Global attractor and threshold dynamics of a reaction–diffusion population model in a polluted environment

Abstract

This paper describes a population model in a polluted environment with heterogeneous parameters and a reaction–diffusion mechanism. We establish the existence of positive solution and a global attractor of the model. By defining the basic reproduction number $\mathcal {R}_0$, the threshold dynamics for the extinction or persistence of the population are discussed. The global attractivity of the unique positive equilibrium $E^*$ is obtained by establishing the Lyapunov function. Finally, some numerical simulations are provided to illustrate the theoretical results.

Appendices

Appendix A: Proof of Lemma 3.2

Proof

We first prove the existence and uniqueness of the equilibrium state. From the second equation of system (3), the equilibrium state $W_2$ satisfies

$$\begin{aligned} \mathcal {L}W_2(x)=-d_3\Delta W_2(x)+h(x)W_2(x). \end{aligned}$$

As $h(x)>0$, we know that the eigenvalue of $\mathcal {L}$ is positive. The operator $\mathcal {L}$ is invertible, and so the second equation of system (3) admits a unique positive equilibrium state $C_e^*(x)$. From Assumption 2, we know that $u(x,t)>0$, and so

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}-d_3\Delta C_e^*(x)+h(x)C_e^*(x)>0,~x\in \Omega , t>0,\\ &{}\frac{\partial C_e^*(x)}{\partial v}=0,~~~~~~~~~~~~~~~~~~~~~~~x\in \partial \Omega , t>0. \end{array}\right. } \end{aligned} \end{aligned}$$

Using the strong extremum principle [43, Theorem 4], we have $C_e^*(x)>0$.

Next, we prove the global stability of $C_e^*(x)$. Define the Lyapunov functional as $\mathcal {L}_2:C({\overline{\Omega }},\mathbb {R})\rightarrow \mathbb {R}$,

$$\begin{aligned} \mathcal {L}_2(W_2(x,t))=\int _\Omega (W_2(x,t)-C_e^*(x))^2dx. \end{aligned}$$

From Green’s formula and the Neumann boundary condition, we obtain

$$\begin{aligned} \mathcal {L}_2'(t)= & {} -2d_3\int _\Omega |\nabla (W_2(x,t)-C_e^*(x))|^2dx\\&-2\int _\Omega h(x)(W_2(x,t)-C_e^*(x))^2dx\le 0. \end{aligned}$$

Furthermore, $\mathcal {L}_2'(t)=0$ if and only if $W_2(x,t)\equiv C_e^*(x)$. Then, from LaSalle’s invariance principle, we know that $C_e^*(x)$ is globally asymptotically stable in $C({\overline{\Omega }},\mathbb {R})$. The global asymptotic stability of $C_0^*(x)$ can be obtained in a similar way. This completes the proof. $\square$

Appendix B: Proof of Lemma 4.2

Proof

From a well-known fact, there exists a positive function $\Psi \in C^2({\overline{\Omega }})$ such that

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}d_1\Delta \Psi +\frac{r_0(x)}{\mathcal {R}_0}\Psi -r_1(x)C_0^*(x)\Psi =0,~x\in \Omega , t>0,\\ &{}\frac{\partial \Psi }{\partial v}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x\in \partial \Omega , t>0. \end{array}\right. } \end{aligned} \end{aligned}$$

(23)

Let $\varphi _0$ be the positive eigenfunction corresponding to $\lambda _0$, that is,

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}d_1\Delta \varphi _0+r_0(x)\varphi _0-r_1(x)C_0^*(x)\varphi _0=\lambda _0\varphi _0,~x\in \Omega ,\\ &{}\frac{\partial \varphi _0}{\partial v}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x\in \partial \Omega . \end{array}\right. } \end{aligned} \end{aligned}$$

(24)

Multiplying Eq. (23) by $\varphi _0$ and Eq. (24) by $\Psi$, and then integrating the resulting expressions on $\Omega$, we have

$$\begin{aligned} \int _\Omega d_1\nabla \varphi _0\cdot \nabla \Psi dx+\int _\Omega \frac{r_0(x)}{\mathcal {R}_0}\Psi \varphi _0dx-\int _\Omega r_1(x)C_0^*(x)\Psi \varphi _0dx=0, \end{aligned}$$

and

$$\begin{aligned} \int _\Omega d_1\nabla \Psi \cdot \nabla \varphi _0dx+\int _\Omega r_0(x)\varphi _0\Psi dx-\int _\Omega r_1(x)C_0^*(x)\varphi _0\Psi dx=\int _\Omega \lambda _0\varphi _0\Psi dx, \end{aligned}$$

respectively. Subtracting one from the other, we obtain

$$\begin{aligned} \left( 1-\frac{1}{\mathcal {R}_0}\right) \int _\Omega r_0(x)\Psi \varphi _0dx=\lambda _0\int _\Omega \varphi _0\Psi dx. \end{aligned}$$

As the integrals on both sides are positive, $1-\frac{1}{\mathcal {R}_0}$ and $\lambda _0$ have the same sign. This completes the proof. $\square$

Appendix C: Proof of Lemma 4.3

Proof

From the first equation of system (2), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial P(x,t)}{\partial t}\ge d_1\Delta P(x,t)-P(x,t)(r_1(x)K_2+f(x)P(x,t)),~x\in {\overline{\Omega }}, t>0,\\ \frac{\partial P(x,t)}{\partial v}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x\in \partial \Omega , t>0. \end{array}\right. } \end{aligned}$$

If $P(x,t;\phi )\not \equiv 0$, for some $t_0>0$, the strong maximum principle [43, Theorem 4] and the Hopf boundary lemma [44, Proposition 13.1] imply that $P(\cdot ,t;\phi )>0,x\in {\overline{\Omega }}, \forall t>t_0$.

From the last two equations of system (2), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial C_0(x,t)}{\partial t}\ge d_2\Delta C_0(x,t)+{\tilde{k}}C_e(x,t)-({\bar{l}}+{\bar{m}})C_0(x,t),~~~~~~x\in {\overline{\Omega }}, t>0,\\ \frac{\partial C_e(x,t)}{\partial t}\ge d_3\Delta C_e(x,t)+b-({\bar{g}}K_3+{\bar{h}})C_e(x,t),~~~~~~~~~~~~~x\in {\overline{\Omega }}, t>0,\\ \frac{\partial C_0(x,t)}{\partial t}=0,\frac{\partial C_e(x,t)}{\partial t}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x\in \partial \Omega , t>0. \end{array}\right. } \end{aligned}$$

Using the comparison principle, we obtain

$$\begin{aligned} \liminf \limits _{t\rightarrow \infty } C_0(\cdot ,t;\phi )\ge \frac{{\tilde{k}}b}{({\bar{g}}K_3+{\bar{h}})({\bar{l}}+{\bar{m}})}, \end{aligned}$$

and

$$\begin{aligned} \liminf \limits _{t\rightarrow \infty } C_e(\cdot ,t;\phi )\ge \frac{b}{{\bar{g}}K_3+{\bar{h}}}, \end{aligned}$$

uniformly for $x\in {\overline{\Omega }}$. This completes the proof. $\square$