Traveling waves of a full parabolic attraction-repulsion chemotaxis systems with logistic sources

TL;DR

This paper establishes the existence and properties of traveling wave solutions in a chemotaxis system with attraction-repulsion dynamics and logistic growth, identifying parameter-dependent wave speeds and their limits.

Contribution

It introduces new conditions for the existence of traveling waves in a chemotaxis system with logistic sources, including parameter-dependent wave speed bounds and asymptotic behaviors.

Findings

Existence of traveling waves for wave speeds within specific bounds.

Wave speed limits tend to infinity as chemotactic sensitivities approach zero.

No traveling wave solutions exist for speeds below a critical threshold.

Abstract

In this paper, we study traveling wave solutions of the chemotaxis systems \begin{equation} \begin{cases} u_{t}=\Delta u -\chi_1\nabla( u\nabla v_1)+\chi_2 \nabla(u\nabla v_2 )+ u(a -b u), \qquad \ x\in\mathbb{R} \\ \tau\partial_tv_1=(\Delta- \lambda_1 I)v_1+ \mu_1 u, \qquad \ x\in\mathbb{R}, \\ \tau\partial v_2=(\Delta- \lambda_2 I)v_2+ \mu_2 u, \qquad \ \ x\in\mathbb{R}, \end{cases} (0.1) \end{equation} where $\tau>0,\chi_{i}> 0,\lambda_i> 0,\ \mu_i>0$ ($i=1,2$) and $\ a>0,\ b> 0$ are constants, and $N$ is a positive integer. Under some appropriate conditions on the parameters, we show that there exist two positive constant $ 0<c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)$ such that for every $c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)$, $(0.1)$ has a traveling wave solution $(u,v_1,v_2)(x,t)=(U,V_1,V_2)(x-ct)$ connecting $(\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2})$ and $(0,0,0)$ satisfying $$ \lim_{z\to \infty}\frac{U(z)}{e^{-\mu z}}=1, $$ where $\mu\in (0,\sqrt a)$ is such that $c=c_\mu:=\mu+\frac{a}{\mu}$. Moreover, $$ \lim_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)=\infty$$ and $$\lim_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)= c_{\tilde{\mu}^*}, $$ where $\tilde{\mu}^*={\min\{\sqrt{a}, \sqrt{\frac{\lambda_1+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_2+\tau a}{(1-\tau)_{+}}}\}}$. We also show that $(0.1)$ has no traveling wave solution connecting $(\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2})$ and $(0,0,0)$ with speed $c<2\sqrt{a}$.

Equations

$$\{\begin{array}{l}{u}_t=\Delta u-{\chi }_1\nabla (u\nabla v_1)+{\chi }_2\nabla (u\nabla v_2)+u(a-bu),x\in \mathbb{R}\\ \tau {\partial }_t{v}_1=(\Delta -{\lambda }_{1}I)v_1+{\mu }_{1}u,x\in \mathbb{R},\\ \tau \partial v_2=(\Delta -{\lambda }_{2}I)v_2+{\mu }_{2}u,x\in \mathbb{R},\end{array}$$

$$\underset{z\to \infty }{\lim }\frac{U(z)}{e^{-\mu z}}=1,$$

$$\underset{({\chi }_1,{\chi }_2)\to (0^{+},0^{+}))}{\lim }{c}^{\ast \ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=\infty$$

$$\underset{({\chi }_1,{\chi }_2)\to (0^{+},0^{+}))}{\lim }{c}^{\ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=c_{{\tilde{\mu }}^{\ast }},$$

$$\{\begin{array}{l}{u}_t=\nabla \cdot (m(u)\nabla u-\chi (u,v)\nabla v)+f(u,v),x\in \Omega \\ \tau v_t=\Delta v+g(u,v),x\in \Omega \end{array}$$

$$\{\begin{array}{l}{\partial }_{t}u=\Delta u-{\chi }_1\nabla (u\nabla v_1)+{\chi }_2\nabla (u\nabla v_2)+u(a-bu),x\in {\mathbb{R}}^N\\ \tau {\partial }_t{v}_1=(\Delta -{\lambda }_{1}I)v_1+{\mu }_{1}u,x\in {\mathbb{R}}^N\\ \tau {\partial }_t{v}_2=(\Delta -{\lambda }_{2}I)v_2+{\mu }_{2}u,\text{in}x\in {\mathbb{R}}^N.\end{array}$$

$$\{\begin{array}{l}{\partial }_{t}u={\partial }_{xx}u+{\partial }_x(u{\partial }_x({\chi }_2{v}_2-{\chi }_1{v}_1))+u(a-bu),x\in \mathbb{R},\\ \tau {\partial }_t{v}_1={\partial }_{xx}{v}_1-{\lambda }_1{v}_1+{\mu }_{1}u,x\in \mathbb{R},\\ \tau {\partial }_t{v}_2={\partial }_{xx}{v}_2-{\lambda }_2{v}_2+{\mu }_{2}u,x\in \mathbb{R},\end{array}$$

$$\{\begin{array}{l}{\partial }_{t}u={\partial }_{xx}u+c{\partial }_{x}u+{\partial }_x(u{\partial }_x({\chi }_2{v}_2-{\chi }_1{v}_1))+u(a-bu),x\in \mathbb{R},\\ 0={\partial }_{xx}{v}_1+c\tau {\partial }_x{v}_1-{\lambda }_1{v}_1+{\mu }_{1}u,x\in \mathbb{R},\\ 0={\partial }_{xx}{v}_2+c\tau {\partial }_x{v}_2-{\lambda }_2{v}_2+{\mu }_{2}u,x\in \mathbb{R},\end{array}$$

$$\overline{M}:=\int_{0}^{\infty}\Big{(}\chi_{2}\lambda_{2}\mu_{2}e^{-\lambda_{2}s}-\chi_{1}\lambda_{1}\mu_{1}e^{-\lambda_{1}s}\Big{)}_{+}ds,$$

$$\underline{M}:=\int_{0}^{\infty}\Big{(}\chi_{2}\lambda_{2}\mu_{2}e^{-\lambda_{2}s}-\chi_{1}\lambda_{1}\mu_{1}e^{-\lambda_{1}s}\Big{)}_{-}ds,$$

$$K:={\int }_0^{\infty }\left|\frac{{\mu }_1{\chi }_1{e}^{-{\lambda }_{1}s}}{2\sqrt{\pi s}}-\frac{{\chi }_2{\mu }_2{e}^{-{\lambda }_{2}s}}{2\sqrt{\pi s}}\right|ds.$$

$$\overline{M}-\underline{M}=\int_{0}^{\infty}\Big{(}\chi_{2}\lambda_{2}\mu_{2}e^{-\lambda_{2}s}-\chi_{1}\lambda_{1}\mu_{1}e^{-\lambda_{1}s}\Big{)}ds=\chi_{2}\mu_{2}-\chi_{1}\mu_{1}.$$

$$\Vert u(\cdot ,t;u_0){\Vert }_{\infty }\le \max \left\{\Vert u_0{\Vert }_{\infty },\frac{a}{b-\underline{M}-c\tau K}\right\},\forall t\ge 0$$

$$\Vert v_i(\cdot ,t;u_0){\Vert }_{\infty }\le \max \left\{\frac{{\mu }_i\Vert u_0{\Vert }_{\infty }}{{\lambda }_i},\frac{a{\mu }_i}{(b-\underline{M}-c\tau K){\lambda }_i}\right\},\forall t\ge 0,i=1,2.$$

$$\underset{t\to \infty }{\lim }\Vert u(\cdot ,t;u_0)-\frac{a}{b}{\Vert }_{\infty }=0$$

$$\underset{t\to \infty }{\lim }\Vert v_i(\cdot ,t;u_0)-\frac{a{\mu }_i}{b{\lambda }_i}{\Vert }_{\infty }=0,i=1,2.$$

$$\overline{M}(s):=({\chi }_2{\mu }_2{\lambda }_2{e}^{-{\lambda }_{2}s}-{\chi }_1{\mu }_1{\lambda }_1{e}^{-{\lambda }_{1}s}{)}_{+},\forall s>0,$$

$$\underline{M}(s):=({\chi }_2{\mu }_2{\lambda }_2{e}^{-{\lambda }_{2}s}-{\chi }_1{\mu }_1{\lambda }_1{e}^{-{\lambda }_{1}s}{)}_{-},\forall s>0,$$

$${\overline{M}}_{\tau ,\mu }(s):=\overline{M}(s)e^{-(\tau \mu c_{\mu }-{\mu }^2)s},\forall \mu ,s>0,{\overline{M}}_{\tau ,\mu }:={\int }_0^{\infty }{\overline{M}}_{\tau ,\mu }(s)ds,$$

$${\underline{M}}_{\tau ,\mu }(s):=\underline{M}(s)e^{-(\tau \mu c_{\mu }-{\mu }^2)s},\forall \mu ,s>0,{\underline{M}}_{\tau ,\mu }:={\int }_0^{\infty }{\underline{M}}_{\tau ,\mu }(s)ds,$$

$$K_{\tau ,\mu }={\int }_0^{\infty }\left|{\chi }_1{\mu }_1{e}^{-{\lambda }_{1}s}-{\chi }_2{\mu }_2{e}^{-{\lambda }_{2}s}\right|\frac{(1+\mu \sqrt{\pi s})e^{-(\tau \mu c_{\mu }-{\mu }^2)s}}{\sqrt{\pi s}}ds,\forall \mu >0.$$

$$f(\mu )=\max \{2(\underline{M}+\tau c_{\mu }K),{\chi }_1{\mu }_1-{\chi }_2{\mu }_2+(\tau c_{\mu }+\mu )K_{\tau ,\mu }+{\overline{M}}_{\tau ,\mu }\},$$

$$c^{\ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=c_{{\mu }^{\ast }}$$

$$c^{\ast \ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=\underset{\mu \to {\mu }^{\ast \ast }+}{\lim }{c}_{\mu }.$$

$$\underset{z\to \infty }{\lim }\frac{U(z)}{e^{-\mu z}}=1,$$

$$\underset{({\chi }_1,{\chi }_2)\to (0^{+},0^{+}))}{\lim }{c}^{\ast \ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=\infty$$

$$\underset{({\chi }_1,{\chi }_2)\to (0^{+},0^{+}))}{\lim }{c}^{\ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=c_{{\tilde{\mu }}^{\ast }},$$

$$u_t=\Delta u+u(a-bu),x\in {\mathbb{R}}^N,t>0,$$

$$u_t=\Delta u+uf(t,x,u),x\in {\mathbb{R}}^N,$$

$$\{\begin{array}{l}{u}_t=u_{xx}+(c+({\chi }_2{v}_2-{\chi }_1{v}_1{)}_x)u_x+u(a+({\chi }_2{\lambda }_2{v}_2-{\chi }_1{\lambda }_1{v}_1)+c\tau ({\chi }_1{v}_1-{\chi }_2{v}_2{)}_x)\\ -(b+{\chi }_2{\mu }_2-{\chi }_1{\mu }_1)u^2,x\in \mathbb{R}\\ 0={\partial }_{xx}{v}_1+c\tau {\partial }_x{v}_1-{\lambda }_1{v}_1+{\mu }_{1}u,x\in \mathbb{R}\\ 0={\partial }_{xx}{v}_2+c\tau {\partial }_x{v}_2-{\lambda }_2{v}_2+{\mu }_{2}u,x\in \mathbb{R}.\end{array}$$

Full text

[TABLE]

where $\tau>0,\chi_{i}>0,\lambda_{i}>0,\ \mu_{i}>0$ ($i=1,2$) and $\ a>0,\ b>0$ are constants, and $N$ is a positive integer. Under some appropriate conditions on the parameters, we show that there exist two positive constant $0<c^{*}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})<c^{**}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})$ such that for every $c^{*}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})\leq c<c^{**}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})$, (0.1) has a traveling wave solution $(u,v_{1},v_{2})(x,t)=(U,V_{1},V_{2})(x-ct)$ connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ satisfying

[TABLE]

where $\mu\in(0,\sqrt{a})$ is such that $c=c_{\mu}:=\mu+\frac{a}{\mu}$. Moreover,

[TABLE]

and

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where $\tilde{\mu}^{*}={\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}}$. We also show that (0.1) has no traveling wave solution connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ with speed $c<2\sqrt{a}$.

Key words. Parabolic-parabolic-parabolic chemotaxis system, logistic source, classical solution, local existence, global existence, asymptotic stability, traveling wave solutions.

2010 Mathematics Subject Classification. 35B35, 35B40, 35K57, 35Q92, 92C17.

1 Introduction and the Statement of the Main Results

Chemotaxis describes the oriented movement of biological cells or organisms in response to chemical gradients. The oriented movement of cells has a crucial role in a wide range of biological phenomena. At the beginning of 1970s, Keller and Segel (see [20], [21]) introduced systems of partial differential equations of the following form to model the time evolution of both the density $u(x,t)$ of a mobile species and the density $v(x,t)$ of a chemoattractant,

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complemented with certain boundary condition on $\partial\Omega$ if $\Omega$ is bounded, where $\Omega\subset{\mathbb{R}}^{N}$ is an open domain; $\tau\geq 0$ is a non-negative constant linked to the speed of diffusion of the chemical; the function $\chi(u,v)$ represents the sensitivity with respect to chemotaxis; and the functions $f$ and $g$ model the growth of the mobile species and the chemoattractant, respectively. In literature, (1.1) is called the Keller-Segel model or a chemotaxis model.

Since the works by Keller and Segel, a rich variety of mathematical models for studying chemotaxis has appeared (see [1, 6, 7, 12, 15, 16, 19, 28, 37, 38, 39, 43, 44, 45, 46, 47, 48, 50, 49, 51], and the references therein). The reader is referred to [14, 17] for some detailed introduction into the mathematics of KS models. In the current paper, we consider chemoattraction-repulsion process on the whole space in which cells undergo random motion and chemotaxis towards attractant and away from repellent [26, 50, 49]. Moreover, we consider the model with proliferation and death of cells. These lead to the model of partial differential equations as follows:

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The objective of the current paper is to study the existence of traveling wave solutions of (1.2) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$. A nonnegative solution $(u(x,t),v_{1}(x,t),v_{2}(x,t))$ of (1.2) defined for every $(x,t)\in{\mathbb{R}}^{N+1}$ is called a traveling wave solution connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ and propagating in the direction $\xi\in S^{N-1}$ with speed $c$ if it is of the form $(u(x,t),v_{1}(x,t),v_{2}(x,t))=(U(x\cdot\xi-ct),V_{1}(x\cdot\xi-ct),V_{2}(x\cdot\xi-ct))$ with $\lim_{z\to-\infty}(U(z),V_{1}(z),V_{2}(z))=(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $\lim_{z\to\infty}(U(z),V_{1}(z),V_{2})=(0,0,0)$.

Observe that, if $(u(x,t),v_{1}(x,t),v_{2}(x,t))=(U(x\cdot\xi-ct),V_{1}(x\cdot\xi-ct),V_{2}(x\cdot\xi-ct))$ $(x\in{\mathbb{R}}^{N},t\in{\mathbb{R}})$ is a traveling wave solution of (1.2) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ and propagating in the direction $\xi\in S^{N-1}$, then $(u,v_{1},v_{2})=(U(x-ct),V_{1}(x-ct),V_{2}(x-ct))$ ($x\in{\mathbb{R}}$) is a traveling wave solution of

[TABLE]

connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$. Conversely, if $(u(x,t),v_{1}(x,t),v_{2}(x,t))=(U(x-ct),V_{1}(x-ct),V_{2}(x,t))$ ($x\in{\mathbb{R}},t\in{\mathbb{R}}$) is a traveling wave solution of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$, then $(u,v_{1},v_{2})=(U(x\cdot\xi-ct),V_{1}(x\cdot\xi-ct),V_{2}(x\cdot\xi-ct))$ $(x\in{\mathbb{R}}^{N})$ is a traveling wave solution of (1.2) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ and propagating in the direction $\xi\in S^{N-1}$. In the following, we will then study the existence of traveling wave solutions of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$.

Observe also that $(u,v_{1},v_{2})=(U(x-ct),V_{1}(x-ct),V_{2}(x-ct))$ is a traveling wave solution of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ with speed $c$ if and only if $(u,v_{1},v_{2})=(U(x),V_{1}(x),V_{2}(x))$ is a stationary solution of the following parabolic-elliptic-elliptic chemotaxis system,

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connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$. In this paper, to study the existence of traveling wave solutions of (1.3), we study the existence of constant $c$’s so that (1.4) has a stationary solution $(U(x),V_{1}(x),V_{2}(x))$ satisfying $(U(-\infty),V_{1}(-\infty),V_{2}(-\infty))=(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(U(\infty),V_{1}(\infty)$, $V_{2}(\infty))=(0,0,0)$.

To this end, we first establish some results on the global existence of classical solutions of (1.4) and the stability of constant solution $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$, which are of independent interest. Note that, for fixed $c$, it can be proved by the similar arguments as those in [34] that for any $u_{0}\in C_{\rm unif}^{b}({\mathbb{R}})$ with $u_{0}\geq 0$, there is $T_{\max}(u_{0})\in(0,\infty]$ such that (1.4) has a unique classical solution $(u(x,t;u_{0}),v_{1}(x,t;u_{0}),v_{2}(x,t;u_{0}))$ on $[0,T_{\max}(u_{0}))$ with $u(x,0;u_{0})=u_{0}(x)$. Furthermore, if $T_{\max}(u_{0})<\infty$, then $\lim_{t\to T_{\max}^{-}(u_{0})}\|u(\cdot,t;u_{0})\|_{\infty}=\infty$.

In [32] together with Wenxian Shen, we studied the existence of traveling wave solutions of (1.3) when $\tau=0$. When $\tau>0$, the dynamics of (1.3) is more complex and most of the techniques developed in [32], for $\tau=0$, can not be adopted directly. So, nontrivial modification and new techniques are needed to handle the full parabolic system (1.3). Also, the results established in [32] make use of the stability of the positive constant equilibria proved in [31]. To our best knowledge, the stability of positive constant equilibria of (1.3) still remains an open problem. In the current paper we established some new results for $\tau=0$, mainly the existence of the so call ”critical wave”, see Theorem C (ii) below.

For clarity of the statements of our main results on the existence of global classical solutions and stability of the steady solution $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ of (1.4), it would be convenience to introduce some definitions. For every real number r, we let $(r)_{+}=\max\{0,r\}$ and $(r)_{-}=\max\{0,-r\}$. Let

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and

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Observe that

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Our main result on the existence of global solution of (1.4) reads as follow.

**Theorem A. ** Suppose that $c\geq 0$ and $b>\underline{M}+c\tau K$ where $\underline{M}$ and $K$ are given by (1.6) and (1.7) respectively. Then for every nonnegative initial function $u_{0}\in C^{b}_{\rm unif}({\mathbb{R}})$, (1.4) has a unique global classical solution $(u(x,t;u_{0}),v_{1}(x,t;u_{0}),v_{2}(x,t;u_{0}))$ satisfying $u(x,0;u_{0})=u_{0}(x)$. Furthermore, it holds that

[TABLE]

and

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Remark 1.1**.**

Note that $\underline{M}\leq\chi_{1}\mu_{1}$ and equality holds if and only if $\chi_{2}=0$. Next, we look at the case case $c=0$ in (1.4). In this case, we have

1) If $b\geq\chi_{1}\mu_{1}$, (1.4) with $c=0$ and $\chi_{2}>0$ always has a unique bounded and nonnegative global classical solution for every given $u_{0}\in C^{b}_{\rm uinf}({\mathbb{R}})$, $u_{0}\geq 0$.

2) If $\lambda_{2}>\lambda_{1}$ and $\chi_{2}\lambda_{2}\mu_{2}\leq\chi_{1}\lambda_{1}\mu_{1}$ then $\underline{M}=\chi_{1}\mu_{1}-\chi_{2}\mu_{2}$.

3) If $\lambda_{2}>\lambda_{1}$ and $\chi_{2}\lambda_{2}\mu_{2}\geq\chi_{1}\lambda_{1}\mu_{1}$ then $\underline{M}=\left(\frac{\lambda_{2}}{\lambda_{1}}-1\right)\chi_{2}\mu_{2}\left(\frac{\chi_{1}\lambda_{1}\mu_{1}}{\chi_{2}\lambda_{2}\mu_{2}}\right)^{\frac{\lambda_{2}}{\lambda_{2}-\lambda_{1}}}.$

4) If $\lambda_{2}<\lambda_{1}$ and $\chi_{2}\lambda_{2}\mu_{2}\leq\chi_{1}\lambda_{1}\mu_{1}$ then $\underline{M}=\chi_{1}\mu_{1}-\chi_{2}\mu_{2}+\left(\frac{\lambda_{1}}{\lambda_{2}}-1\right)\chi_{1}\mu_{1}\left(\frac{\chi_{2}\lambda_{2}\mu_{2}}{\chi_{1}\lambda_{1}\mu_{1}}\right)^{\frac{\lambda_{1}}{\lambda_{1}-\lambda_{2}}}$.

5) If $\lambda_{2}<\lambda_{1}$ and $\chi_{2}\lambda_{2}\mu_{2}\geq\chi_{1}\lambda_{1}\mu_{1}$ then $\underline{M}=0.$

6) If $\lambda_{1}=\lambda_{2}$ then $\underline{M}=(\chi_{2}\mu_{2}-\chi_{1}\mu_{1})_{-}$.

Therefore, in the case $c=0$, Theorem A improves Theorem A in [31].

Next, we state our result on the stability of the positive constant equilibrium solution of (1.4).

Theorem B. Suppose that $c\geq 0$ and $b>2(\underline{M}+c\tau K)$ where $\overline{M}$ and $K$ are given by (1.6) and (1.7) respectively. Then for every initial function $u_{0}\in C^{b}_{\rm unif}({\mathbb{R}})$, with $\inf_{x\in{\mathbb{R}}}u_{0}(x)>0$, the unique global classical solution $(u(x,t;u_{0}),v_{1}(x,t;u_{0}),v_{2}(x,t;u_{0}))$ satisfying $u(x,0;u_{0})=u_{0}(x)$ of (1.4) satisfies

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and

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To state our main results on the existence of traveling wave solutions of (1.3) as well for our results in the subsequent sections, we introduce a few more notations. Let, for $0<\mu\leq\sqrt{a}$ and $\tau>0$, $c_{\mu}=\mu+\frac{a}{\mu}$,

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and

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Observe that $\overline{M}=\int_{0}^{\infty}\overline{M}(s)ds$ and $\underline{M}=\int_{0}^{\infty}\underline{M}(s)ds$ where $\overline{M}$ and $\underline{M}$ are given by (1.5) and (1.6) respectively.

Next, we state our results on the existence of traveling wave solutions of (1.3). Let us consider the auxiliary function $f\ :\ (0,\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\})\to{\mathbb{R}}$ defined by

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where $c_{\mu}=\mu+\frac{a}{\mu}$, $\underline{M}$, $K$, $\overline{M}_{\tau,\mu}$, and $K_{\tau,\mu}$ are given by (1.6), (1.7), (1.15) and (1.17) respectively. Clearly, the function $f$ is continuous. We suppose that the following standing assumption holds.

(H) $b>\inf_{\mu}f(\mu)$.

Assuming that (H) holds, we let $(\mu^{**}_{\tau},\mu^{*}_{\tau})$ denotes the right maximal open connected component of the open set $O=\{\mu\in(0,\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\})\ :\ b>f(\mu)\}$ and set

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and

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**Theorem C. ** Assume (H). Let $c^{*}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})$ and $c^{**}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})$ be given (1.18) and (1.19) respectively. The following hold.

(i)

For every $c^{*}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})<c<c^{**}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})$, (1.3) has a traveling wave solution $(u,v_{1},v_{2})(x,t)=(U,V_{1},V_{2})(x-ct)$ connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ satisfying

[TABLE]

where $\mu\in(0,\sqrt{a})$ is such that $c=c_{\mu}:=\mu+\frac{a}{\mu}=c$. Moreover,

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and

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where $\tilde{\mu}^{*}={\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}}$

(ii)

There is a traveling wave solution $(u,v_{1},v_{2})(x,t)=(U,V_{1},V_{2})(x-ct)$ of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ with speed $c^{*}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})$.

(iii)

There is no traveling wave solution $(u,v_{1},v_{2})(x,t)=(U,V_{1},V_{2}))(x-ct)$ of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ with speed $c<2\sqrt{a}$.

Let us make few comments about Theorem C. We first remark that $\tau=0$ is allowed in Theorem C. However, our result Theorem C (i) in this case is hard to compare with the results in [32]. The result in Theorem C (ii) is new in the case of $\tau=0$. Note also that when $\tau\geq 1$ and $b$ is sufficiently large, we have that $c^{*}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})=2\sqrt{a}$, which is the minimal wave of the classical Fisher-KPP equation. Hence in this case, it follows from Theorem C (ii) $\&$ (iii) that the presence of the chemotaxis signal does not affect the minimal wave speed. It remains open whether $c^{*}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})$ can be taken to be $2\sqrt{a}$ in general. It also remains open whether $c^{**}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})$ can be taken to be infinity. These questions are related to whether (1.3) has a minimum and/or maximal traveling wave speeds. Note that in the absence of the chemotaxis effect, that is $\chi_{1}=\chi_{2}=0$, the first equation in (1.3) becomes the following scalar reaction diffusion equation,

[TABLE]

which is referred to as Fisher or KPP equations due to the pioneering works by Fisher ([8]) and Kolmogorov, Petrowsky, Piscunov ([22]) on the spreading properties of (1.21). It follows from the works [8], [22], and [41] that $c^{*}_{-}$ and $c^{*}_{+}$ in Theorem C and Theorem D, respectively, can be chosen so that $c^{\ast}_{-}=c^{\ast}_{+}=2\sqrt{a}$ ($c^{*}:=2\sqrt{a}$ is called the spatial spreading speed of (1.21) in literature), and that (1.21) has traveling wave solutions $u(t,x)=\phi(x-ct)$ connecting $\frac{a}{b}$ and [math] (i.e. $(\phi(-\infty)=\frac{a}{b},\phi(\infty)=0)$) for all speeds $c\geq c^{*}$ and has no such traveling wave solutions of slower speed. Since the pioneering works by Fisher [8] and Kolmogorov, Petrowsky, Piscunov [22], a huge amount research has been carried out toward the spreading properties of reaction diffusion equations of the form,

[TABLE]

where $f(t,x,u)<0$ for $u\gg 1$, $\partial_{u}f(t,x,u)<0$ for $u\geq 0$ (see [2, 3, 4, 5, 9, 10, 23, 24, 27, 29, 30, 35, 36, 41, 42, 52], etc.). The existence of minimal wave speeds becomes a natural question to study as it is related to whether the presence of the chemotaxis speeds up or slows down the minimal wave speed. A partial and satisfactory answer to this is obtained by Theorem C when $\tau\geq 1$ and the self-limitation rate $b$ of the mobile species is sufficiently large. We plan to study these questions in our future works.

The rest of the paper is organized as follows. In section 2 we study the dynamics of classical solutions of (1.4) and prove Theorems A $\&$ B. Section 3 is to develop the machinery and set up the right frame work to study the existence of traveling solution. Finally, based on the results established in section 3, we complete the proof of Theorem C in section 4.

2 Dynamics of the induced parabolic-elliptic-elliptic chemotaxis system

In this section, we study the global existence of classical solutions of (1.4) with given nonnegative initial functions and the stability of the constant solution $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ of (1.4), and prove Theorems A and B.

For fixed $c$, it can be proved by the similar arguments as those in [34] that for any $u_{0}\in C_{\rm unif}^{b}({\mathbb{R}})$ with $u_{0}\geq 0$, there is $T_{\max}(u_{0})\in(0,\infty]$ such that (1.4) has a unique classical solution $(u(x,t;u_{0}),v_{1}(x,t;u_{0}),v_{1}(x,t;u_{0}))$ on $[0,T_{\max}(u_{0}))$ with $u(x,0;u_{0})=u_{0}(x)$.

Observe that (1.4) is equivalent to

[TABLE]

Proof of Theorem A.

For given $u_{0}\in C_{\rm unif}^{b}({\mathbb{R}})$ and $T>0$, choose $C_{0}>0$ such that $0\leq u_{0}\leq C_{0}$ and $C_{0}\geq\frac{a}{b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-\overline{M}-c\tau K}$, and let

[TABLE]

For given $u\in\mathcal{E}(u_{0},T)$, let $v_{1}(x,t;u)$ and $v_{2}(x,t;u)$ be the solutions of the second and third equations in (2.1). Then for every $u\in\mathcal{E}(u_{0},T)$, $x\in{\mathbb{R}},t\geq 0$, we have that

[TABLE]

and

[TABLE]

Using the fact that $\int_{{\mathbb{R}}}e^{-z^{2}}dz=\sqrt{\pi}$ and $\int_{0}^{\infty}ze^{-z^{2}}dz=\frac{1}{2}$, it follows from (2.2) and (2.3) that

[TABLE]

and

[TABLE]

Thus

[TABLE]

For given $u\in\mathcal{E}(u_{0},T)$, let $\tilde{U}(x,t;u)$ be the solution of the initial value problem

[TABLE]

Since $u_{0}\geq 0$, comparison principle for parabolic equations implies that $\tilde{U}(x,t)\geq 0$ for every $x\in{\mathbb{R}},\ t\in[0,T]$. It follows from (2.4) that

[TABLE]

Hence, comparison principle for parabolic equations implies that

[TABLE]

where $\tilde{u}$ is the solution of the ODE

[TABLE]

Since $b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}>0$, the function $\tilde{u}(\cdot,\|u_{0}\|_{\infty})$ is defined for all time and satisfies $0\leq\tilde{u}(t,\|u_{0}\|_{\infty})\leq\max\{\|u_{0}\|_{\infty},\ \frac{a+(\overline{M}+c\tau K)C_{0}}{b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}}\}$ for every $t\geq 0$. This combined with (2.6) yields that

[TABLE]

Note that the second inequality in (2.7) follows from the fact $\frac{a+(\overline{M}+c\tau K)C_{0}}{b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}}\leq C_{0}$ whenever $C_{0}\geq\frac{a}{b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-\overline{M}-c\tau K}$. It then follows that $\tilde{U}(\cdot,\cdot;u)\in\mathcal{E}(u_{0},T)$.

Following the proof of Lemma 4.3 in [33], we can prove that the mapping $\mathcal{E}(u_{0},T)\ni u\mapsto\tilde{U}(\cdot,\cdot;u)\in\mathcal{E}(u_{0},T)$ has a fixed point $\tilde{U}(x,t;u)=u(x,t)$. Note that if $\tilde{U}(\cdot,\cdot,u)=u$, then $(u(\cdot,\cdot),v_{1}(\cdot,\cdot;u),v_{2}(\cdot,\cdot;u))$ is a solution of (1.4). Since $u(\cdot,\cdot;u_{0})$ is the only solution of (1.4), thus $u(\cdot,\cdot,u_{0})=u(\cdot,\cdot)$. Hence, it follows from (2.7) that for any $T>0$,

[TABLE]

This implies that $T_{\max}(u_{0})=\infty$. Recall fron (1.8) that $\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-\overline{M}=-\underline{M}$. Thus, inequalities (1.9) and (1.12) follow from (2.8) with $C_{0}=\max\{\|u_{0}\|_{\infty},\frac{a}{b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-\overline{M}-c\tau K}\}$. ∎

In the next result, we prove the stability of the positive constant solution $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ of (1.4).

Proof of Theorem B.

Let $u_{0}\in C^{b}_{\rm uinf}({\mathbb{R}})$ with $\inf_{x\in{\mathbb{R}}}u_{0}(x)>0$ be given. By Theorem A, we have that $\sup_{x\in{\mathbb{R}},\ t\geq 0}u(x,t;u_{0})<\infty$. Hence

[TABLE]

Thus, we have that

[TABLE]

Therefore, comparison principle for parabolic equations implies that

[TABLE]

where $U^{l}(t)$ is the solution of the ODE

[TABLE]

Since $\inf_{x}u_{0}(x)>0$, we have that $U^{l}(t)>0$ for every $t\geq 0$. Thus

[TABLE]

Next, let us define

[TABLE]

Then for any $\epsilon>0$, there is $T_{\epsilon}>0$ such that

[TABLE]

It follows from (2.2) and (2.10) that for every $x\in{\mathbb{R}}$ and $t\geq T_{\varepsilon}$ we have that

[TABLE]

Similar arguments leading to the last inequality yield that

[TABLE]

Similarly, using (2.3) and (2.10), we have for every $x\in{\mathbb{R}}$ and $t\geq T_{\varepsilon}$,

[TABLE]

and

[TABLE]

Thus, using inequalities (2) and (2), for every $t\geq T_{\varepsilon}$, we have that

[TABLE]

Thus, comparison principle for parabolic equations implies that

[TABLE]

where $\overline{U}^{\varepsilon}(t)$ is the solution of the ODE

[TABLE]

Since $\|u(\cdot,T_{\varepsilon};u_{0})\|_{\infty}>0$ and $b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}>0$, we have that $\overline{U}^{\varepsilon}(t)$ is defined for all time and satisfies

[TABLE]

Thus, it follows from (2.15) that

[TABLE]

Letting $\varepsilon\to 0$ in the last inequality, we obtain

[TABLE]

But, note that $(a+\overline{u}(c\tau K+\overline{M})-\underline{u}(c\tau K+\underline{M}))_{+}=0$ would implies that $\underline{u}=\overline{u}=0$. Which in turn yields that

[TABLE]

This is impossible since $a>0$. Hence $(a-\chi\underline{u}+\frac{\chi\tau c}{2}(\overline{u}-\underline{u}))_{+}>0$. Whence

[TABLE]

Next, using again inequalities (2.12) and (2.14), for every $t\geq T_{\varepsilon}$, we have that

[TABLE]

Thus, comparison principle for parabolic equations implies that

[TABLE]

where $\underline{U}^{\varepsilon}(t)$ is the solution of the ODE

[TABLE]

From (2.9) we know that $\inf_{x\in{\mathbb{R}}}u(x,T_{\varepsilon},u_{0})>0$. Thus, using the fact $b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}>0$, we have that $\underline{U}^{\varepsilon}$ is defined for all time and satisfies

[TABLE]

Thus, it follows from (2.17) that

[TABLE]

Letting $\varepsilon$ tends to 0 in the last inequality, we obtain that

[TABLE]

It follows from inequality (2.16) and (2.18) that

[TABLE]

Which is equivalent to

[TABLE]

Observe that $b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-2c\tau K-\overline{M}-\underline{M}=b+2\chi_{2}\mu_{2}-2\chi_{1}\mu_{1}-2c\tau K>0$. Thus, it follows from (2.19) that $\overline{u}=\underline{u}$. Thus it follows from (2.16) and (2.18) that $(b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-\overline{M}+\underline{M})\overline{u}=a$. Combining this with (1.8), we conclude that $\overline{u}=\underline{u}=\frac{a}{b}$. ∎

3 Super- and Sub- solutions

In this section, we will construct super- and sub-solutions of some related equations of (1.4), which will be used to prove the existence of traveling wave solutions of (1.3) in next section. Throughout this section we suppose that $a>0$ and $b>0$ are given positive real numbers.

Note that, for given $c$, to show the existence of a traveling wave solution of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ is equivalent to show the existence of a stationary solution connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$.

For every $\tau>0,\ 0<\mu<\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}$ and $x\in{\mathbb{R}}$ define

[TABLE]

and set

[TABLE]

Note that for every fixed $\tau>0$ and $0<\mu<\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}$, $\lambda_{i}+\tau\mu c_{\mu}-\mu^{2}>0$ and the function $\varphi_{\tau,\mu}$ is decreasing, infinitely many differentiable, and satisfies

[TABLE]

and

[TABLE]

For every $C_{0}>0,\ \tau>0$ and $0<\mu<\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}$ define

[TABLE]

Since $\varphi_{\tau,\mu}$ is ia non decreasing, then the functions $U^{+}_{\tau,\mu,C_{0}}$ is non-increasing. Furthermore, the functions $U^{+}_{\tau,\mu,C_{0}}$ belongs to $C^{\delta}_{\rm unif}({\mathbb{R}})$ for every $0\leq\delta<1$, $\tau>0$, $0<\mu<\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}$, and $C_{0}>0$.

Let $C_{0}>0,\ \tau>0$ and $0<\mu<\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}$ be fixed. Next, let $\mu<\tilde{\mu}<\min\{2\mu,\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}$ and $d>\max\{1,C_{0}^{\frac{\mu-\tilde{\mu}}{\mu}}\}$. The function $\varphi_{\tau,\mu}-d\varphi_{\tau,\tilde{\mu}}$ attains its maximum value at $\bar{a}_{\mu,\tilde{\mu},d}:=\frac{\ln(d\tilde{\mu})-\ln(\mu)}{\tilde{\mu}-\mu}$ and takes the value zero at $\underline{a}_{\mu,\tilde{\mu},d}:=\frac{\ln(d)}{\tilde{\mu}-\mu}$. Define

[TABLE]

From the choice of $d$, it follows that $0\leq U_{\tau,\mu,C_{0}}^{-}\leq U^{+}_{\tau,\mu,C_{0}}\leq C_{0}$ and $U_{\tau,\mu,C_{0}}^{-}\in C^{\delta}_{\rm unif}({\mathbb{R}})$ for every $0\leq\delta<1$. Finally, let us consider the set $\mathcal{E}_{\tau,\mu}(C_{0})$ defined by

[TABLE]

It should be noted that $U_{\tau,\mu,C_{0}}^{-}$ and $\mathcal{E}_{\tau,\mu}(C_{0})$ all depend on $\tilde{\mu}$ and $d$. Later on, we shall provide more information on how to choose $d$ and $\tilde{\mu}$ whenever $\tau$, $\mu$ and $C_{0}$ are given.

For every $u\in C_{\rm unif}^{b}({\mathbb{R}})$, consider

[TABLE]

where

[TABLE]

For given $u\in C_{\rm unif}^{b}({\mathbb{R}})$, it is well known that the function $V_{1}(x;u)$ and $V_{2}(x,u)$ are the solutions of the second and third equations of (1.4) in $C^{b}_{\rm unif}({\mathbb{R}})$ respectively.

For given open intervals $D\subset{\mathbb{R}}$ and $I\subset{\mathbb{R}}$, a function $U(\cdot,\cdot)\in C^{2,1}(D\times I,{\mathbb{R}})$ is called a super-solution (respectively sub-solution) of (3) on $D\times I$ if

[TABLE]

(respectively

[TABLE]

)

Next, we state the main result of this section. For convenience, we introduce the following standing assumption.

(H) $\tau>0$, $0<\mu<\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}$, $b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}>(\tau c_{\mu}+\mu)K_{\tau,\mu}+\overline{M}_{\tau,\mu}$, and

[TABLE]

where $\overline{M}$, $K$, $\overline{M}_{\tau,\mu}$, and $K_{\tau,\mu}$ are given by (1.5), (1.7), (1.15), and (1.17) respectively.

Theorem 3.1**.**

Assume (H). Then the following hold.

• (1)

There is a positive real number $\tilde{C}_{0}>0$, $\tilde{C}_{0}=\tilde{C}_{0}(\tau,\chi_{1},\lambda_{1},\mu_{1},\chi_{2},\lambda_{2},\mu_{2},\mu)$, such for every $C_{0}\geq\tilde{C}_{0}$, and for every $u\in\mathcal{E}_{\tau,\mu}(C_{0})$, we have that $U(x,t)=C_{0}$ is supper-solutions of (3) on ${\mathbb{R}}\times{\mathbb{R}}$.

• (2)

For every $C_{0}>0$ and for every $u\in\mathcal{E}_{\tau,\mu}(C_{0})$, $U(x,t)=\varphi_{\tau,\mu}(x)$ is a supper-solutions of (3) on ${\mathbb{R}}\times{\mathbb{R}}$.

• (3)

For every $C_{0}>0$, there is $d_{0}>\max\{1,C_{0}^{\frac{\mu-\tilde{\mu}}{\mu}}\}$, $d_{0}=d_{0}(\tau,\chi_{1},\lambda_{1},\mu_{1},\chi_{2},\lambda_{2},\mu_{2},\mu)$, such that for every $u\in\mathcal{E}_{\tau,\mu}(C_{0})$, we have that $U(x,t)=U_{\tau,\mu,C_{0}}^{-}(x)$ is a sub-solution of (3) on $(\underline{a}_{\mu,\tilde{\mu},d},\infty)\times{\mathbb{R}}$ for all $d\geq d_{0}$ and $\mu<\tilde{\mu}<\min\{\sqrt{a},\ \sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\ \sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}},\ 2\mu,\mu+\frac{b+2\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-(\tau c_{\mu}+\mu)K_{\tau,\mu}}{1+K_{\tau,\mu}}\}$.

• (4)

Let $\tilde{C}_{0}$ be given by (1), then for every $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$, $U(x,t)=U_{\tau,\mu,\tilde{C}_{0}}^{-}(x_{\delta})$ is a sub-solution of (3) on ${\mathbb{R}}\times{\mathbb{R}}$ for $0<\delta\ll 1$, where $x_{\delta}=\underline{a}_{\mu,\tilde{\mu},d}+\delta$.

To prove Theorem 3.1, we first establish some estimates on $V_{i}(\cdot;u)$ and $\frac{d}{dx}V_{i}(\cdot;u)$.

It follows from (3.7) that

[TABLE]

Furthermore, let

[TABLE]

For every $i\in\{1,2\}$ and $u\in C_{\rm unif}^{b}({\mathbb{R}})$, $u\geq 0,$ we have that $V_{i}(\cdot;u)\in C^{2,b}_{\rm unif}({\mathbb{R}})$ with $V_{i}(\cdot;u)\geq 0$ and

[TABLE]

Combining this with inequality (3.9), we obtain that

[TABLE]

The next Lemma provides pointwise and uniform estimates for $(\chi_{2}\lambda_{2}V_{2}-\chi_{1}\lambda_{1}V_{1})(\cdot;u)$ whenever $u\in\mathcal{E}_{\tau,\mu}(C_{0})$.

Lemma 3.2**.**

For every $i\in\{1,2\}$, $C_{0}>0$, $\tau>0$, $0<\mu<\min\{\sqrt{a},\ \sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}$ and $u\in\mathcal{E}_{\tau,\mu}(C_{0})$, let $V_{i}(\cdot;u)$ be defined as in (3.7), then

[TABLE]

Proof.

For every $u\in\mathcal{E}_{\tau,\mu}(C_{0})$, since $0\leq U^{-}_{\tau,\mu,C_{0}}\leq u\leq U^{+}_{\tau,\mu,C_{0}}$ then for every $x\in{\mathbb{R}}$, setting $Z=\chi_{2}\lambda_{2}V_{2}-\chi_{1}\lambda_{1}V_{1}$, we have that

[TABLE]

and

[TABLE]

Observe that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

The Lemma follows from inequalities (3), (3), (3), (3),(3), and (3). ∎

Next, we present a pointwise/uniform estimate for $\frac{d}{dx}(\chi_{1}V_{1}-\chi_{2}V_{2})(\cdot;u)$ whenever $u\in\mathcal{E}_{\tau,\mu}(C_{0}).$

Lemma 3.3**.**

Let $\tau>0,\ C_{0}>0$ and $0<\mu<\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}$ be fixed. Let $u\in C^{b}_{\rm unif}({\mathbb{R}})$ and $V_{i}(\cdot;u)\in C^{2,b}_{\rm unif}({\mathbb{R}})$ be defined by (3.7), then

[TABLE]

for every $x\in{\mathbb{R}}$ and every $u\in\mathcal{E}_{\tau,\mu}(C_{0})$, .

Proof.

Let $u\in\mathcal{E}_{\tau,\mu}(C_{0})$ and fix any $x\in{\mathbb{R}}$.

[TABLE]

Note that we have used the following fact in the last inequality

[TABLE]

On the other hand, we have that

[TABLE]

Similarly, we have that

[TABLE]

The Lemma follows from (3), (3), and (3.21). ∎

Remark 3.4**.**

It follows from Lemma 3.2 that

[TABLE]

and

[TABLE]

Now we are ready to present the proof of Theorem 3.1.

Proof of Theorem 3.1.

For every $U\in C^{2,1}({\mathbb{R}}\times{\mathbb{R}}_{+})$, let

[TABLE]

(1) First, using inequality (3.22), we have that

[TABLE]

Since $\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-\overline{M}=-\underline{M}$, then it follows from (H) that $b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-\overline{M}-c_{\mu}\tau K$. Thus taking $\tilde{C}_{0}:=\frac{a}{b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-\overline{M}-c_{\mu}\tau K}$, it follows from inequality (3.25) that for every $C_{0}\geq\tilde{C}_{0}$, we have that

[TABLE]

Hence, for every $C_{0}\geq\tilde{C}_{0}$, we have that $U(x,t)=C_{0}$ is a super-solution of (3) on ${\mathbb{R}}\times{\mathbb{R}}$.

(2) It follows from Lemma 3.3, and inequality (3.23), and (3.8) that

[TABLE]

Hence $U(x,t)=\varphi_{\tau,\mu}(x)$ is also a super-solution of (3) on ${\mathbb{R}}\times{\mathbb{R}}$.

(3) Let $C_{0}>0$ and $O=(\underline{a}_{\mu,\tilde{\mu},d},\infty)$. Then for $x\in O$, $U_{\tau,\mu,C_{0}}^{-}(x)>0$. For $x\in O$, it follows from Lemma 3.3 , and inequality (3.23), and (3.8) that

[TABLE]

Note that $U_{\tau,\mu,C_{0}}^{-}(x)>0$ is equivalent to $\varphi_{\tau,\mu}(x)>d\varphi_{\tau,\tilde{\mu}}(x)$, which is again equivalent to

[TABLE]

Since $A_{1}>0$, thus for $x\in O$, we have

[TABLE]

Note also that, by (3.8),

[TABLE]

whenever $(\tilde{\mu}-\mu)K_{\tau,\mu}\leq b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-(\tau c_{\mu}+\mu)K_{\tau,\mu}$. Observe that

[TABLE]

Furthermore, we have that $U_{\tau,\mu,C_{0}}^{-}(x)>0$ implies that $x>0$ for $d\geq\max\{1,C_{0}^{\frac{\mu-\tilde{\mu}}{\mu}}\}$. Thus, for every $d\geq d_{0}:=\max\{1,\frac{A_{1}}{A_{0}},C_{0}^{\frac{\mu-\tilde{\mu}}{\mu}}\}$, we have that

[TABLE]

whenever $x\in O$ and $\mu<\tilde{\mu}<\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}},2\mu,\mu+\frac{b+\chi_{2}\mu_{2}-\chi_{1}\mu_{1}-(\tau c_{\mu}+\mu)K_{\tau,\mu}}{1+K_{\tau,\mu}}\}$. Hence $U(x,t)=U_{\tau,\mu,C_{0}}^{-}(x)$ is a sub-solution of (3) on $(\underline{a}_{\mu,\tilde{\mu},d},\infty)\times{\mathbb{R}}$.

(4) Since $\overline{M}-\underline{M}=\chi_{2}\mu_{2}-\chi_{1}\mu_{1}$, thus, it follows from (3.8) that

[TABLE]

Hence, for $0<\delta\ll 1$, we have that

[TABLE]

where $x_{\delta}=\underline{a}_{\mu,\tilde{\mu},d}+\delta$. This implies that $U(x,t)=U_{\tau,\mu,\tilde{C}_{0}}^{-}(x_{\delta})$ is a sub-solution of (3) on ${\mathbb{R}}\times{\mathbb{R}}$. ∎

4 Traveling wave solutions

In this section we study the existence and nonexistence of traveling wave solutions of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$, and prove Theorems C and D.

4.1 Proof of Theorem C

In this subsection, we prove Theorem C. To this end, we first prove the following important result.

Theorem 4.1**.**

Assume (H). Then (1.3) has a traveling wave solution $(u(x,t),v_{1}(x,t),v_{2}(x,t))=(U(x-c_{\mu}t),V_{1}(x-c_{\mu}t),V_{2}(x-c_{\mu}t))$ satisfying

[TABLE]

where $c_{\mu}=(\mu+\frac{a}{\mu})$.

In order to prove Theorem 4.1, we first prove some lemmas. These Lemmas extend some of the results established in [33], so some details might be omitted in their proofs. The reader is referred to the proofs of Lemmas 3.2, 3.3, 3.5 and 3.6 in [33] for more details.

In the remaining part of this subsection we shall suppose that (H) holds and $\tilde{\mu}$ is fixed, where $\tilde{\mu}$ satisfies

[TABLE]

Furthermore, we choose $\tilde{C}_{0}=\tilde{C}_{0}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2},\mu)$ and $d=d_{0}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2},\mu)$ to be the constants given by Theorem 3.1 and to be fixed and set $U^{+}_{\tau,\mu,\tilde{C}_{0}}:=U^{+}_{\mu}$ and $U^{-}_{\tau,\mu,\tilde{C}_{0}}:=U^{-}_{\mu}$. Fix $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$. For given $u_{0}\in C_{\rm unif}^{b}({\mathbb{R}})$, let $U(x,t;u_{0},u)$ be the solution of (3) with $U(x,0;u_{0},u)=u_{0}(x)$. By the arguments in the proofs of Theorem 1.1 and Theorem 1.5 in [34], we have $U(x,t;U_{\mu}^{+},u)$ exists for all $t>0$ and ${U(\cdot,\cdot;{U_{\mu}^{+}},u)}\in C([0,\infty),C^{b}_{\rm unif}({\mathbb{R}}))\cap C^{1}((0\ ,\ \infty),C^{b}_{\rm unif}({\mathbb{R}}))\cap C^{2,1}({\mathbb{R}}\times(0,\ \infty))$ satisfying

[TABLE]

for $0<\theta,\nu\ll 1$.

Lemma 4.2**.**

Assume (H). Then for every $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$, the following hold.

(i)

$0\leq U(\cdot,t;U_{\mu}^{+},u)\leq U_{\mu}^{+}(\cdot)$* for every $t\geq 0.$*

(ii)

$U(\cdot,t_{2};U_{\mu}^{+},u)\leq U(\cdot,t_{1};U_{\mu}^{+},u)$* for every $0\leq t_{1}\leq t_{2}$.*

Proof.

(i) Note that $0\leq U^{+}_{\mu}(\cdot)\leq\tilde{C}_{0}$. Then by comparison principle for parabolic equations and Theorem 3.1(1), we have

[TABLE]

Similarly, note that $0\leq U_{\mu}^{+}(x)\leq\varphi_{\mu}(x)$. Then by comparison principle for parabolic equations and Theorem 3.1(2) again, we have

[TABLE]

Thus $U(\cdot,t;U_{\mu}^{+},u)\leq U^{+}_{\mu}$. This completes the proof of (i).

(ii) For $0\leq t_{1}\leq t_{2}$, since

[TABLE]

and by (i), $U(\cdot,t_{2}-t_{1};U_{\mu}^{+},u)\leq U^{+}_{\mu}$, (ii) follows from comparison principle for parabolic equations. ∎

Let us define $U(x;u)$ to be

[TABLE]

By the a priori estimates for parabolic equations, the limit in (4.2) is uniform in $x$ in compact subsets of ${\mathbb{R}}$ and $U(\cdot;u)\in C_{\rm unif}^{b}({\mathbb{R}})$. Next we prove that the function $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})\to U(\cdot;u)\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$.

Lemma 4.3**.**

Assume (H). Then,

[TABLE]

for every $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$, $x\in{\mathbb{R}}$, and $0<\delta\ll 1$.

Proof.

Let $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ be fixed. Let $O=(\underline{a}_{\mu,\tilde{\mu},d},\infty)$. Note that $U_{\mu}^{-}(\underline{a}_{\mu,\tilde{\mu},d})=0$. By Theorem 3.1(3), $U_{\mu}^{-}(x)$ is a sub-solution of (3) on $O\times(0,\infty)$. Note also that $U_{\mu}^{+}(x)\geq U_{\mu}^{-}(x)$ for $x\geq\underline{a}_{\mu,\tilde{\mu},d}$ and $U(\underline{a}_{\mu,\tilde{\mu},d},t;U_{\mu}^{+},u)>0$ for all $t\geq 0$. Then by comparison principle for parabolic equations, we have that

[TABLE]

Now for any $0<\delta\ll 1$, by Theorem 3.1(4), $U(x,t)=U_{\mu}^{-}(x_{\delta})$ is a sub-solution of (3) on ${\mathbb{R}}\times{\mathbb{R}}$. Note that $U_{\mu}^{+}(x)\geq U_{\mu}^{-}(x_{\delta})$ for $x\leq x_{\delta}$ and $U(x_{\delta},t;U_{\mu}^{+},u)\geq U_{\mu}^{-}(x_{\delta})$ for $t\geq 0$. Then by comparison principle for parabolic equations again,

[TABLE]

The lemma then follows. ∎

Remark 4.4**.**

It follows from Lemmas 4.2 and 4.3 that if (H) holds, then

[TABLE]

for every $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$, $t\geq 0$ and $0\leq\delta\ll 1$, where

[TABLE]

This implies that

[TABLE]

for every $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$. Hence $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})\mapsto U(\cdot;u)\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0}).$

Lemma 4.5**.**

Assume (H). Then for every $\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ the associated function $U(\cdot;u)$ satisfied the elliptic equation,

[TABLE]

Proof.

The following arguments generalized the arguments used in the proof of Lemma 4.6 in [33]. Hence we refer to [33] for the proofs of the estimates stated below.

Let $\{t_{n}\}_{n\geq 1}$ be an increasing sequence of positive real numbers converging to $\infty$. For every $n\geq 1$, define $U_{n}(x,t)=U(x,t+t_{n};U_{\mu}^{+},u)$ for every $x\in{\mathbb{R}},\ t\geq 0$. For every $n$, $U_{n}$ solves the PDE

[TABLE]

Let $\{T(t)\}_{t\geq 0}$ be the analytic semigroup on $C^{b}_{\rm unif}({\mathbb{R}})$ generated by $\Delta-I$ and let $X^{\beta}={\rm Dom}((I-\Delta)^{\beta})$ be the fractional power spaces of $I-\Delta$ on $C_{\rm unif}^{b}({\mathbb{R}})$ ($\beta\in[0,1]$).

The variation of constant formula and the fact that $\partial_{xx}V_{i}(\cdot;u)-\lambda_{i}V=-\mu_{i}u$ yield that

[TABLE]

Let $0<\beta<\frac{1}{2}$ be fixed. There is a positive constant $C_{\beta}$, (see [13]), such that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Note that we have used Lemma 3.3, mainly the fact that $|\partial_{x}(\chi_{2}V_{2}-\chi_{1}V_{1})(\cdot;u)|\leq K\tilde{C}_{0}$, to obtain the uniform upper bound estimates for $\|I_{2}(t)\|_{X^{\beta}}$. Therefore, for every $T>0$ we have that

[TABLE]

where

[TABLE]

Hence, it follows from (4.6) that

[TABLE]

Next, for every $t,h\geq 0$ and $n\geq 1$, we have that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

It follows from inequalities (4.8), (4.9), (4.11), (4.10) and (4.12), the functions $U_{n}:[0,\infty)\to X^{\beta}$ are uniformly bounded and equicontinuous. Since $X^{\beta}$ is continuously imbedded in $C^{\nu}({\mathbb{R}})$ for every $0\leq\nu<2\beta$ (See [13]), therefore, the Arzela-Ascoli Theorem and Theorem 3.15 in [11], imply that there is a function $\tilde{U}(\cdot,\cdot;u)\in C^{2,1}({\mathbb{R}}\times(0,\infty))$ and a subsequence $\{U_{n^{\prime}}\}_{n\geq 1}$ of $\{U_{n}\}_{n\geq 1}$ such that $U_{n^{\prime}}\to\tilde{U}$ in $C^{2,1}_{loc}({\mathbb{R}}\times(0,\infty))$ as $n\to\infty$ and $\tilde{U}(\cdot,\cdot;u)$ solves the PDE

[TABLE]

But $U(x;u)=\lim_{t\to\infty}U(x,t;U_{\mu}^{+},u)$ and $t_{n^{\prime}}\to\infty$ as $n\to\infty$, hence $\tilde{U}(x,t;u)=U(x;u)$ for every $x\in{\mathbb{R}},\ t\geq 0$. Hence $U(\cdot;u)$ solves (4.5). ∎

Lemma 4.6**.**

Assume (H). Then, for any given $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$, (4.5) has a unique bounded non-negative solution satisfying that

[TABLE]

The proof of Lemma 4.6 follows from [33, Lemma 3.6].

We now prove Theorem 4.1.

Proof of Theorem 4.1.

Following the proof of Theorem 3.1 in [33], let us consider the normed linear space $\mathcal{E}=C^{b}_{\rm unif}({\mathbb{R}})$ endowed with the norm

[TABLE]

For every $u\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ we have that $\|u\|_{\ast}\leq\tilde{C}_{0}.$ Hence $\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ is a bounded convex subset of $\mathcal{E}$. Furthermore, since the convergence in $\mathcal{E}$ implies the pointwise convergence, then $\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ is a closed, bounded, and convex subset of $\mathcal{E}$. Furthermore, a sequence of functions in $\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ converges with respect to norm $\|\cdot\|_{\ast}$ if and only if it converges locally uniformly on ${\mathbb{R}}$.

We prove that the mapping $\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})\ni u\mapsto U(\cdot;u)\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ has a fixed point. We divide the proof in three steps.

Step 1. In this step, we prove that the mapping $\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})\ni u\mapsto U(\cdot;u)\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ is compact.

Let $\{u_{n}\}_{n\geq 1}$ be a sequence of elements of $\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$. Since $U(\cdot;u_{n})\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ for every $n\geq 1$ then $\{U(\cdot;u_{n})\}_{n\geq 1}$ is clearly uniformly bounded by $\tilde{C}_{0}$. Using inequality (4.6), we have that

[TABLE]

for all $n\geq 1$ where $M_{1}$ is given by (4.7). Therefore there is $0<\nu\ll 1$ such that

[TABLE]

for every $n\geq 1$ where $\tilde{M_{1}}$ is a constant depending only on $M_{1}$. Since for every $n\geq 1$ and every $x\in{\mathbb{R}}$, we have that $U(x,t;U_{\mu}^{+},u_{n})\to U(x;u_{n})$ as $t\to\infty,$ then it follows from (4.14) that

[TABLE]

for every $n\geq 1$. Which implies that the sequence $\{U(\cdot;u_{n})\}_{n\geq 1}$ is equicontinuous. The Arzela-Ascoli’s Theorem implies that there is a subsequence $\{U(\cdot;u_{n^{\prime}})\}_{n\geq 1}$ of the sequence $\{U(\cdot;u_{n})\}_{n\geq 1}$ and a function $U\in C({\mathbb{R}})$ such that $\{U(\cdot;u_{n^{\prime}})\}_{n\geq 1}$ converges to $U$ locally uniformly on ${\mathbb{R}}$. Furthermore, the function $U$ satisfies inequality (4.15). Combining this with the fact $U_{\mu}^{-}(x)\leq U(x;u_{n^{\prime}})\leq U_{\mu}^{+}(x)$ for every $x\in{\mathbb{R}}$ and $n\geq 1$, by letting $n$ goes to infinity, we obtain that $U\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$. Hence the mapping $\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})\ni u\mapsto U(\cdot;u)\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ is compact.

Step 2. In this step, we prove that the mapping $\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})\ni u\mapsto U(\cdot;u)\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ is continuous. This follows from the arguments used in the proof of Step 2, Theorem 3.1, [33]

Now by Schauder’s Fixed Point Theorem, there is $U\in\mathcal{E}_{\tau,\mu}(\tilde{C}_{0})$ such that $U(\cdot;U)=U(\cdot)$. Then $(U(x),V(x;U))$ is a stationary solution of (1.4) with $c=c_{\mu}$. It is clear that

[TABLE]

Step 3. We claim that

[TABLE]

For otherwise, we may assume that there is $x_{n}\to-\infty$ such that $U(x_{n})\to\lambda\not=\frac{a}{b}$ as $n\to\infty$. Define $U_{n}(x)=U(x+x_{n})$ for every $x\in{\mathbb{R}}$ and $n\geq 1$. By the arguments of Lemma 4.5, there is a subsequence $\{U_{n^{\prime}}\}_{n\geq 1}$ of $\{U_{n}\}_{n\geq 1}$ and a function $U^{*}\in C_{\rm unif}^{b}({\mathbb{R}})$ such that $\|U_{n^{\prime}}-U^{*}\|_{\ast}\to 0$ as $n\to\infty$. Moreover, $(U^{*},V_{1}(\cdot;U^{*}),V_{2}(\cdot;U^{*}))$ is also a stationary solution of (1.4) with $c=c_{\mu}$.

Claim 1. $\inf_{x\in{\mathbb{R}}}U^{*}(x)>0$.

Indeed, let $0<\delta\ll 1$ be fixed. For every $x\in{\mathbb{R}}$, there $N_{x}\gg 1$ such that $x+x_{n^{\prime}}<x_{\delta}$ for all $n\geq N_{x}$. Hence, It follows from Remark 4.4 that

[TABLE]

Letting $n\to\infty$ in the last inequality, we obtain that $U_{\mu,\delta}^{-}(x_{\delta})\leq U^{*}(x)$ for every $x\in{\mathbb{R}}$. The claim thus follows.

Claim 2. $U^{*}(x)\equiv\frac{a}{b}$.

Note also that the function $(\tilde{U}(x,t),\tilde{V}_{1}(x,t),\tilde{V}_{2}(x,t))=(U^{*}(x-c_{\mu}t),V_{1}(x-c_{\mu}t,U^{*}),V_{2}(x-c_{\mu}t,U^{*}))$ solves (1.4). Then by Theorem B and Claim 1,

[TABLE]

This implies that $U^{*}(x)=\frac{a}{b}$ for any $x\in{\mathbb{R}}$ and the claim thus follows.

By Claim 2, we have $\lim_{n\to\infty}U(x_{n})=U^{*}(0)=\frac{a}{b}$, which contracts to $\lim_{n\to\infty}U(x_{n})=U^{*}(0)=\lambda\not=\frac{a}{b}$. ∎

Now, we are ready to prove Theorem C.

Proof of Theorem C (i).

Let us set

[TABLE]

Note that under the assumption of Theorem A, we have that

[TABLE]

Hence the open set

[TABLE]

is nonempty. Let $(\mu^{**}_{\tau},\mu^{*}_{\tau})$ denotes the right maximal open connected component of the open set $O_{\tau}$. That is $(\mu^{**}_{\tau},\mu^{*}_{\tau})$ is the open connected component of $O_{\tau}$ with maximal length and, closed to $\sqrt{a}$. Since the open interval \Big{(}0\ ,\ \min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}\Big{)} has finite length, then $(\mu^{**}_{\tau},\mu^{*}_{\tau})$ is well defined. Next, we set

[TABLE]

where $c_{\mu}=\mu+\frac{a}{\mu}$. Let $c^{*}(\tau,\chi_{1},\mu_{1},\lambda_{1}$, $\chi_{2},\mu_{2},\lambda_{2})<c<c^{**}(\tau,\chi_{1},\mu_{1},\lambda_{1}$, $\chi_{2},\mu_{2},\lambda_{2})$ be given and let $\mu\in(\mu^{**},\mu^{*})$ be the unique solution of the equation $c_{\mu}=c$. It follows from Theorem 4.1, that (1.2) has a traveling wave solution $(U(x,t),V_{1}(x,t),V_{2}(x,t))=(U(x-ct),V_{1}(x-ct),V_{2}(x-ct))$ with speed c connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$. Moreover $\lim_{z\to\infty}\frac{U(z)}{e^{-\mu z}}=1$.

Observe that for every $\lambda_{i},\mu_{i}>0$, $i=1,2$ and $\tau>0$ we have

[TABLE]

Thus we have that

[TABLE]

and

[TABLE]

Thus

[TABLE]

and

[TABLE]

where $\tilde{\mu}^{*}={\min\{\sqrt{a},\sqrt{\frac{\lambda_{1}+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_{2}+\tau a}{(1-\tau)_{+}}}\}}$. ∎

Next, we present the proof of Theorem C (ii).

Proof of Theorem C (ii).

For every $c_{n}>c^{*}$ with $c_{n}\to c^{*}=c^{*}(\tau,\chi_{1},\mu_{1},\lambda_{1},\chi_{2},\mu_{2},\lambda_{2})$, let $(U^{c_{n}}(x),V_{1}^{c_{n}}(x),V_{2}^{c_{n}}(x))$ be the traveling wave solution solution of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ with speed $c_{n}$ given by Theorem C (i). For each $n\geq 1$, note that the set $\{x\in{\mathbb{R}}\ :\ U^{c_{n}}(x_{n})=\frac{a}{2b}\}$ is compact and nonempty, so there is $x_{n}\in{\mathbb{R}}$ such that

[TABLE]

Since $\sup_{n}\|U^{c_{n}}\|_{\infty}<\infty$, hence by estimates for parabolic equations, without loss of generality, we may suppose that that $U^{c_{n}}(x+x_{n})\to U^{*}(x)$ as $n\to\infty$ locally uniformly. Furthermore, taking $V_{i}^{*}(x)=V_{i}(x,U^{*})$, $i=1,2$, it holds that $(U^{*},V_{1}^{*},V_{2}^{*})$ solves

[TABLE]

$U^{*}(0)=\frac{a}{2b}$ and $U^{*}(x)\geq\frac{a}{2b}$ for every $x\leq 0$. Next we claim that

[TABLE]

Suppose on the contrary that (4.17) does not hold. Then there is a sequence $\{z_{n}\}_{n\geq 1}$ such that $z_{n}<z_{n+1}$ for every $n$, $z_{n}\to\infty$, $z_{1}=0$, and

[TABLE]

For every $n\geq 1$ let $\{y_{n}\}_{n\geq 1}$ be the sequence defined by

[TABLE]

Observe that

[TABLE]

Since $(U^{*}(x-c^{*}t),V_{1}^{*}(x-c^{*}t),V_{2}^{*}(x-c^{*}t))$ is a positive entire solution of (1.3) with

[TABLE]

then by the stability of the constant equilibrium $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ provided by Theorem B, we obtain that

[TABLE]

Therefore, without loss of generality, we may suppose that $y_{n}\in(x_{n},x_{n+1})$ for every $n\geq 1$ with

[TABLE]

Note also that

[TABLE]

Thus for $n$ large enough, we have that

[TABLE]

which contradicts to (4.16). Therefore, (4.17) holds.

It follows again from

[TABLE]

and the stability of the constant equilibrium $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ provided by Theorem B that

[TABLE]

Therefore $(u^{c^{*}}(t,x),v^{c^{*}}(t,x))=(U^{*}(x-c^{*}t),V^{*}(x-c^{*}t))$ is a traveling wave solution of (1.3) with speed $c^{*}$ connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$.

∎

For clarity sake in the proof of Theorem C (iii), we first present the following lemma.

Lemma 4.7**.**

Let $(u,v_{1},v_{2})(x,t)=(U(x-ct),V_{1}(x-ct),V_{2}(x-ct))$ be traveling wave solution of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ with speed $c$. There is $X_{0}\gg 1$ such that

[TABLE]

Proof.

Suppose by contradiction that (4.18) does not hold. There exists a sequence $\{x_{n}\}_{n\geq 1}$ with $x_{n}\to\infty$ such that $x_{n}$ is a local minimum point of $U$. Hence,

[TABLE]

Since $U(\infty)=0$, then it holds that

[TABLE]

Therefore, there is $n\gg 1$ such that

[TABLE]

which contradicts to (4.5). Thus, (4.18) holds.

∎

Next, we present the proof of Theorem C(iii).

Proof of Theorem C (iii).

Let $(u,v_{1},v_{2})(x,t)=(U(x-ct),V_{1}(x-ct),V_{2}(x-ct))$ be traveling wave solution of (1.3) connecting $(\frac{a}{b},\frac{a\mu_{1}}{b\lambda_{1}},\frac{a\mu_{2}}{b\lambda_{2}})$ and $(0,0,0)$ with speed $c$. Let $X_{0}\gg 1$ be given by Lemma 4.7. We shall show that $c\geq 2\sqrt{a}$. Suppose by contradiction that $c<2\sqrt{a}$. Choose $q\in(\max\{c,0\},2\sqrt{a})$ and $0<\varepsilon\ll 1$ satisfying

[TABLE]

Since $(U,V_{1},V_{2})(\infty)=(0,0,0)$, there is $X_{\varepsilon}>X_{0}$ such that

[TABLE]

Hence, since $U^{\prime}(x-ct)\leq 0$ for $x-ct\geq X_{\varepsilon}$, the function $u(x,t)=U(x-ct)$ satisfies

[TABLE]

Observe that with $L:=\frac{2\pi}{\sqrt{4(a-\varepsilon)-(q+\varepsilon)^{2}}}$ the function

[TABLE]

with $m_{0}=\min\{U(x-ct_{0})\ :\ 0\leq x\leq L+qt_{0}\}$ where $t_{0}:=\frac{X_{\varepsilon}}{q-c}$ satisfies $0\leq\underline{u}(t,x)\leq m_{0}$,

[TABLE]

[TABLE]

and

[TABLE]

Therefore, by comparison principle for parabolic equations, it follows from (4.19)-(4.22) that

[TABLE]

In particular, for $x=qt+\frac{L}{2}$, $t\geq t_{0}$, we have from the last inequality that

[TABLE]

Thus $q\leq c$, since $U(\infty)=0$. Which contradicts to $q>c$. Therefore we must have $c\geq 2\sqrt{a}$. ∎