Invasion traveling wave solutions of a competitive system with dispersal

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].

$J_i:\mathbb{R}\to \mathbb{R}$, $i=1,2$, 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.

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.

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.

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

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

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.

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 ${sup}_{\xi \in \mathbb{R}}{\underline{\varphi }}_1(\xi )>0$.

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 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.

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

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

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

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

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

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

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.

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

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

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

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

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

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 ${\hat{\varphi }}_1^{-}=0$.

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. □

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.

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.

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

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

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).