Invasion traveling wave solutions of a competitive system with dispersal

Abstract

This paper is concerned with the invasion traveling wave solutions of a Lotka-Volterra type competition system with nonlocal dispersal, the purpose of which is to formulate the dynamics between the resident and the invader. By constructing upper and lower solutions and passing to a limit function, the existence of traveling wave solutions is obtained if the wave speed is not less than a threshold. When the wave speed is smaller than the threshold, the nonexistence of invasion traveling wave solutions is proved by the theory of asymptotic spreading.

MSC: 35C07, 35K57, 37C65.

1 Introduction

In the past decades, much attention has been paid to the spatial propagation modes of the following Lotka-Volterra type diffusion system:

$$\{\begin{array}{c}\frac{\partial u_1(x,t)}{\partial t}=d_1\mathrm{\Delta }{u}_1(x,t)+r_1{u}_1(x,t)[1-u_1(x,t)-b_1{u}_2(x,t)],\hfill \\ \frac{\partial u_2(x,t)}{\partial t}=d_2\mathrm{\Delta }{u}_2(x,t)+r_2{u}_2(x,t)[1-u_2(x,t)-b_2{u}_1(x,t)],\hfill \end{array}$$

(1.1)

in which all the parameters are positive and $x\in \mathbb{R}$, $t>0$, $u_1$, $u_2$ are two competitors. Many investigators considered its traveling wave solutions connecting different spatial homogeneous steady states such as the existence, monotonicity, minimal wave speed and stability; see [1–16].

In particular, if $b_1<1<b_2$ holds in (1.1), then the corresponding reaction system has a stable equilibrium $(1,0)$ and an unstable one $(0,1)$. With the condition $b_1<1<b_2$, many papers including [2, 3, 5, 6, 8, 16] studied the traveling wave solutions connecting $(1,0)$ with $(0,1)$. These traveling wave solutions can formulate the spatial exclusive process between the resident $u_2$ and the invader $u_1$ so that the minimal wave speed reflecting the invasion speed of the invader becomes a hot topic in these works; we refer to Shigesada and Kawasaki [17] for some examples of the corresponding biological records and the literature importance of invasion speed. Moreover, the similar problem was also discussed in different spatial media such as the lattice differential systems in Guo and Liang [4], Guo and Wu [18].

In this paper, we consider the minimal wave speed of traveling wave solutions in the following nonlocal dispersal system (see Yu and Yuan [19]):

$$\{\begin{array}{c}\frac{\partial u_1(x,t)}{\partial t}=d_1[{\int }_{\mathbb{R}}{J}_1(x-y)u_1(y,t)dy-u_1(x,t)]\hfill \\ \phantom{\frac{\partial u_1(x,t)}{\partial t}=}+r_1{u}_1(x,t)[1-u_1(x,t)-b_1{u}_2(x,t)],\hfill \\ \frac{\partial u_2(x,t)}{\partial t}=d_2[{\int }_{\mathbb{R}}{J}_2(x-y)u_2(y,t)dy-u_2(x,t)]\hfill \\ \phantom{\frac{\partial u_2(x,t)}{\partial t}=}+r_2{u}_2(x,t)[1-u_2(x,t)-b_2{u}_1(x,t)],\hfill \end{array}$$

(1.2)

in which $x\in \mathbb{R}$, $t>0$, $u_1(x,t)$ and $u_2(x,t)$ denote the densities of two competitors at time t and location $x\in \mathbb{R}$, all the parameters are positive and

$$b_1<1<b_2,$$

(1.3)

$J_i:\mathbb{R}\to \mathbb{R}$, $i=1,2$, are probability functions formulating the random dispersal of individuals and satisfy the following assumptions:

(J1) $J_i$ is nonnegative and Lebesgue measurable for each $i=1,2$;

(J2) for any $\lambda \in \mathbb{R}$, ${\int }_{\mathbb{R}}{J}_i(y)e^{\lambda y}dy<\mathrm{\infty }$, $i=1,2$;

(J3) ${\int }_{\mathbb{R}}{J}_i(y)dy=1$, $J_i(y)=J_i(-y)$, $y\in \mathbb{R}$, $i=1,2$.

In (1.2), the spatial migration of individuals is formulated by the so-called dispersal operator, which has significant sense in population dynamics. For example, in the patch models of population dynamics [20], the rate of immigration into a patch from a particular other patch is usually taken as proportional to the local population, and the dispersal can be regarded as the extension of these ideas to a continuous media model. Such a diffusion mechanism also arises from physics processes with long range effect and other disciplines [13], and the dynamics of evolutionary systems with dispersal effect has been widely studied in recent years; we refer to [13, 21–32] and the references cited therein.

Hereafter, a traveling wave solution of (1.2) is a special solution of the form

$$u_1(x,t)={\varphi }_1(\xi ),u_2(x,t)={\varphi }_2(\xi ),\xi =x+ct,$$

where $c>0$ is the wave speed at which the wave profile $({\varphi }_1,{\varphi }_2)\in C^1(\mathbb{R},{\mathbb{R}}^2)$ propagates in spatial media ℝ. Thus, $({\varphi }_1,{\varphi }_2)$ with $c>0$ must satisfy

$$\{\begin{array}{c}c{\varphi }_1^{\mathrm{\prime }}(\xi )=d_1[{\int }_{\mathbb{R}}{J}_1(\xi -y){\varphi }_1(y)dy-{\varphi }_1(\xi )]\hfill \\ \phantom{c{\varphi }_1^{\mathrm{\prime }}(\xi )=}+r_1{\varphi }_1(\xi )[1-{\varphi }_1(\xi )-b_1{\varphi }_2(\xi )],\xi \in \mathbb{R},\hfill \\ c{\varphi }_2^{\mathrm{\prime }}(\xi )=d_2[{\int }_{\mathbb{R}}{J}_2(\xi -y){\varphi }_2(y)dy-{\varphi }_2(\xi )]\hfill \\ \phantom{c{\varphi }_2^{\mathrm{\prime }}(\xi )=}+r_2{\varphi }_2(\xi )[1-{\varphi }_2(\xi )-b_2{\varphi }_1(\xi )],\xi \in \mathbb{R}.\hfill \end{array}$$

(1.4)

Moreover, we also require the following asymptotic boundary conditions:

$$\underset{\xi \to -\mathrm{\infty }}{lim}({\varphi }_1(\xi ),{\varphi }_2(\xi ))=(0,1),\underset{\xi \to \mathrm{\infty }}{lim}({\varphi }_1(\xi ),{\varphi }_2(\xi ))=(1,0).$$

(1.5)

From the viewpoint of ecology, a traveling wave solution satisfying (1.4)-(1.5) can model the population invasion process: at any fixed $x\in \mathbb{R}$, only $u_2$ (the resident) can be found long time ago ($t\to -\mathrm{\infty }$ such that $x+ct\to -\mathrm{\infty }$), but after a long time ($t\to \mathrm{\infty }$ such that $x+ct\to \mathrm{\infty }$), only $u_1$ (the invader) can be seen. Therefore, we call a traveling wave solution satisfying (1.4)-(1.5) an invasion traveling wave solution.

To obtain the existence of (1.4)-(1.5) if the wave speed is larger than a threshold depending on $J_1$, $d_1$, $r_1$ and $b_1$, we construct proper upper and lower solutions and use the results in Pan et al. [33]. If the wave speed is the threshold, the existence of traveling wave solutions is proved by passing to a limit function. Finally, when the wave speed is smaller than the threshold, the nonexistence of traveling wave solutions is established by the theory of asymptotic spreading developed by Jin and Zhao [34]. For more results on the traveling wave solutions of evolutionary systems with nonlocal dispersal, we refer to Bates et al. [22], Coville and Dupaigne [35, 36], Li et al. [37], Lv [38], Pan [39], Pan et al. [33, 40], Sun et al. [41], Wu and Liu [42], Xu and Weng [43], Zhang et al. [44]. In particular, when $b_1,b_2\in (0,1)$ hold in (1.2), Yu and Yuan [19] established the existence of traveling wave solutions connecting $(0,0)$ with

$$(\frac{1-b_2}{1-b_1{b}_2},\frac{1-b_1}{1-b_1{b}_2}).$$

In addition, Li and Lin [45] and Zhang et al. [46] investigated the existence of positive traveling wave solutions of (1.2) for $b_1<0$, $b_2<0$ and $b_1{b}_2<0$, respectively.

The rest of this paper is organized as follows. In Section 2, we give some preliminaries. By constructing upper and lower solutions and using a limit process, the existence of traveling wave solutions is established in Section 3. In the last section, we obtain the nonexistence of traveling wave solutions.

2 Preliminaries

In this paper, we shall use the standard partial order in ${\mathbb{R}}^2$. Moreover, denote

then X is a Banach space equipped with the standard supremum norm. If $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^2$ with $\mathbf{a}\le \mathbf{b}$, then

$$X_{[\mathbf{a},\mathbf{b}]}=\{\mathbf{u}\in X:\mathbf{a}\le \mathbf{u}(\xi )\le \mathbf{b},\xi \in \mathbb{R}\}.$$

In order to apply the comparison principle, we first make a change of variables to obtain a cooperative system. Let ${\varphi }_1^{\ast }={\varphi }_1$, ${\varphi }_2^{\ast }=1-{\varphi }_2$, and drop the star for the sake of convenience, then (1.4) becomes

$$\{\begin{array}{c}c{\varphi }_1^{\mathrm{\prime }}(\xi )=d_1[{\int }_{\mathbb{R}}{J}_1(\xi -y){\varphi }_1(y)dy-{\varphi }_1(\xi )]+r_1{\varphi }_1(\xi )[1-b_1-{\varphi }_1(\xi )+b_1{\varphi }_2(\xi )],\hfill \\ c{\varphi }_2^{\mathrm{\prime }}(\xi )=d_2[{\int }_{\mathbb{R}}{J}_2(\xi -y){\varphi }_2(y)dy-{\varphi }_2(\xi )]+r_2[1-{\varphi }_2(\xi )][b_2{\varphi }_1(\xi )-{\varphi }_2(\xi )].\hfill \end{array}$$

(2.1)

At the same time, (1.5) will be

$$\underset{\xi \to -\mathrm{\infty }}{lim}({\varphi }_1(\xi ),{\varphi }_2(\xi ))=(0,0),\underset{\xi \to \mathrm{\infty }}{lim}({\varphi }_1(\xi ),{\varphi }_2(\xi ))=(1,1).$$

(2.2)

Take $\beta =2(d_1+d_2+r_1+r_2+1)(1+b_1+b_2)$ and

then $H_i$ is monotone in the functional sense if $({\varphi }_1,{\varphi }_2)\in X_{[\mathbf{0},\mathbf{1}]}$. Applying these notations, we further define an operator $F=(F_1,F_2):X_{[\mathbf{0},\mathbf{1}]}\to X_{[\mathbf{0},\mathbf{1}]}$ as follows:

$$F_i({\varphi }_1,{\varphi }_2)(\xi )=\frac{1}{c}{\int }_{-\mathrm{\infty }}^{\xi }{e}^{-\frac{\beta }{c}(\xi -s)}{H}_i({\varphi }_1,{\varphi }_2)(s)ds,i=1,2.$$

Clearly, a fixed point of $(F_1,F_2)$ in X satisfies (2.1), and a solution of (2.1) is also a fixed point of F. To continue our discussion, we also introduce the following definition.

Definition 2.1 Assume that $({\rho }_1,{\rho }_2)\in X_{[\mathbf{0},\mathbf{1}]}$. If ${\rho }_1$, ${\rho }_2$ are differentiable on $\mathbb{R}\setminus \mathbb{T}$, here $\mathbb{T}$ contains finite points, and the derivatives are essentially bounded so that

$$\{\begin{array}{c}c{\rho }_1^{\mathrm{\prime }}(\xi )\ge (\le )d_1[{\int }_{\mathbb{R}}{J}_1(\xi -y){\rho }_1(y)dy-{\rho }_1(\xi )]\hfill \\ \phantom{c{\rho }_1^{\mathrm{\prime }}(\xi )\ge (\le )}+r_1{\rho }_1(\xi )[1-b_1-{\rho }_1(\xi )+b_1{\rho }_2(\xi )],\hfill \\ c{\rho }_2^{\mathrm{\prime }}(\xi )\ge (\le )d_2[{\int }_{\mathbb{R}}{J}_2(\xi -y){\rho }_2(y)dy-{\rho }_2(\xi )]\hfill \\ \phantom{c{\rho }_2^{\mathrm{\prime }}(\xi )\ge (\le )}+r_2[1-{\rho }_2(\xi )][b_2{\rho }_1(\xi )-{\rho }_2(\xi )]\hfill \end{array}$$

(2.3)

for $\xi \in \mathbb{R}\setminus \mathbb{T}$, then it is an upper (a lower) solution of (2.1).

Using Pan et al. [33], Theorem 3.2, we obtain the following conclusion.

Lemma 2.2 Assume that $({\overline{\varphi }}_1(\xi ),{\overline{\varphi }}_2(\xi ))$ is an upper solution of (2.1), while $({\underline{\varphi }}_1(\xi ),{\underline{\varphi }}_2(\xi ))$ is a lower solution of (2.1). Also, suppose that

(P1) $({\underline{\varphi }}_1(\xi ),{\underline{\varphi }}_2(\xi ))\le ({\overline{\varphi }}_1(\xi ),{\overline{\varphi }}_2(\xi ))$;

(P2) ${lim}_{\xi \to -\mathrm{\infty }}({\overline{\varphi }}_1(\xi ),{\overline{\varphi }}_2(\xi ))=(0,0)$, ${lim}_{\xi \to \mathrm{\infty }}({\overline{\varphi }}_1(\xi ),{\overline{\varphi }}_2(\xi ))=(1,1)$;

(P3) ${sup}_{s<\xi }{\underline{\varphi }}_i(s)\le {inf}_{s>\xi }{\overline{\varphi }}_i(s)$ for all $\xi \in \mathbb{R}$, $i=1,2$, and ${sup}_{\xi \in \mathbb{R}}{\underline{\varphi }}_1(\xi )>0$.

Then (2.1)-(2.2) has a positive monotone solution $({\varphi }_1(\xi ),{\varphi }_2(\xi ))$ such that

$$({\underline{\varphi }}_1(\xi ),{\underline{\varphi }}_2(\xi ))\le ({\varphi }_1(\xi ),{\varphi }_2(\xi ))\le ({\overline{\varphi }}_1(\xi ),{\overline{\varphi }}_2(\xi )).$$

We now consider the following initial value problem:

$$\{\begin{array}{c}\frac{\partial u(x,t)}{\partial t}=d[{\int }_{\mathbb{R}}J(x-y)u(y,t)dy-u(x,t)]+ru(x,t)[1-u(x,t)],\hfill \\ u(x,0)=\varphi (x),x\in \mathbb{R},\hfill \end{array}$$

(2.4)

where J satisfies (J1) to (J3), $d>0$ and $r>0$ are constants, and the initial value $\varphi (x)\in C(\mathbb{R},\mathbb{R})$ with

In addition, let $C^{+}$ be a subset of C defined by

$$C^{+}=\{\varphi \in C:\varphi (x)\ge 0,x\in \mathbb{R}\}.$$

In Jin and Zhao [34], the authors investigated the asymptotic spreading of a periodic population model with spatial dispersal. Note that the parameters in (2.4) are positive constants, then [34], Theorem 2.1, implies the following result.

Lemma 2.3 Assume that $\varphi (x)\in C^{+}$. Then (2.4) has a unique solution $u(x,t)$ such that

$$u(x,t)\ge 0,x\in \mathbb{R},t>0.$$

In particular, if $\varphi (x)\in C_{[0,a]}$ with some $a\ge 1$, then

$$0\le u(x,t)\le a,x\in \mathbb{R},t>0.$$

Furthermore, we can also apply the results of Jin and Zhao [34], Theorem 3.5, since the assumptions (H1) and (H2) of [34] are clear. Define

$$c_1=\underset{\lambda >0}{inf}\frac{d[{\int }_{\mathbb{R}}J(y)e^{\lambda y}dy-1]+r}{\lambda }.$$

Then Jin and Zhao [34], Theorem 3.5, indicates the following conclusion.

Lemma 2.4 Assume that $\varphi (x)\in C^{+}$ admits nonempty support. Then

$$\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\underset{|x|<ct}{inf}u(x,t)=\underset{t\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{|x|<ct}{sup}u(x,t)=1\mathit{\text{for any }}c<c_1,$$

where $u(x,t)$ is defined by (2.4).

3 Existence of traveling wave solutions

In this section, we shall prove the existence of positive solutions of (2.1)-(2.2). Let

$${\mathrm{\Delta }}_1(\lambda ,c)=d_1[{\int }_{\mathbb{R}}{J}_1(y)e^{\lambda y}dy-1]-c\lambda +r_1(1-b_1)$$

for any $\lambda \ge 0$, $c>0$.

Lemma 3.1 There exists a constant $c^{\ast }>0$ such that the following items hold.

• (1) For each $c>c^{\ast }$, ${\mathrm{\Delta }}_1(\lambda ,c)=0$ has two positive real roots ${\lambda }_1(c)<{\lambda }_2(c)$.

• (2) If $c=c^{\ast }$, then there exists $\lambda (c^{\ast })>0$ such that ${\mathrm{\Delta }}_1(\lambda (c^{\ast }),c^{\ast })=0$ and ${\mathrm{\Delta }}_1(\lambda ,c^{\ast })>0$ for any $\lambda \ne \lambda (c^{\ast })$.

• (3) If $c<c^{\ast }$, then ${\mathrm{\Delta }}_1(\lambda ,c)>0$ for any $\lambda \ge 0$.

The above result is clear and we omit the proof here. Using these constants, we can prove the following conclusion.

Theorem 3.2 Assume that $c>c^{\ast }$ and one of the following two items holds.

• (1) $b_1{b}_2>1$ and

  $$d_2[{\int }_{\mathbb{R}}{J}_2(y)e^{{\lambda }_1(c)y}dy-1]-c{\lambda }_1(c)+r_2(b_1{b}_2-1)\le 0.$$

  (3.1)

• (2) $b_1{b}_2\le 1$ and

  $$d_2[{\int }_{\mathbb{R}}{J}_2(y)e^{{\lambda }_1(c)y}dy-1]-c{\lambda }_1(c)\le 0.$$

  (3.2)

Then (2.1)-(2.2) has a monotone solution.

Proof Define continuous functions as follows:

$${\overline{\varphi }}_1(\xi )=min\{e^{{\lambda }_1(c)\xi },1\},{\overline{\varphi }}_2(\xi )=min\{e^{{\lambda }_1(c)\xi }/b_1,1\}.$$

Claim A: $({\overline{\varphi }}_1(\xi ),{\overline{\varphi }}_2(\xi ))$ is an upper solution to (2.1).

Moreover, let ${\underline{\varphi }}_2(\xi )=0$ hold and ${\underline{\varphi }}_1(\xi )$ satisfy

$$c{\varphi }_1^{\mathrm{\prime }}(\xi )=d_1[{\int }_{\mathbb{R}}{J}_1(\xi -y){\varphi }_1(y)dy-{\varphi }_1(\xi )]+r_1{\varphi }_1(\xi )[1-b_1-{\varphi }_1(\xi )]$$

and

$$\underset{\xi \to -\mathrm{\infty }}{lim}{\varphi }_1(\xi )e^{-{\lambda }_1(c)\xi }=1.$$

Evidently, $({\underline{\varphi }}_1(\xi ),{\underline{\varphi }}_2(\xi ))$ is a lower solution to (2.1) (for the existence of ${\underline{\varphi }}_1(\xi )$ and ${\underline{\varphi }}_1(\xi )\le min\{e^{{\lambda }_1(c)\xi },1-b_1\}$, we refer to Pan et al. [33]). By Lemma 2.2, we see that (2.1)-(2.2) has a monotone solution $({\varphi }_1(\xi ),{\varphi }_2(\xi ))$. Now, it suffices to prove Claim A.

If ${\overline{\varphi }}_1(\xi )=1$ or ${\overline{\varphi }}_2(\xi )=1$, the result is clear. If $\xi \le 0$, then

$${\overline{\varphi }}_2(\xi )\le e^{{\lambda }_1(c)\xi }/b_1$$

such that

$$\begin{array}{c}{d}_1[{\int }_{\mathbb{R}}{J}_1(\xi -y){\overline{\varphi }}_1(y)dy-{\overline{\varphi }}_1(\xi )]-c{\overline{\varphi }}_1^{\mathrm{\prime }}(\xi )+r_1{\overline{\varphi }}_1(\xi )[1-b_1-{\overline{\varphi }}_1(\xi )+b_1{\overline{\varphi }}_2(\xi )]\hfill \\ \le d_1[{\int }_{\mathbb{R}}{J}_1(\xi -y)e^{{\lambda }_1(c)y}dy-e^{{\lambda }_1(c)\xi }]-c{\lambda }_1(c)e^{{\lambda }_1(c)\xi }\hfill \\ +r_1{e}^{{\lambda }_1(c)\xi }[1-b_1-e^{{\lambda }_1(c)\xi }+b_1{e}^{{\lambda }_1(c)\xi }/b_1]\hfill \\ =e^{{\lambda }_1(c)\xi }{\mathrm{\Delta }}_1({\lambda }_1(c),c)=0,\hfill \end{array}$$

which completes the proof on ${\overline{\varphi }}_1(\xi )$ for $\xi \ne 0$.

We now consider ${\overline{\varphi }}_2(\xi )<1$ with $\xi <0$. If $b_1{b}_2\ge 1$, then $b_2{e}^{{\lambda }_1(c)\xi }\ge e^{{\lambda }_1(c)\xi }/b_1$ such that

$$b_2{\overline{\varphi }}_1(\xi )-{\overline{\varphi }}_2(\xi )=b_2{e}^{{\lambda }_1(c)\xi }-\frac{e^{{\lambda }_1(c)\xi }}{b_1}\ge 0$$

and

$$r_2[1-{\overline{\varphi }}_2(\xi )][b_2{\overline{\varphi }}_1(\xi )-{\overline{\varphi }}_2(\xi )]\le r_2[b_2{e}^{{\lambda }_1(c)\xi }-\frac{e^{{\lambda }_1(c)\xi }}{b_1}].$$

Therefore, (3.1) leads to

$$\begin{array}{c}{d}_2[{\int }_{\mathbb{R}}{J}_2(\xi -y){\overline{\varphi }}_2(y)dy-{\overline{\varphi }}_2(\xi )]-c{\overline{\varphi }}_2^{\mathrm{\prime }}(\xi )+r_2[1-{\overline{\varphi }}_2(\xi )][b_2{\overline{\varphi }}_1(\xi )-{\overline{\varphi }}_2(\xi )]\hfill \\ \le d_2[{\int }_{\mathbb{R}}{J}_2(\xi -y){\overline{\varphi }}_2(y)dy-{\overline{\varphi }}_2(\xi )]-c{\overline{\varphi }}_2^{\mathrm{\prime }}(\xi )+r_2[b_2{\overline{\varphi }}_1(\xi )-{\overline{\varphi }}_2(\xi )]\hfill \\ \le \frac{e^{{\lambda }_1(c)\xi }}{b_1}[d_2[{\int }_{\mathbb{R}}{J}_2(y)e^{{\lambda }_1(c)y}dy-1]-c{\lambda }_1(c)+r_2(b_1{b}_2-1)]\hfill \\ \le 0.\hfill \end{array}$$

If $b_1{b}_2<1$, then $b_2{\overline{\varphi }}_1(\xi )-{\overline{\varphi }}_2(\xi )\le 0$ and (3.2) imply that

$$\begin{array}{c}{d}_2[{\int }_{\mathbb{R}}{J}_2(\xi -y){\overline{\varphi }}_2(y)dy-{\overline{\varphi }}_2(\xi )]-c{\overline{\varphi }}_2^{\mathrm{\prime }}(\xi )+r_2[1-{\overline{\varphi }}_2(\xi )][b_2{\overline{\varphi }}_1(\xi )-{\overline{\varphi }}_2(\xi )]\hfill \\ \le d_2[{\int }_{\mathbb{R}}{J}_2(\xi -y){\overline{\varphi }}_2(y)dy-{\overline{\varphi }}_2(\xi )]-c{\overline{\varphi }}_2^{\mathrm{\prime }}(\xi )\hfill \\ \le \frac{e^{{\lambda }_1(c)\xi }}{b_1}[d_2{\int }_{\mathbb{R}}{J}_2(y)e^{{\lambda }_1(c)y}dy-d_2-c{\lambda }_1(c)]\hfill \\ \le 0.\hfill \end{array}$$

Therefore, Claim A is true. The proof is complete. □

Theorem 3.3 Assume that one of the following items holds.

• (1) $b_1{b}_2>1$ and

  $$d_2[{\int }_{\mathbb{R}}{J}_2(y)e^{{\lambda }_1(c^{\ast })y}dy-1]-c^{\ast }{\lambda }_1\left(c^{\ast }\right)+r_2(b_1{b}_2-1)<0.$$

  (3.3)

• (2) $b_1{b}_2\le 1$ and

  $$d_2[{\int }_{\mathbb{R}}{J}_2(y)e^{{\lambda }_1(c^{\ast })y}dy-1]-c^{\ast }{\lambda }_1\left(c^{\ast }\right)<0.$$

  (3.4)

Then (2.1)-(2.2) has a monotone solution with $c=c^{\ast }$.

Proof If (3.3) or (3.4) holds, then there exists a decreasing sequence ${\{c_n\}}_{n=1}^{\mathrm{\infty }}$ with $c_n\to c^{\ast }$, $n\to \mathrm{\infty }$ such that for each $c_n$, (2.1)-(2.2) has a positive monotone solution $({\varphi }_1^n,{\varphi }_2^n)$. Note that a traveling wave solution is invariant in the sense of phase shift, so we can assume that

$${\varphi }_2^n(0)=1/2$$

(3.5)

for any n. By the Ascoli-Arzela lemma and a standard nested subsequence argument (see, e.g., Thieme and Zhao [47]), there exists a subsequence of ${\{c_n\}}_{n=1}^{\mathrm{\infty }}$, which is still denoted by ${\{c_n\}}_{n=1}^{\mathrm{\infty }}$ without confusion, such that $({\varphi }_1^n(\xi ),{\varphi }_2^n(\xi ))$ converges uniformly on every bounded interval, and hence pointwise on ℝ to a continuous function $({\hat{\varphi }}_1(\xi ),{\hat{\varphi }}_2(\xi ))$. Moreover, for each $c_n$, we have

$$\frac{1}{c_n}{e}^{-\frac{\beta }{c_n}(\xi -s)}\to \frac{1}{c^{\ast }}{e}^{-\frac{\beta }{c^{\ast }}(\xi -s)}\text{for any }\xi \in \mathbb{R},s\le \xi ,$$

and the convergence in s is uniform for $s\le \xi$. Letting $n\to \mathrm{\infty }$ and using the dominated convergence theorem in $(F_1,F_2)$, we know that $({\hat{\varphi }}_1(\xi ),{\hat{\varphi }}_2(\xi ))$ also satisfies (2.1) with $c=c^{\ast }$. In addition, the following items are also clear.

(T1) ${\hat{\varphi }}_2(0)=1/2$ (by (3.5));

(T2) ${\hat{\varphi }}_1(\xi )$, ${\hat{\varphi }}_2(\xi )$ are nondecreasing in ξ;

(T3) $0\le {\hat{\varphi }}_1(\xi )$, ${\hat{\varphi }}_2(\xi )\le 1$, $\xi \in \mathbb{R}$.

The items (T1) to (T3) further indicate that ${lim}_{\xi \to \pm \mathrm{\infty }}{\hat{\varphi }}_i(\xi )$ exists for $i=1,2$. Denote

$$\underset{\xi \to -\mathrm{\infty }}{lim}{\hat{\varphi }}_i(\xi )={\hat{\varphi }}_i^{-},\underset{\xi \to \mathrm{\infty }}{lim}{\hat{\varphi }}_i(\xi )={\hat{\varphi }}_i^{+},i=1,2.$$

From (T1), it is clear that

$$0\le {\hat{\varphi }}_2^{-}\le \frac{1}{2}\le {\hat{\varphi }}_2^{+}\le 1.$$

If ${\hat{\varphi }}_2^{-}\in (0,1/2]$, then the dominated convergence theorem in $F_2$ implies that

$$b_2{\hat{\varphi }}_1^{-}={\hat{\varphi }}_2^{-}.$$

Using the dominated convergence theorem in $F_1$ for $\xi \to -\mathrm{\infty }$, we get the following possible conclusions:

(L1) ${\hat{\varphi }}_1^{-}=0$;

(L2) $1-b_1-{\hat{\varphi }}_1^{-}+b_1{\hat{\varphi }}_2^{-}=1-b_1-{\hat{\varphi }}_1^{-}+b_1{b}_2{\hat{\varphi }}_1^{-}=0$.

If (L1) is true, then the dominated theorem in $F_2$ tells us

$${\hat{\varphi }}_2^{-}[1-{\hat{\varphi }}_2^{-}]=0,$$

which implies a contradiction. If (L2) is true, then $b_1{b}_2>b_1$ leads to

$$0=1-b_1-{\hat{\varphi }}_1^{-}+b_1{b}_2{\hat{\varphi }}_1^{-}>1-b_1-{\hat{\varphi }}_1^{-}+b_1{\hat{\varphi }}_1^{-}=(1-{\hat{\varphi }}_1^{-})(1-b_1),$$

which is also a contradiction. What we have done implies that ${\hat{\varphi }}_2^{-}=0$. Using the dominated convergence theorem in $F_2$ again, we see that $b_2{\hat{\varphi }}_1^{-}={\hat{\varphi }}_2^{-}=0$ and ${\hat{\varphi }}_1^{-}=0$.

If ${\hat{\varphi }}_2^{+}\in [1/2,1)$, then a discussion similar to that on ${\hat{\varphi }}_2^{-}$ can be presented and we omit it here. Because ${\hat{\varphi }}_2^{+}=1$, then the dominated convergence in $F_1$ as $\xi \to +\mathrm{\infty }$ indicates that ${\hat{\varphi }}_1^{+}=0$ or ${\hat{\varphi }}_1^{+}=1$. If ${\hat{\varphi }}_1^{+}=0$ is true, then ${\varphi }_1(\xi )\equiv 0$ holds and

$$\{\begin{array}{c}c{\varphi }_2^{\mathrm{\prime }}(\xi )=d_2[{\int }_{\mathbb{R}}{J}_2(\xi -y){\varphi }_2(y)dy-{\varphi }_2(\xi )]-r_2{\varphi }_2(\xi )[1-{\varphi }_2(\xi )],\hfill \\ {lim}_{\xi \to -\mathrm{\infty }}{\varphi }_2(\xi )=0,{lim}_{\xi \to \mathrm{\infty }}{\varphi }_2(\xi )=1\hfill \end{array}$$

has a monotone solution, which is impossible. Therefore, ${\hat{\varphi }}_1^{+}=1$ holds.

Thus, $({\hat{\varphi }}_1(\xi ),{\hat{\varphi }}_2(\xi ))$ is a positive monotone solution of (2.1)-(2.2) with $c=c_{\ast }$, the proof is complete. □

4 Nonexistence of traveling wave solutions

In this section, we shall formulate the nonexistence of invasion traveling wave solutions of (1.2) by the theory of asymptotic spreading. Before this, we first present a comparison principle formulated by Jin and Zhao [34], Theorem 2.3.

Lemma 4.1 Assume that $\varphi (x)\in C^{+}$. If $w(x,t)\ge 0$, $x\in \mathbb{R}$, $t>0$ is bounded such that

$$\{\begin{array}{c}\frac{\partial w(x,t)}{\partial t}\ge (\le )d[{\int }_{\mathbb{R}}J(x-y)w(y,t)dy-w(x,t)]+rw(x,t)[1-w(x,t)],\hfill \\ w(x,0)\ge (\le )\varphi (x),x\in \mathbb{R},\hfill \end{array}$$

(4.1)

then $w(x,t)\ge (\le )u(x,t)$, $x\in \mathbb{R}$, $t>0$.

We now give the main result of this section.

Theorem 4.2 If $c<c^{\ast }$, then (2.1)-(2.2) has no positive solutions.

Proof Define

$$c_2=\underset{\lambda >0}{inf}\left\{\frac{d_1[{\int }_{\mathbb{R}}{J}_1(y)e^{\lambda y}dy-1]+r_1(1-b_1)}{\lambda }\right\}.$$

Then $c_2=c^{\ast }$ is evident.

If (2.1)-(2.2) has a positive solution $({\varphi }_1(\xi ),{\varphi }_2(\xi ))$ for some $c=\overline{c}<c^{\ast }$, then

$${\varphi }_2(\xi )={\varphi }_2(x+\overline{c}t)\ge 0,x\in \mathbb{R},t>0,\xi \in \mathbb{R}$$

implies that ${\varphi }_1(\xi )$ also satisfies

$$\overline{c}{\varphi }_1^{\mathrm{\prime }}(\xi )\ge d_1[{\int }_{\mathbb{R}}{J}_1(\xi -y){\varphi }_1(y)dy-{\varphi }_1(\xi )]+r_1{\varphi }_1(\xi )[1-b_1-{\varphi }_1(\xi )]$$

(4.2)

with the following asymptotic boundary condition:

$$\underset{\xi \to -\mathrm{\infty }}{lim}{\varphi }_1(\xi )=0,\underset{\xi \to \mathrm{\infty }}{lim}{\varphi }_1(\xi )=1.$$

(4.3)

Recalling the definition of traveling wave solutions, we see that $w(x,t)={\varphi }_1(x+\overline{c}t)$ also satisfies

$$\frac{\partial w(x,t)}{\partial t}\ge d_1[{\int }_{\mathbb{R}}{J}_1(x-y)w(y,t)dy-w(x,t)]+r_{1}w(x,t)[1-b_1-w(x,t)]$$

(4.4)

and

$$0\le w(x,t)\le 1,x\in \mathbb{R},t\ge 0,\underset{x\to \mathrm{\infty }}{lim}w(x,0)=1.$$

(4.5)

Using Lemmas 2.4 and 4.1, we see that

$$\underset{t\to \mathrm{\infty }}{lim}\underset{2|x|=(\overline{c}+c^{\ast })t}{inf}w(x,t)\ge 1-b_1$$

(4.6)

since $\overline{c}+c^{\ast }<2c^{\ast }$.

However, the boundary condition (4.3) indicates that

$$\xi =x+\overline{c}t\to -\mathrm{\infty }\text{with }-2x=(\overline{c}+c^{\ast })t,t\to \mathrm{\infty }$$

and

$$\underset{t\to \mathrm{\infty },-2x=(\overline{c}+c^{\ast })t}{lim}w(x,t)=0,$$

(4.7)

which implies a contradiction between (4.6) and (4.7). The proof is complete. □

Remark 4.3 Under proper assumptions, we have obtained the threshold of the existence of positive solutions to (2.1)-(2.2).