1}\bigg |\bigg \{\int _{-\infty }^{z}e^{\beta _1(z-t)}+\int _{z}^{\infty }e^{\beta _2(z-t)}\bigg \}[F(C_1, N_1)(t)]dt\bigg |\\&\le \beta _2< \infty , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \bigg |\frac{d[P_2(C_1, N_1)(z)]}{dz}\bigg |=|-\frac{\lambda }{s}C_1(1-kN_1)|<\infty . \end{aligned}$$

This means that $P_1(C_1, N_1)(z)$ and $P_2(C_1, N_1)(z)$ are equicontinuous. Similar to the analysis of [26, Lemma 3.2] and [27, Lemma 3.6], there is a positive solution (C, N)(z) of (1.5) satisfying (3.4) by the Ascoli-Arzela theorem and the schauder fixed point theorem. Thus, the proof is complete. $\square$

Theorem 3.4

Assume that MN holds. For any $s>s_0$, system (1.5)–(1.6) has a solution (C(z), N(z)) such that

$$\begin{aligned} C(z)>0, \quad N(z)>0 \ \text {for all} \ z\in \mathbb {R}. \end{aligned}$$

Proof

The existence of (C(z), N(z)) satisfying (1.5) is guaranteed by Lemma 3.3. Due to $(C(z), N(z))\in \Gamma$ is a fixed point of the operator P, we have

$$\begin{aligned} \begin{aligned} C(z)=P_1(C,N)(z)&\ge P_1(\underline{C}, \bar{N})\\&=\frac{1}{\beta _2-\beta _1}\bigg \{\int _{-\infty }^{z}e^{\beta _1(z-t)}F(\underline{C}, \bar{N})(t)dt+\int _{z}^{\infty }e^{\beta _2(z-t)}F(\underline{C},\bar{N})(t)dt\bigg \}\\&>0, \quad z\in \mathbb {R}, \end{aligned} \end{aligned}$$

where $\underline{C}$ and $\bar{N}$ are defined in (3.2). The expression (2.3) of N(z) implies $N(z)>0$ for all $z\in \mathbb {R}$. It only needs to prove the boundary condition (1.6). By (3.2), we have $\lim \limits _{z\rightarrow \infty }(C, N)(z)=(0,0)$. Note that $\liminf _{z\rightarrow -\infty }C(z)> 0$ because of the choice of $\underline{C}$. Then by (2.3), we have $\lim \limits _{z\rightarrow -\infty }N(z)=\frac{1}{k}$. Define

$$\begin{aligned} \bar{u}=\limsup _{z\rightarrow -\infty } C(z), \quad \underline{u}=\liminf _{z\rightarrow -\infty } C(z) \end{aligned}$$

By taking lim sup on both sides of the integral equation (2.8), we obtain

$$\begin{aligned} \bar{u}\le \frac{\alpha \bar{u}+\bar{u}\big [1-\lambda -\bar{u}+(\lambda k-\gamma )\frac{1}{k}\big ]}{\alpha }. \end{aligned}$$

(3.9)

Similarly, by taking lim inf of (2.8), we have

$$\begin{aligned} \underline{u}\ge \frac{\alpha \underline{u}+\underline{u}\big [1-\lambda -\underline{u}+(\lambda k-\gamma )\frac{1}{k}\big ]}{\alpha } \end{aligned}$$

(3.10)

It follows from (3.9) and (3.10) that

$$\begin{aligned} \bar{u}=\underline{u}=1-\frac{\gamma }{k}. \end{aligned}$$

Thus, $C(-\infty )=1-\frac{\gamma }{k}$. Therefore, the proof is complete. $\square$

Theorem 3.5

Assume that MN holds. For $s=s_0$, system (1.5)–(1.6) has a non-negative solution (C(z), N(z)). This means that the minimal wave speed of system (1.5)–(1.6) is linearly selected.

Proof

Let $\{s^n\}_{n\in \mathbb {N}}$ be a decreasing sequence with $s^n\rightarrow s_0$ as $n\rightarrow \infty$, where $s^0=2s_0$. By Theorem 3.4, system (1.5)–(1.6) has a positive solution $(C^n(z), N^n(z))$ for all $s=s^n$, where $N^n(z)$ satisfies (2.3) with $C(z)=C^n(z)$. Choosing $m>1$ so that we can fix $C^n(0)=\frac{1}{2m}(1-\frac{\gamma }{k})$ for each n and $C^n(z)<\frac{1}{2\,m}(1-\frac{\gamma }{k})$ for all $z>0$ and $n\in \mathbb {N}$. Note that

$$\begin{aligned} \bigg |\frac{dC^n(z)}{dz}\bigg |=\bigg |\frac{dP_1(C^n,N^n)(z)}{dz}\bigg |\le \beta _2<\infty . \end{aligned}$$

Thus, $\{C^n(z)\}$ is equicontinuous. By the Ascoli-Arzela theorem, there is a subsequence $\{s^{n_j}\}$ such that $\{C^{n_j}(z)\}$ converges uniformly to a limit function, say $C^0(z)$, satisfying the integral equation (2.8). Let $N^0(z)$ be given by the right-hand side of (2.3) with $C(z)=C^0(z)$ for $s=s_0$. Therefore, the formula $(C^0, N^0)(z)$ satisfies (1.5) with $s=s_0$,

$$\begin{aligned} C^0(0)=\frac{1}{2m}\left( 1-\frac{\gamma }{k}\right) , \quad C^0(z)\le \frac{1}{2m}\left( 1-\frac{\gamma }{k}\right) \ \text {for all}\ z>0, \end{aligned}$$

(3.11)

and

$$\begin{aligned} N^0(\infty )=0, \quad 0\le N^0(z)\le \frac{1}{k}. \end{aligned}$$

(3.12)

It means that $c(t, x)=C^0(x-s_0t)$ and $n(t, x)=N^0(x-s_0t)$ satisfy

$$\begin{aligned} \left\{ \begin{aligned}&c_t=c_{xx}+c\big [1-\lambda -c+(\lambda k-\gamma ) n\big ], \\&n_t=\lambda c(1-kn), \\&c(t,0)=C^0(x), \ \ n(t,0)=N^0(x), \end{aligned} \right. \end{aligned}$$

(3.13)

In view of $N^0\le \frac{1}{k}$, thus, $c(t, x)=C^0(x-s_0t)$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&c_t\ge c_{xx}+c(1-\frac{\gamma }{k}-c), \\&c(t,0)=C^0(x). \end{aligned} \right. \end{aligned}$$

(3.14)

By [26, Lemmas 2.2$-$2.4], for each fixed $x\in \mathbb {R}$, we have

$$\begin{aligned} \liminf _{t\rightarrow \infty } C^0(x-s_0t)\ge 1-\frac{\gamma }{k}, \end{aligned}$$

which implies $\liminf _{z\rightarrow -\infty }C^0(z)\ge 1-\frac{\gamma }{k}$. Due to (2.3), it further follows that $N^0(-\infty )=\frac{1}{k}$. By a similar argument in Theorem 3.4, we have $C^0(-\infty )=1-\frac{\gamma }{k}$. Then it suffices to verify $C^0(\infty )=0$. By contradiction, assume there exists $\bar{\alpha }\in (0,\frac{1}{2m}(1-\frac{\gamma }{k}))$ such that $\limsup _{z\rightarrow \infty }C^0(z)=\bar{\alpha }$. Then there exists a positive sequence $\{z_i\}_{i\in \mathbb {N}}$ so that

$$\begin{aligned} \lim _{i\rightarrow \infty }z_i=+\infty , \ C^0(z_i)>\bar{\alpha }/2 \ \text {for all }\ i\in \mathbb {N}. \end{aligned}$$

Since $C^0(z)$ is uniform continuous, there exists $\bar{\delta }>0$ so that

$$\begin{aligned} C^0(z)>\bar{\alpha }/4,\ \ z\in (z_i-\bar{\delta }, z_i+\bar{\delta }),\ i\in \mathbb {N}. \end{aligned}$$

Consider the initial value problem

$$\begin{aligned} \left\{ \begin{aligned}&v_t=v_{xx}+v(1-\frac{\gamma }{k}-v), \ x\in \mathbb {R}, \ t>0,\\&v(0,x)=v_0(x), \ x\in \mathbb {R}, \end{aligned} \right. \end{aligned}$$

(3.15)

where $v_0(x)$ satisfies

• (v1) $v_0(x)$ is bounded and uniformly continuous,

• (v2) $v_0(x)=\bar{\alpha }/4$, $x\in [-\bar{\delta }/2, \bar{\delta }/2]$,

• (v3) $v_0(x)=0$ if $|x|\ge \bar{\delta }$,

• (v4) $v_0(x)$ is decreasing if $x\in (\bar{\delta }/2, \bar{\delta })$, and increasing if $x\in (-\bar{\delta }, -\bar{\delta }/2)$.

Note that (3.15) is a Fisher-KPP equation. It follows from [4, 24] that there is $T>0$ so that

$$\begin{aligned} v(t,0)>\frac{1}{2m}\left( 1-\frac{\gamma }{k}\right) , \ t>T. \end{aligned}$$

(3.16)

For any $\bar{L}>0$, we can choose $i_1<i_2$ such that

$$\begin{aligned} z_{i_1}>\bar{L}, \ z_{i_2}-z_{i_1}>s_0T,\ C^0(z_{i_1})>\bar{\alpha }/2,\ C^0(z_{i_2})>\bar{\alpha }/2. \end{aligned}$$

Let $x=z_{i_2}$. By (3.16) and [26, Lemma 2.4], we know that $c(t,x)=C^0(x-s_0t)$ satisfies

$$\begin{aligned} c((z_{i_2}-z_{i_1})/s_0,z_{i_2})=C^0(z_{i_1})>v((z_{i_2}-z_{i_1})/s_0,0)>\frac{1}{2m}(1-\frac{\gamma }{k}), \end{aligned}$$

which contradicts (3.11). Thus, we have $C^0(\infty )=0$. Therefore, the proof is complete. $\square$

4 Bistable Wave in the Case BM: Existence and Uniqueness

In this section, we consider system (1.2) under the condition BM. Clearly, $e_\beta$ is stable, and $e_0$ is non-asymptotically stable or strongly stable. Since $e_0$ is neither isolated nor strongly-stable, the monotonic semiflow theorey in [12] can not be used directly to prove the existence of the bistable monotone traveling wave solution of (1.5)–(1.6). Here, we introduce a new idea to prove it, which may be applicable to other models with non-isolated equilibrium points.

Obviously(simply by the comparison principle) it can be seen from (1.5)–(1.6) (or (2.4)) that this model has no continuous and positive traveling wave with speed $s\le 0$. Consider the wave speed $s>0$ in the reaction term of (2.4) as a parameter. We would like to construct an auxiliary parabolic partial differential equation as follows

$$\begin{aligned} c_t=c_{xx}+c\left\{ 1-\lambda -c+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{x}^{\infty }\frac{\lambda c(t,x)}{s}dx}\right] \right\} . \end{aligned}$$

(4.1)

Note that (4.1) has two constant equilibria $c=0$ and $c=1-\frac{\gamma }{k}$. Due to the non-local term in the right side, its initial function needs to be restricted in a specific weighted functional space. For traveling waves to (4.1), we mean

$$\begin{aligned} c(t,x)=U(z), \ \ z=x-S(s)t, \end{aligned}$$

where S(s) is the wave speed dependent on s. After a substitution, (4.1) becomes

$$\begin{aligned} U''+S(s)U'+U\left\{ 1-\lambda -U+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda U(\tau )}{s}d\tau }\right] \right\} =0, \end{aligned}$$

(4.2)

subject to the boundary conditions

$$\begin{aligned} U(-\infty )=1-\frac{\gamma }{k},\ \ \ U(\infty )=0. \end{aligned}$$

(4.3)

We will prove the existence and uniqueness of wavefront (S(s), U), with S(s) being non-increasing function of s.

Let $\mathcal {C}=BC(\mathbb {R}, \mathbb {R})$ be the space of all bounded, continuous and $\infty$-integrable functions from $\mathbb {R}$ to $\mathbb {R}$, and $\mathcal {M}\subset \mathcal {C}$ be the set of all continuous and non-increasing functions from $\mathbb {R}$ to $\mathbb {R}$. $\mathcal {C}_+=\{ \varphi \in \mathcal {C}: \varphi (x) \ge 0, \forall x\in \mathbb {R} \}$. Define $\mathcal {M}_\beta =\{\varphi \in \mathcal {M}: 0\le \varphi \le \beta \}$, where $\beta =1-\frac{\gamma }{k}$.

Denote $\{Q_t\}_{t\ge 0}$ as the solution semiflow associated with (4.1) on $\mathcal {C}_+$. That is,

$$\begin{aligned} Q_t[\varphi ](x)=c(t,x,\varphi ), \ \forall \varphi \in BC(\mathbb {R}, \mathbb {R}_+), \ x\in \mathbb {R}, \ t\ge 0, \end{aligned}$$

where $c(t,x,\varphi )$ is the unique solution of (4.1) satisfying $c(0,\cdot ,\varphi )=\varphi$. Let E be the set of all fixed points of $Q_t$ for $t>0$ restricted to $\mathbb {R}$. Then $E=\{0, \beta \}$. We use Q to denote $Q_1$, and $Q^n$ is the n-th iteration of Q.

Assume that $\underline{\psi }$ and $\bar{\psi }$ are nonincreasing functions in $\mathcal {M}_\beta$ satisfying

$$\begin{aligned} \underline{\psi }(x)= \left\{ \begin{aligned}&(\beta -\tilde{\delta })(1-e^{\tilde{\mu } x}),&x<0,\\&0,&x\ge 0,\\ \end{aligned}\right. \quad \bar{\psi }(x)= \left\{ \begin{aligned}&\beta ,&x\le x_1,\\&e^{-\sqrt{\lambda -1}x},&x>x_1,\\ \end{aligned}\right. \end{aligned}$$

(4.4)

where $\tilde{\delta }>0$ is small, $\tilde{\mu }$ is a positive constant, and $x_1=-\frac{1}{\sqrt{\lambda -1}}\ln \beta >0$. Clearly, $\underline{\psi }\le \bar{\psi }$.

Lemma 4.1

Assume that BM holds. There exists a positive number $\tilde{S}>0$ such that for any $S\ge \tilde{S}$, we have

$$\begin{aligned} Q[\underline{\psi }](x)\ge \underline{\psi }(x+S) \ \ \text {and} \ \ Q[\bar{\psi }](x)\le \bar{\psi }(x-S) \ \text {for all} \ x\in \mathbb {R}. \end{aligned}$$

(4.5)

Proof

Let $\tilde{z}_1=x-St$. For $\tilde{z}_1\le x_1$, we have $\bar{\psi }(\tilde{z}_1)=\beta$, and then

$$\begin{aligned} \begin{aligned}&\bar{\psi }''+S\bar{\psi }'+\bar{\psi }\left\{ 1-\lambda -\bar{\psi } +\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{\tilde{z}_1}^{\infty }\frac{\lambda \bar{\psi }(\tau )}{s}d\tau }\right] \right\} \\&\quad \le \beta \{1-\lambda -\beta +\left( \lambda -\frac{\gamma }{k}\right) \}=0. \end{aligned} \end{aligned}$$

(4.6)

For $\tilde{z}_1>x_1$, we get $\bar{\psi }(\tilde{z}_1)=e^{-\sqrt{\lambda -1} \tilde{z}_1}$. For any $S\ge \tilde{S}_1$, where $\tilde{S}_1$ satisfies $-\sqrt{\lambda -1}\tilde{S}_1+\lambda -\frac{\gamma }{k}=0$, it gives

$$\begin{aligned} \begin{aligned}&\bar{\psi }''+S\bar{\psi }'+\bar{\psi }\left\{ 1-\lambda -\bar{\psi }+ \left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{\tilde{z}_1}^{\infty }\frac{\lambda \bar{\psi }(\tau )}{s}d\tau }\right] \right\} \\&\quad =e^{-\sqrt{\lambda -1} \tilde{z}_1}\left\{ -\sqrt{\lambda -1} S-e^{-\sqrt{\lambda -1} \tilde{z}_1}+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-\frac{ks}{\lambda \sqrt{\lambda -1}}e^{-\sqrt{\lambda -1} \tilde{z}_1}}\right] \right\} \\&\quad \le 0. \end{aligned} \end{aligned}$$

(4.7)

Therefore, $Q[\bar{\psi }](x)\le \bar{\psi }(x-S)$.

In order to prove $Q[\underline{\psi }](x)\ge \underline{\psi }(x+S)$, we let $\tilde{z}_2=x+St$. Choose $\tilde{\epsilon }>0$ such that

$$\begin{aligned} \tilde{\delta }-\left( \lambda -\frac{\gamma }{k}\right) \tilde{\epsilon }>0. \end{aligned}$$

(4.8)

There exists $M_0(s)$ so that $k\int _{\tilde{z}_2}^{\infty }\frac{\lambda \underline{\psi }(\tau )}{s}d\tau >-\ln \tilde{\epsilon }$ when $\tilde{z}_2<M_0(s)$. That is,

$$\begin{aligned} e^{-k\int _{\tilde{z}_2}^{\infty }\frac{\lambda \underline{\psi }(\tau )}{s}d\tau }<\tilde{\epsilon }\quad \text {for}\quad \tilde{z}_2<M_0(s). \end{aligned}$$

(4.9)

Take $\tilde{\eta }:=\frac{\underline{\psi }(M_0(s))}{\beta -\tilde{\delta }}\in (0,1)$. There are two cases to analyze: $M_0(s)<0$ and $M_0(s)\ge 0$. We first assume $M_0(s)<0$. For $\tilde{z}_2<M_0(s)<0$, if $S\ge \tilde{\mu }$, then by (4.8) and (4.9), we have

$$\begin{aligned} \begin{aligned}&\underline{\psi }''-S\underline{\psi }'+\underline{\psi }\left\{ 1-\lambda -\underline{\psi }+ \left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{\tilde{z}_2}^{\infty }\frac{\lambda \underline{\psi }(\tau )}{s}d\tau }\right] \right\} \\&\quad \ge \underline{\psi }''-S\underline{\psi }'+\underline{\psi }\left\{ 1-\lambda -\underline{\psi }+ \left( \lambda -\frac{\gamma }{k}\right) (1-\tilde{\epsilon })\right\} \\&\quad =(\beta -\tilde{\delta })\left\{ \tilde{\mu }e^{\tilde{\mu } \tilde{z}_2} (S-\tilde{\mu }) +(1-e^{\tilde{\mu } \tilde{z}_2})\left[ \tilde{\delta }-\left( \lambda -\frac{\gamma }{k}\right) \tilde{\epsilon } +(\beta -\tilde{\delta })e^{\tilde{\mu } \tilde{z}_2}\right] \right\} \ge 0.\\ \end{aligned} \end{aligned}$$

(4.10)

For $\tilde{z}_2\in [M_0(s),0)$, we have $\underline{\psi }(\tilde{z}_2)\in (0, \tilde{\eta }(\beta -\tilde{\delta })]$ and $e^{\tilde{\mu } \tilde{z}_2}\in [1-\tilde{\eta },1)$. If $S\ge \frac{\tilde{\mu }^2+\tilde{\eta }[\tilde{\eta }(\beta -\tilde{\delta })+\lambda -1]}{\tilde{\mu }(1-\tilde{\eta })}$, then

$$\begin{aligned} \begin{aligned}&\underline{\psi }''-S\underline{\psi }'+\underline{\psi }\left\{ 1-\lambda -\underline{\psi }+ \left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{\tilde{z}_2}^{\infty }\frac{\lambda \underline{\psi }(\tau )}{s}d\tau }\right] \right\} \\&\quad \ge \underline{\psi }''-S\underline{\psi }'+\underline{\psi }\big \{1-\lambda -\underline{\psi }\big \}\\&\quad \ge -(\beta -\tilde{\delta })\tilde{\mu }^2e^{\tilde{\mu } \tilde{z}_2}+(\beta -\tilde{\delta })S\tilde{\mu }e^{\tilde{\mu } \tilde{z}_2}+\tilde{\eta }(\beta -\tilde{\delta })[1-\lambda -\tilde{\eta }(\beta -\tilde{\delta })]\\&\quad \ge (\beta -\tilde{\delta })\bigg \{ -\tilde{\mu }^2+S\tilde{\mu }(1-\tilde{\eta })+\tilde{\eta }[1-\lambda -\tilde{\eta }(\beta -\tilde{\delta })] \bigg \}\ge 0\\ \end{aligned} \end{aligned}$$

(4.11)

For $\tilde{z}_2\ge 0$, we have $\underline{\psi }(\tilde{z}_2)=0$, and thus

$$\begin{aligned} \underline{\psi }''-S\underline{\psi }'+\underline{\psi }\left\{ 1-\lambda -\underline{\psi }+ \left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{\tilde{z}_2}^{\infty }\frac{\lambda \underline{\psi }(\tau )}{s}d\tau }\right] \right\} =0. \end{aligned}$$

(4.12)

Assume $M_0(s)\ge 0$. For $\tilde{z}_2<0\le M_0(s)$, we can derive (4.10) by the same way. For $\tilde{z}_2\ge 0$, (4.12) holds. Therefore, $Q[\underline{\psi }](x)\ge \underline{\psi }(x+S)$ for all $S\ge \tilde{S}_2:=\max \{\frac{\tilde{\mu }^2+\tilde{\eta }[\tilde{\eta }(\beta -\tilde{\delta })+\lambda -1]}{\tilde{\mu }(1-\tilde{\eta })}, \tilde{\mu } \}$. Let $\tilde{S}=\max \{\tilde{S}_1, \tilde{S}_2\}$. Thus, the proof is complete. $\square$

This lemma implies that $\bar{\psi }(x-St)$ and $\underline{\psi }(x+St)$ form a pair of upper/lower solutions to the auxiliary equation (4.1) for large S.

For each n, make a shift to each upper and lower solutions and define

$$\begin{aligned} \underline{\psi }_n(x)=\underline{\psi }(x+n+\tilde{S}) \quad \text {and} \quad \bar{\psi }_n(x)=\bar{\psi }(x+x_1-(n+\tilde{S})). \end{aligned}$$

Obviously by way of the invariance of translation, both of them are still upper/lower solutions of Q,

Let $\kappa _n:=(n+\tilde{S})/n$. Define $A_{\xi }[\phi ](x)=\phi (\xi x)$ for all $x\in \mathbb {R}$, $\xi \in \mathbb {R}$ and $\phi \in \mathcal {M}_{\beta }$.

Similar to the proof of [12, Lemma 3.3], we have the following lemma.

Lemma 4.2

Assume that BM holds. For each $n\in \mathbb {N}$, $G_n:=Q\circ A_{\kappa _n}$ has a fixed point $\phi _n$ in $\mathcal {M}_{\beta }$ such that $\phi _n(x)$ is nonincreasing in x and $\underline{\psi }_n\le \phi _n\le \bar{\psi }_n$.

Theorem 4.3

Assume that BM holds. There exists a $S\in \mathbb {R}$ such that $\{Q^n\}_{n\ge 1}$ admits a nonincreasing traveling wave connecting $\beta$ to 0 with speed S.

Proof

Choose $\tilde{M}>x_1>0$ such that $e^{-\sqrt{\lambda -1}\tilde{M}}<\frac{1}{2}(\beta -\tilde{\delta })$, and it follows

$$\begin{aligned}{} & {} e^{-\sqrt{\lambda -1}\tilde{M}}=\bar{\psi }(\tilde{M})=\bar{\psi }_n(\tilde{M}-x_1+n+\tilde{S})\ge \phi _n(\tilde{M}-x_1+n+\tilde{S}),\\{} & {} \frac{1}{2}(\beta -\tilde{\delta })=\underline{\psi }\left( -\frac{1}{\tilde{\mu }}\ln 2\right) = \underline{\psi }_n\left( -\frac{1}{\tilde{\mu }}\ln 2-(n+\tilde{S})\right) \le \phi _n\left( -\frac{1}{\tilde{\mu }}\ln 2-(n+\tilde{S})\right) . \end{aligned}$$

Define $b_n:=\sup \limits _{}\{x: \phi _n(x)\in [\frac{1}{2}(\beta -\tilde{\delta }),\beta ]\}$. In other word, $b_n$ is a point so that $\phi _n(b_n)=\frac{1}{2} (\beta -\tilde{\delta })$. Then $-\frac{1}{\tilde{\mu }}\ln 2-(n+\tilde{S})\le b_n\le \tilde{M}-x_1+n+\tilde{S}$. Let $\phi _{+,n}(x):=\phi _n(x+b_n)$, and we have

$$\begin{aligned} \phi _{+,n}=\phi _n(\cdot +b_n)=G_n[\phi _n](\cdot +b_n)=Q[\phi _n(\kappa _n(\cdot +b_n))]\in Q[C_{\beta }]. \end{aligned}$$

Due to the compactness of Q, there exist a subsequence (denoted by n), a nonincreasing function $\phi _+$, and $\xi _+$ such that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{b_n}{n}=\xi _+, \quad \lim \limits _{n\rightarrow \infty } \phi _{+,n}=\phi _+. \end{aligned}$$

Define $S_+:=\tilde{S}\xi _+.$ Observe that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\kappa _n(x+b_n)-b_n=\lim \limits _{n\rightarrow \infty }\left( x+\tilde{S}\cdot \frac{x+b_n}{n}\right) =x+S_+ \end{aligned}$$

holds uniformly for x in any bounded subset of $\mathbb {R}$. For any $x\in \mathbb {R}$, we have

$$\begin{aligned} \begin{aligned} \phi _+(x-S_+)&=\lim \limits _{n\rightarrow \infty } \phi _{+,n}(x-S_+)=\lim \limits _{n\rightarrow \infty }\phi _n(x-S_++b_n)=\lim \limits _{n\rightarrow \infty }G_n[\phi _n](x-S_++b_n)\\&=\lim \limits _{n\rightarrow \infty }Q[\phi _n(\kappa _n(\cdot +b_n))](x-S_+)=\lim \limits _{n\rightarrow \infty }Q[\phi _{+,n}(\kappa _n(\cdot +b_n)-b_n)](x-S_+)\\&=Q[\phi _+](x). \end{aligned} \end{aligned}$$

(4.13)

Therefore, $\phi _+(-\infty )$ and $\phi _+(\infty )$ are fixed points of Q. Note that $\frac{1}{2}(\beta -\tilde{\delta })\le \lim \limits _{n\rightarrow \infty }\phi _n(b_n)=\phi _+(0)<\beta$. It follows that $\phi _+(-\infty )=\beta$ and $\phi _+(\infty )=0$. Hence, the proof is complete. $\square$

Since the solution of (4.1) is continuous in its parameter s, it is easy to know that $S_+$ is a continuous function of s. Returning to the equation (4.1) with $S_+=S(s)$, we are ready to have the following theorem.

Theorem 4.4

Assume that BM holds. There exists $S(s)\in \mathbb {R}$ and $U\in \mathcal {M}_\beta$ connecting $\beta$ to 0 such that $Q_t[U](x)=U(x-S(s)t)$ for all $x\in \mathbb {R}$, that is, (4.1) admits a non-increasing traveling wave $U(x-S(s)t)$ connecting $\beta$ to 0. Furthermore, this bistable traveling wave U(z) is strictly decreasing in z with $U'(z)<0$.

Proof

We only prove $U'(z)<0$ by a contradiction argument. Assume $U'(z_0)=0$ for some $z_0\in \mathbb {R}$. The strong maximum principle implies that $U'(z)\equiv 0$ for $z\in \mathbb {R}$. It contradicts the boundary condition (4.3). $\square$

In the following, we want to establish the uniqueness of the above traveling waves in an exponentially decay function space. We now use the bistable traveling wave U(z) to construct a pair of upper and lower solutions of (4.1). We first define two continuous functions

$$\begin{aligned} P_+(x)=\left\{ \begin{aligned}&P^+,&x\le -\eta _1<0,\\&e^{-\mu x},&x>-\eta _1,\\ \end{aligned}\right. \quad \quad P_-(x)=\left\{ \begin{aligned}&P^-,&x\le -\eta _2<0,\\&e^{-\mu x},&x>-\eta _2.\\ \end{aligned}\right. \end{aligned}$$

(4.14)

Here $P^{\pm }>1$, and $\mu >0$ is small to be determined. $\eta _1$ and $\eta _2$ are chosen so that the above two functions are continuous.

Lemma 4.5

Assume that BM holds. Define

$$\begin{aligned} \phi ^{\pm }(z,t)=U(z\pm \xi _0\pm \sigma _1\delta (1-e^{-\rho t}))\pm \delta P_{\pm }(z\pm \xi _0)e^{-\rho t}, \quad z=x-S(s)t, \quad t>0. \end{aligned}$$

Then there exists $\sigma _1<0$, $P^{\pm }>0$, $\rho >0$ and $\delta _0>0$ such that for any $\xi _0\in \mathbb {R}$, $\delta \in (0,\delta _0)$, and $\mu =\sqrt{\delta }$, the formulas $\phi ^{\pm }(z,t)$ are upper and lower solutions of (4.1) respectively. It further follows that $\phi ^{-}(z,t)\le c(x,t)\le \phi ^{+}(z,t)$, if $\phi ^{-}(x,0)\le c(x,0)\le \phi ^{+}(x,0)$.

Proof

We only prove $\phi ^+(z,t)$ is an upper solution of (4.1), and the lower solution $\phi ^-(z,t)$ can be verified by a similar argument. Let $\xi =z+ \xi _0+ \sigma _1\delta (1-e^{-\rho t})$ and $\eta =z+\xi _0$. We choose $\delta$ and $\rho$ that are small enough. Take $M_1>0$ and sufficiently small $\epsilon >0$ such that $e^{-k\int _{\xi }^{\infty }\frac{\lambda U(\tau )}{s}d\tau }>1-\epsilon$ for $\xi >M_1$ and $U(M_1)=\mu ^2(1-\frac{\gamma }{k})$. Take $M_2>0$ and sufficiently small $\epsilon _1>0$ such that $e^{-k\int _{\xi }^{\infty }\frac{\lambda U(\tau )}{s}d\tau }<\epsilon _1$ for $\xi <-M_2$ and $U(-M_2)=\theta _1(1-\frac{\gamma }{k})>\frac{1}{2}(1-\frac{\gamma }{k})$.

Let $M>\max \{\eta _1, \eta _2, M_1, M_2\}$. Set

$$\begin{aligned} \begin{aligned}&L_0:=\int _{\eta }^{\infty }\frac{\lambda P_+(\tau )}{s}d\tau ,\\&L_1:=\max _{-M\le \xi \le M}U'(\xi ),\\&L_2:=\max \left\{ \max _{\eta \in [-M, M]}P'(\eta ), \max _{\eta \in [-M, M]}P''(\eta ) \right\} ,\\&\sigma _1<\frac{L_2+S(s)L_2+P^+\rho + P^+(1-\frac{\gamma }{k})+(1-\frac{\gamma }{k})(\lambda k-\gamma )L_0}{\rho L_1}. \end{aligned} \end{aligned}$$

(4.15)

Substituting $\phi ^+$ into (4.1), we get

$$\begin{aligned} \begin{aligned}&\phi ^+_{zz}+S(s)\phi ^+_{z}-\phi ^+_{t}+\phi ^+\left\{ 1-\lambda -\phi ^++\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda \phi ^+(\tau ,t)}{s}d\tau }\right] \right\} \\&\quad =U''+\delta P_+''e^{-\rho t}+S(s)U'+S(s)\delta P_+'e^{-\rho t}-\sigma _1\rho \delta U'e^{-\rho t}+\delta P_+\rho e^{-\rho t} +(U+\delta P_+e^{-\rho t})\\&\qquad \times \left\{ 1-\lambda -U-\delta P_+ e^{-\rho t}+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{\xi }^{\infty }\frac{\lambda U(\tau )}{s}d\tau } e^{-k\int _{\eta }^{\infty }\frac{\lambda P_+(\tau )\delta e^{-\rho t}}{s}d\tau }\right] \right\} \\&\quad =\delta e^{-\rho t}\bigg \{ P_+''+S(s) P_+'+ P_+\rho -\sigma _1\rho U'\bigg \}\\&\qquad +\delta P_+ e^{-\rho t}=\left\{ 1-\lambda -U-\delta P_+ e^{-\rho t}+(\lambda -\frac{\gamma }{k})\left[ 1-e^{-k\int _{\xi }^{\infty }\frac{\lambda U(\tau )}{s}d\tau }e^{-k\int _{\eta }^{\infty }\frac{\lambda P_+(\tau )\delta e^{-\rho t}}{s}d\tau }\right] \right\} \\&\qquad +U\left\{ -\delta P_+ e^{-\rho t}+\left( \lambda -\frac{\gamma }{k}\right) e^{-k\int _{\xi }^{\infty }\frac{\lambda U(\tau )}{s}d\tau }\left[ 1-e^{-k\int _{\eta }^{\infty }\frac{\lambda P_+(\tau )\delta e^{-\rho t}}{s}d\tau }\right] \right\} \\&\qquad :=I+II+III, \quad \forall t>0. \end{aligned} \end{aligned}$$

(4.16)

Here we use the fact U is the biatable traveling wave of (4.1) and satisfies (4.2). In order to prove $\phi ^+(z,t)$ is an upper solution of (4.1), we consier three cases:

Case (i): $|\xi |\le M$. Accroding to (4.15), (4.16), and $\delta$ is sufficiently small, we have

$$\begin{aligned}{} & {} I\le \delta e^{-\rho t}[L_2+S(s)L_2+P^+\rho -\sigma _1\rho L_1], \\{} & {} II\le P^+\delta e^{-\rho t}\left[ 1-\lambda +\left( \lambda -\frac{\gamma }{k}\right) \right] = P^+\delta e^{-\rho t}\left( 1-\frac{\gamma }{k}\right) , \end{aligned}$$

and

$$\begin{aligned} III\le \left( 1-\frac{\gamma }{k}\right) \left\{ \left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{\eta }^{\infty }\frac{\lambda P_+(\tau )\delta e^{-\rho t}}{s}d\tau }\right] \right\} \le \left( 1-\frac{\gamma }{k}\right) \left( \lambda -\frac{\gamma }{k}\right) kL_0\delta e^{-\rho t}. \end{aligned}$$

Then it gives

$$\begin{aligned} \begin{aligned}&\phi ^+_{zz}+S(s)\phi ^+_{z}-\phi ^+_{t}+\phi ^+\left\{ 1-\lambda -\phi ^++\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda \phi ^+(\tau ,t)}{s}d\tau }\right] \right\} \\&\quad \le \delta e^{-\rho t}[L_2+S(s)L_2+P^+\rho -\sigma _1\rho L_1]+P^+\delta e^{-\rho t}\left( 1-\frac{\gamma }{k}\right) +\left( 1-\frac{\gamma }{k}\right) \left( \lambda -\frac{\gamma }{k}\right) kL_0\delta e^{-\rho t}\\&\quad =\delta e^{-\rho t}\{L_2+S(s)L_2+P^+\rho -\sigma _1\rho L_1+ P^+\left( 1-\frac{\gamma }{k}\right) +\left( 1-\frac{\gamma }{k}\right) (\lambda k-\gamma )L_0 \}\le 0. \end{aligned} \end{aligned}$$

(4.17)

The last inequality sign in (4.17) holds by (4.15).

Case (ii): $\xi >M$. Note that $\xi >M_1$ and $\xi >-\eta _1$. Then $\eta >-\eta _1$ since $\delta$ is sufficiently small. It means $P_+(\eta )=e^{-\mu \eta }$. Accroding to (4.15), (4.16), $\mu =\sqrt{\delta }$, and $\delta$ is sufficiently small, we have

$$\begin{aligned}{} & {} I\le \delta e^{-\rho t}[\mu ^2 e^{-\mu \eta }-\mu S(s)e^{-\mu \eta }+\rho e^{-\mu \eta }], \\{} & {} II\le e^{-\mu \eta }\delta e^{-\rho t}\left\{ 1-\lambda +\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-(1-\epsilon )e^{-\frac{\sqrt{\delta }k\lambda }{s} e^{-\mu \eta }e^{-\rho t}}\right] \right\} , \end{aligned}$$

and

$$\begin{aligned} III\le & {} \mu ^2(1-\frac{\gamma }{k})\left\{ \left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{\eta }^{\infty }\frac{\lambda P_+(\tau )\delta e^{-\rho t}}{s}d\tau }\right] \right\} \\\le & {} \mu \left( 1-\frac{\gamma }{k}\right) \left\{ \left( \lambda -\frac{\gamma }{k}\right) \frac{k\lambda }{s}e^{-\mu \eta }\delta e^{-\rho t} \right\} . \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned}&\phi ^+_{zz}+S(s)\phi ^+_{z}-\phi ^+_{t}+\phi ^+\left\{ 1-\lambda -\phi ^++\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda \phi ^+(\tau ,t)}{s}d\tau }\right] \right\} \\&\quad \le \delta e^{-\rho t}(\mu ^2 e^{-\mu \eta }-\mu S(s)e^{-\mu \eta }+\rho e^{-\mu \eta })\\&\qquad +e^{-\mu \eta }\delta e^{-\rho t}\left\{ 1-\lambda +\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-(1-\epsilon )e^{-\frac{\sqrt{\delta }k\lambda }{s} e^{-\mu \eta }e^{-\rho t}}\right] \right\} \\&\qquad +\mu \left( 1-\frac{\gamma }{k}\right) \left\{ \left( \lambda -\frac{\gamma }{k}\right) \frac{k\lambda }{s}e^{-\mu \eta }\delta e^{-\rho t} \right\} \\&\quad =\delta e^{-\rho t}e^{-\mu \eta }\left\{ \mu ^2-\mu S(s)+\rho +1-\lambda +\bar{\delta }+\frac{\mu \lambda }{s}\left( 1-\frac{\gamma }{k}\right) (\lambda k-\gamma ) \right\} \le 0. \end{aligned} \end{aligned}$$

(4.18)

Here $\bar{\delta }=(\lambda -\frac{\gamma }{k})[1-(1-\epsilon )e^{-\frac{\sqrt{\delta }k\lambda }{s} e^{-\mu \eta }e^{-\rho t}}]$ and $\mu$ are small enough since $\delta$ and $\epsilon$ are sufficiently small. Since $\lambda >1$ and $\rho$ small enough, then the last inequality in (4.18) holds.

Case (iii): $\xi <-M$. Note that $\xi <-M_2$ and $\xi <-\eta _1$. Then $\eta <-\eta _1$. Thus $P(\eta )=P^+$. Accroding to (4.15), (4.16), and $\delta$ is sufficiently small, we have

$$\begin{aligned}{} & {} I\le \delta e^{-\rho t}P^+\rho , \\{} & {} II\le P^+\delta e^{-\rho t}\left\{ 1-\lambda -U+\left( \lambda -\frac{\gamma }{k}\right) \right\} \le P^+\delta e^{-\rho t}\left\{ 1-\lambda -\theta _1\left( 1-\frac{\gamma }{k}\right) +\left( \lambda -\frac{\gamma }{k}\right) \right\} , \end{aligned}$$

and

$$\begin{aligned} III\le -\theta _1\left( 1-\frac{\gamma }{k}\right) \delta P^+ e^{-\rho t} \end{aligned}$$

since $(\lambda -\frac{\gamma }{k})e^{-k\int _{\xi }^{\infty }\frac{\lambda U(\tau )}{s}d\tau }[1-e^{-k\int _{\eta }^{\infty }\frac{\lambda P_+(\tau )\delta e^{-\rho t}}{s}d\tau }] \bigg \}\rightarrow 0$.

Then equation (4.16) becomes

$$\begin{aligned} \begin{aligned}&\phi ^+_{zz}+S(s)\phi ^+_{z}-\phi ^+_{t}+\phi ^+\left\{ 1-\lambda -\phi ^++\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda \phi ^+(\tau ,t)}{s}d\tau }\right] \right\} \\&\quad \le \delta e^{-\rho t}(P^+\rho )+P^+\delta e^{-\rho t}\left\{ 1-\lambda -\theta _1\left( 1-\frac{\gamma }{k}\right) +\left( \lambda -\frac{\gamma }{k}\right) \right\} - \theta _1\left( 1-\frac{\gamma }{k}\right) \delta P^+ e^{-\rho t} \\&\quad =P^+\delta e^{-\rho t}\left\{ \rho +(1-2\theta _1)\left( 1-\frac{\gamma }{k}\right) \right\} \le 0, \end{aligned} \end{aligned}$$

(4.19)

since $\theta _1>\frac{1}{2}$ and $\rho$ sufficiently small. Combining cases (i)–(iii), we have $\phi ^+_{zz}+S(s)\phi ^+_{z}-\phi ^+_{t}+\phi ^+\big \{1-\lambda -\phi ^++(\lambda -\frac{\gamma }{k})[1-e^{-k\int _{z}^{\infty }\frac{\lambda \phi ^+(\tau ,t)}{s}d\tau }]\big \}\le 0$ for $z\in \mathbb {R}$ and $t\ge 0$. Therefore $\phi ^+(z, t)$ is an upper solution of (4.1). $\square$

Next, we shall establish the uniqueness of the bistable traveling wave of system (4.1) by using Lemma 4.5.

Theorem 4.6

Assume that BM holds. If $U^*(x-S^*(s)t)$ is a bistable traveling wave of system (4.1) connecting $\beta$ to 0, then there exists $z^*\in \mathbb {R}$ such that $U^*(x-S^*(s)t)=U(x-S(s)t+z^*)$ and $S^*(s)=S(s)$, where $U(x-S(s)t)$ is the solution in Theorem 4.4.

Proof

There exist $\xi _1\in \mathbb {R}$ and $h\gg 1$ such that

$$\begin{aligned} U(x+\xi _1)-\delta P_-(x)\le U^*(x)\le U(x+\xi _1-h)+\delta P_+(x) \end{aligned}$$

since the decay rate $\mu$ in $P_+(x)$ and $P_-(x)$ is sufficiently small. Then by Lemma 4.5 and the comparison theorem, we have

$$\begin{aligned} \begin{aligned}&U(x-S(s)t+\xi _1-\sigma _1\delta (1-e^{-\rho t}))-\delta P_-(x-S(s)t)e^{-\rho t}\le U^*(x-S^*(s)t)\\&\quad \le U(x-S(s)t+\xi _1-h+\sigma _1\delta (1-e^{-\rho t}))+\delta P_+(x-S(s)t)e^{-\rho t}. \end{aligned} \end{aligned}$$

(4.20)

Fix $\bar{z}=x-S^*(s)t$ such that $U^*(\bar{z})>0$. If $S^*(s)> S(s)$, then

$$\begin{aligned} U^*(\bar{z})\le U(\bar{z}+(S^*(s)-S(s))t+\xi _1-h+\sigma _1\delta (1-e^{-\rho t}))+\delta P_+(z)e^{-\rho t}\rightarrow 0, \text {as} \ t\rightarrow \infty . \end{aligned}$$

It contradicts the fact $U^*(\bar{z})>0$. Thus $S^*(s)\le S(s)$. Similarly, by the left inequality in (4.20), we have $S^*(s)\ge S(s)$. Therefore, $S^*(s)= S(s)$.

Let $t\rightarrow \infty$ in (4.20). We have

$$\begin{aligned} U(z+\xi _1-\sigma _1\delta )\le U^*(z)\le U(z+\xi _1-h+\sigma _1\delta ), \ \forall z\in \mathbb {R}. \end{aligned}$$

By using a similar method in [7] (see Step 2 on page 133), we can prove the wave profile is unique up to translation. $\square$

In order to prove the existence and uniqueness of bistable monotone traveling wave of (1.5)–(1.6) under BM, we need to show $S(s)=s$ has only one positive root. Before proving this, we give the following critical lemma, which reveals the monotonicity of S(s) with respect to s.

Lemma 4.7

Assume that BM holds. The wave speed S(s) is non-increasing with respect to s in (4.2), that is, $S(s_1)\ge S(s_2)$ if $0<s_1\le s_2$.

Proof

Take $0<s_1\le s_2$. To the contrary, assume $S(s_1)<S(s_2)$. Let $V_2(z)$ be the traveling wave with speed $S(s_2)$, satisfying

$$\begin{aligned} V_2''+S(s_2)V_2'+V_2\left\{ 1-\lambda -V_2+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda V_2(\tau )}{s_2}d\tau }\right] \right\} =0. \end{aligned}$$

(4.21)

Consider

$$\begin{aligned} \left\{ \begin{aligned}&c_t=c_{xx}+c\left\{ 1-\lambda -c+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{x}^{\infty }\frac{\lambda c(x,t)}{s_2}dx}\right] \right\} , \\&c(x,0)=V_2(x), \\ \end{aligned} \right. \end{aligned}$$

(4.22)

and it has a solution

$$\begin{aligned} c(t,x)=V_2(z),\ \ \ z= x-S(s_2)t. \end{aligned}$$

Let $V_1(z)=V_1(x-S(s_1)t)=\bar{c}(t,x)$ satisfy

$$\begin{aligned} V_1''+S(s_1)V_1'+V_1\left\{ 1-\lambda -V_1+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda V_1(\tau )}{s_1}d\tau }\right] \right\} =0. \end{aligned}$$

(4.23)

Note that the reaction term is non-increasing with respect to s. Thus, we have

$$\begin{aligned} \left\{ \begin{aligned}&\bar{c}_t\ge \bar{c}_{xx}+\bar{c}\left\{ 1-\lambda -\bar{c}+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{x}^{\infty }\frac{\lambda \bar{c}(t,x)}{s_2}dx}\right] \right\} , \\&\bar{c}(x,0)=V_1(x). \\ \end{aligned} \right. \end{aligned}$$

(4.24)

Due to the fact that

$$\begin{aligned}{} & {} V_1(z)\sim 1-\frac{\gamma }{k}-b_2e^{\zeta (S(s_1)) z}, \ V_2(z)\sim 1-\frac{\gamma }{k}-b_3e^{\zeta (S(s_2)) z},\ \text {as} \ z\rightarrow -\infty , \\{} & {} V_1(z)\sim b_4e^{-\nu (S(s_1))z}, \ V_2(z)\sim b_5e^{-\nu (S(s_2)) z},\ \text {as} \ z\rightarrow \infty , \end{aligned}$$

where $0<\zeta (S(s_2))<\zeta (S(s_1)), 0<\nu (S(s_1))<\nu (S(s_2))$, and $b_i, i=2,3,4, 5,$ are positive constants, and by translation if necessary, we have $V_2(x)\le V_1(x)$ for all $x\in \mathbb {R}$. Using the comparison principle [14], we have

$$\begin{aligned} V_2(x-S(s_2)t)\le V_1(x-S(s_1)t). \end{aligned}$$

Fix $z_1^*=x-S(s_2)t$ such that $V_2(z_1^*)>0$. It follows that

$$\begin{aligned} V_2(x-S(s_2)t) \le V_1(x-S(s_1)t)=V_1(z_1^*+(S(s_2)-S(s_1))t)\rightarrow 0,\ \text {as}\ t\rightarrow \infty , \end{aligned}$$

which contradicts $V_2(z_1^*)>0$. Therefore, $S(s_1)\ge S(s_2)$. $\square$

Theorem 4.8

Assume that BM holds. There exists a unique monotone bistable traveling wave (s, C(z), N(z)), $z=x-st$, of system (1.5)–(1.6), where s is positive.

Proof

Theorems 4.4 to 4.6 gives the existence and uniqueness of the monotone bistable traveling wave solution of (4.2)–(4.3). We also know that S(s) is nonincreasing function of s. To return to (1.5)–(1.6) for the existence of waves, we need to show that $S(s)=s$ has a positive root. To this end, it suffices to show $S(0_+)>0$. Let

$$\begin{aligned} f(U)=U\left\{ 1-\lambda -U+\left( \lambda -\frac{\gamma }{k}\right) \left[ 1-e^{-k\int _{z}^{\infty }\frac{\lambda U(\tau )}{s}d\tau }\right] \right\} . \end{aligned}$$

Then (4.2) can be written as

$$\begin{aligned} U''+S(s,f)U'+f(U)=0. \end{aligned}$$

(4.25)

Here we just use S(s, f) to indicate that the speed is dependent on the reaction term. Take a small $\hat{\varepsilon }>0$ such that $U(M_1)=\hat{\varepsilon }$ for some constant $M_1$. Consider $s\rightarrow 0$ in (4.25). Then there exists a sufficiently small $\delta >0$ such that $e^{-k\int _{z}^{\infty }\frac{\lambda U(\tau )}{s}d\tau }<\delta$ for $z< M_1$. It follows that

$$\begin{aligned} f(U)\ge g(U):=\left\{ \begin{aligned}&U\left( 1-\frac{\gamma }{k}-\delta \left( \lambda -\frac{\gamma }{k}\right) -U\right) ,\quad&U> \hat{\varepsilon }, \\&U(1-\lambda -U),\ {}&U< \hat{\varepsilon }.\\ \end{aligned} \right. \end{aligned}$$

(4.26)

Consider

$$\begin{aligned} U''+S(s,g)U'+g(U)=0. \end{aligned}$$

(4.27)

The existence of U(z) in (4.27) connecting $1-\frac{\gamma }{k}-\delta (\lambda -\frac{\gamma }{k})$ to 0 can be guaranteed by the classical work [14]. We multiply the left and right sides of (4.27) by $U'$, and integrate both sides from $-\infty$ to $\infty$ with respect to z. It follows that the wave speed S(s, g) satisfies

$$\begin{aligned} S(s,g)=\frac{\int _{0}^{1-\frac{\gamma }{k}-\delta \left( \lambda -\frac{\gamma }{k}\right) }g(U)dU}{\int _{-\infty }^{\infty }(U')^2dz}. \end{aligned}$$

Since $\hat{\varepsilon }$ and $\delta$ are small enough, we have

$$\begin{aligned} \int _{0}^{1-\frac{\gamma }{k}-\delta \left( \lambda -\frac{\gamma }{k}\right) }g(U)dU=\frac{1}{6}\left( 1-\frac{\gamma }{k}- \delta (\lambda -\frac{\gamma }{k})\right) ^3-\frac{1}{2}(1-\delta )\left( \lambda -\frac{\gamma }{k}\right) \hat{\varepsilon }^2>0. \end{aligned}$$

It means $S(s,g)>0$. By (4.25), (4.26), (4.27), and using a similar argument in the proof of Lemma 4.7, we have $S(s,f)\ge S(s,g)>0$ when $s \rightarrow 0$. By Lemma 4.7, the equation $S(s)=s$ has a unique positive solution. Thus, system (2.4)–(2.5) has a unique monotonic solution (s, C(z)), that is, there exists a unique monotone bistable traveling wave (s, C(z), N(z)) of system (1.5)–(1.6). It completes the proof. $\square$

5 Numerical Simulations

In this section, we will use MATLAB software to simulate our model in three cases MM-BM. The function “pdepe” is used to approximate the PDE system with Neumann boundary conditions. The numerical results show that the solution reaches its equilibria when |x| is large in a short time, which means the domain size has almost no impact on the simulation of the spreading speeds, as long as it is properly large. We shall compute the spreading speeds by using the Level Set Method. Let the following step functions be the initial data:

$$\begin{aligned} c(0,x)=\left\{ \begin{aligned}&\frac{1}{2}\left( \lambda -\frac{\gamma }{k}\right) ,&x<-95,\\&0,\ {}&x\ge -95, \end{aligned}\right. , \quad n(0,x)=\left\{ \begin{aligned}&\frac{1}{2k},&x<-95,\\&0,&x\ge -95,\\ \end{aligned}\right. \end{aligned}$$

(5.1)

where parameters $\lambda , \gamma$ and k are to be chosen.

Example 5.1

Let $\lambda =0.5, \gamma =0.5, k=2$. The initial data are defined in (5.1).

In the case of this parameter choice, we have $e_{\beta }=(0.75,0.5), e_0=(0,0)$. Note that MM holds. The detailed simulations can be seen in Fig. 1. The 3-D figures of c(t, x) and n(t, x) are shown in the right column. The left column is the same movement from the top view. These figures show that the solution propagates to the right with a constant speed, which is consistent with the result when $e_{\beta }$ is stable and $e_0$ is unstable under MM. We can use the level sets shown in Fig. 2 to calculate the numerical speeds. To compute the numerical moving speed $s_c^*$ of c(t, x), we shall find the level set x(t) so that $c(t,x(t))=\frac{1}{2}(\lambda -\frac{\gamma }{k})=0.375$ for different value of t. Similarly, we find x(t) so that $n(t,x(t))=\frac{1}{2k}=0.25$, which can be used to compute the numerical moving speed $s_n^*$ of n(t, x). By finding the difference in x coordinates in the level set and dividing by the corresponding time difference at each snapshot time, we will get the numerical calculation speed. Taking snapshots of c and n when $t=350,360,370,...450$, respectively, shown in Fig. 2, we can derive the moving speeds at different times. When the speed difference among them is less than $10^{-3}$, we stop the computation and take their value as the final result. By computation, we have $s_c^*=1.4145$ and $s_n^*=1.4140$. Note that the linear speed $s_0=2\sqrt{1-\lambda }=1.4142$. Therefore, we obtain $s_c^*\simeq s_n^*\simeq s_0$.

Fig. 1

(Color online) Simulation of (c, n)(t, x) under the case MM. Here, $\lambda =0.5, \gamma =0.5, k=2$. The left column are the 2-D figures from the top view. Figures in the right column are the 3-D movements of c and n as time t increases

Fig. 2

(Color online) Snapshots of c(t, x) and n(t, x)’s movements. The graph on the left is c(t, x), and the right one is n(t, x) at different time $t=350,360,370,...450$

Fig. 3

(Color online) Simulation of (c, n)(t, x) in the case MN. Here, $\lambda =0.5, \gamma =1.5, k=2$. The left column are the 2-D figures from the top view. Figures in the right column are the 3-D movements of c and n as time t increases

Example 5.2

Let $\lambda =0.5, \gamma =1.5, k=2$. The initial data are defined in (5.1).

In this case, we get $e_{\beta }=(0.25,0.5), e_0=(0,0)$. Note that MN holds. The detailed simulations can be seen in Fig. 3. The 3-D figures of c(t, x) and n(t, x) are shown in the right column. The left column is the same movement from the top view. Clearly, the solution propagates to the right and stablizes into a traveling wave. After $t=400$, we compute the numerical moving speeds $s_c^*=1.4150$ and $s_n^*=1.4160$ with a error in $5\times 10^{-3}$. Note that $s_0= 1.4142$. Therefore, we get $s_c^*\simeq s_n^*\simeq s_0.$ Linear selection is realized in this case. We should mention that the solution c is not monotone in x and there is a hump appearing near the sharp change of solution value.

Example 5.3

Let $\lambda =1.5, \gamma =0.5, k=2.$ The initial data are defined as in (5.1).

Note that $e_{\beta }=(0.75,0.5), e_0=(0,0)$ and BM holds. The detailed simulations can be seen in Fig. 4. The 3-D figures of c(t, x) and n(t, x) are shown in the right column. The left column is the same movement from the top view. Under BM, $e_{\beta }$ and $e_0$ are both stable. From Fig. 4, the solution propagates to the right (that is $s>0$), which means that the cell population in the system is persistent when time t is large.

Fig. 4

(Color online) Simulation of (c, n)(t, x) in the case BM. Here, $\lambda =1.5, \gamma =0.5, k=2.$ The left column are the 2-D figures from the top view. Figures in the right column are the 3-D movements of c and n as time t increases

Fig. 5

(Color online) In the left figure, we have $\gamma =\lambda =0.5$. It shows the relashionship between the numerical moving speed $s^*$ and k for fixed $\gamma$ and $\lambda$. The transition of speed selection in terms of $k^*$ over $\gamma$, when $\lambda =0.6$, is in the right figure

For the left picture in Fig. 5, we choose $\gamma =\lambda =0.5$. We notice that the numerical speed $s^*$ is non-decreasing with respect to k for fixed $\gamma$ and $\lambda$, see the dot-sign (red) curve. The straight (blue) line is the linear speed, and $s_0=2\sqrt{1-\lambda }\approx 1.4142$. There exists a turning point $k^*$ such that two curves coincide (with $s^*=s_0$) for $k<k^*$, and $s^*>s_0$ for $k>k^*$. When $\gamma =\lambda =0.5$, our numerical result shows $k^*\approx 6.5$.

Finally, we want to provide the transition of speed selection in terms of two parameters k and $\gamma$. There are three curves in the right picture of Fig. 5. The plus-sign (blue) curve is $\gamma =k$. The values of $\gamma$ and k should be chosen below this curve due to the condition $\frac{\gamma }{k}<1$. The circle-sign (black) curve is $\gamma =\lambda k=0.6k$ which corresponds to the KPP-Fisher case. If $\gamma$ and k are chosen between the plus-sign (blue) curve and the circle-sign (black) curve, then we have $\lambda k-\gamma <0$ (non-monotonic case) and the minimal speed is always the linear speed; When $\gamma$ and k are chosen below the circular(black) curve, then $\lambda k-\gamma >0$ (monotonic case). The dot-sign (red) curve shows the relationship $k^*$ on $\gamma$, where $k^*$, defined in Theorem 2.9, is the transition (or turning point) for the minimal speed selection. If the parameter value is above the dot-sign (red) curve, the minimal wave speed is linearly selected; Otherwise, if the parameter value is below the dot-sign (red) curve, the minimal wave speed is nonlinearly selected.

6 Discussion

In this work, we investigated the existence of traveling waves and the minimal wave speed selection to a model of precursor and differentiated cells (1.2). Due to the non-isolated equilibrium $e_0$, most current theories can not be directly applied to prove the existence of traveling waves. Analyzing the propagation dynamics of system (1.2) becomes tricky and challenging. Since model (1.2) has rich and complex propagation properties when the system parameters are in various value regions, we discussed it in three cases, MM-BM. Developing new ideas such as continuation via perturbation in a weighted functional space and construction of auxiliary parabolic non-local model, we successfully established the existence of traveling waves under monostable monotone, monostable non-monotone, and bistable monotone cases.

For the case MM, we also derived necessary and sufficient conditions to distinguish between linear and nonlinear selection of the minimal speed (see Theorems 2.7 and 2.9), and found the decay rate of the minimal traveling wave as $z\rightarrow \infty$ in terms of the parameter of k (see Theorem 2.13). Here, Theorems 2.7 and 2.8 are a development of the abstract results of speed selection in [31] to this system with infinite equlibria. We proved the existence of the transition point $k^*$ for the minimal speed selection when $\lambda$ and $\gamma$ are fixed. This result can also be seen in the picture of Fig. 5. Two explicit estimates about $k^*$(see Theorems 2.11 and 2.12) were provided. For the case MN, we proved that the minimal wave speed is always linearly selected without any extra requirements on system parameters. For the case BM, we showed the existence and uniqueness of the bistable wavefront (see Theorems 4.6 and 4.8), with a positive wave speed.

Back to the original system (1.1), the parameter $k=\frac{\rho _1}{\rho _2}$ is the ratio of the carrying capacities of the precursor and differentiated cell populations. In view of Theorem 2.9, the spreading speed becomes faster when the carrying capacitiy $\rho _1$ of the precursor cell population becomes bigger, and when the carrying capacitiy $\rho _2$ of the differentiated cell population becomes smaller. This is in agreement with biological interpretation. Otherwise, the spreading speed is slower.

Our system (1.2) is closely related, but different from the following model with Belousov–Zhabotinsky reaction [17, 22, 23, 36, 37, 43, 44]

$$\begin{aligned} \left\{ \begin{aligned}&u_t=D_uu_{xx}+u(1-u-rv), \\&v_t=D_vv_{xx}-buv.\\ \end{aligned} \right. \end{aligned}$$

(6.1)

which is always monotonic, while system (1.2) exhibits the so-called non-monotonic dynamics in some parameter region. We shall show the novelty of our results in comparison to previous works through the following items.

• 1. Gibbs [17] studied the existence of the traveling wave to the system (6.1) with $D_u=1$ and $D_v=0$. Note that letting $\gamma =0$, our system (1.2) can be changed to (6.1) by setting $D_u=1$ and $D_v=0$. This means the model in [17] is a special case of our system (1.2). Furthermore, determining the value of the minimal wave speed $s_{\min }$ is a crucial and difficult problem, which has not been touched in [17]. In fact, there is limited research involving this problem of the system (6.1) with $D_u=1$ and $D_v=0$. By letting $\gamma =0$, our results (Theorems 2.7–2.9 and 2.11–2.12) can give novel estimates for $s_{\min }$ of this system.

• 2. The novel ideas we developed here to prove the existence of the traveling wave for both monostable and bistable cases can also be extended and applied to various new systems, especially high dimensional systems with non-isolated equilibrium points and systems with non-local term. For example, if the diffusion rate in n-equation to system (1.1) is non-zero, our method still works by defining an implicit operator for the solution n in n-equation and transforming the original system to a non-local equation. However, the shooting method used in [17, 22, 23] is not suitable for high dimensional system or the system with non-local term. The regular super-solutions method developed in [43, 44] may not be applied to our degenerate system (1.2), and it is also challenging to apply it to other more complex new systems due to the need to construct a suitable upper solution and to prove the boundary conditions at infinity.

• 3. Hou[18] studied the special case $\gamma =0$ of system (1.2), which can be transformed into system (6.1) with $D_u=1$ and $D_v=0$. The author focused exclusively on the monostable case and provided a sufficient condition for the linear selection, outlined as:

  $$\begin{aligned} 0<\lambda <\frac{2}{k+2+\sqrt{2}k}. \end{aligned}$$

  (6.2)

  By Theorem 2.11, we have

  $$\begin{aligned} 0<\lambda <\frac{2}{k+2} \end{aligned}$$

  (6.3)

  for the linear selection when $\gamma =0$ and $\lambda <2/3$. Thus, in this case, our linear selection condition (6.3) is better than (6.2) in [18].

• 4. The exact value of the minimal wave speed $s_{\min }$ for the system (6.1) with $D_u=D_v=1$ or the delayed Belousov–Zhabotinsky reaction model [43] cannot be worked out[43, 44]. Our methods in Theorems 2.7–2.9 and 2.11–2.12 can be applied to these two systems to get new bound estimates for $s_{\min }$.