Effects of Prey’s Diffusion on Predator–Prey Systems with Two Patches

Abstract

This paper considers predator–prey systems in which the prey can move between source and sink patches. First, we give a complete analysis on global dynamics of the model. Then, we show that when diffusion from the source to sink is not large, the species would coexist at a steady state; when the diffusion is large, the predator goes to extinction, while the prey persists in both patches at a steady state; when the diffusion is extremely large, both species go to extinction. It is derived that diffusion in the system could lead to results reversing those without diffusion. That is, diffusion could change species’ coexistence if non-diffusing, to extinction of the predator, and even to extinction of both species. Furthermore, we show that intermediate diffusion to the sink could make the prey reach total abundance higher than if non-diffusing, larger or smaller diffusion rates are not favorable. The total abundance, as a function of diffusion rates, can be both hump-shaped and bowl-shaped, which extends previous theory. A novel finding of this work is that there exist diffusion scenarios which could drive the predator into extinction and make the prey reach the maximal abundance. Diffusion from the sink to source and asymmetry in diffusion could also lead to results reversing those without diffusion. Meanwhile, diffusion always leads to reduction of the predator’s density. The results are biologically important in protection of endangered species.

Appendices

Appendix A: Results for Sect. 2

Proposition 5.1

Solutions of system (1) are nonnegative and bounded.

Proof

Since $x=0$ is a solution of (1), $y_1y_2$-plane is invariant, which implies $x(t)\ge 0$ as $t> 0$.

On the boundary $y_1=0$, from the second equation of (1) we have $\mathrm{d}y_1/\mathrm{d}t = D y_2$. If $y_2>0$, then $\mathrm{d}y_1/\mathrm{d}t >0$, which implies that $y_1(t)$ is nonnegative when t increases. Assume $y_2=0$. Since x-axis is an invariant set of system (1), no orbit could pass through the invariant set, which implies that $y_1(t) \equiv 0$. Thus, $y_1(t)\ge 0$ when $t>0$. Similarly, $y_2(t)\ge 0$ when $t>0$. Thus, solutions of systems (1) are nonnegative.

Assume that $(x(t),y_1(t), y_2(t))$ is a solution of (1) with $x(0) \ge 0, y_i(0) \ge 0, i=1,2$. Let ${\bar{r}}=\min \{ r,r_1, r_2 \}$. By adding the three equations of (1), we obtain

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \left( y_1+y_2+\frac{a_{21} }{a_{12} }x\right)&=- \frac{r a_{21} }{a_{12} } x + r_1y_1\left( 1-\frac{y_1}{K_1}\right) -r_2y_2\\&\le r_1y_1\left( 2-\frac{y_1}{K_1}\right) - {\bar{r}} \left( y_1+y_2+\frac{a_{21} }{a_{12} } x\right) \\&\le r_1 K_1 - {\bar{r}} (y_1+y_2+\frac{a_{21} }{a_{12} } x). \end{aligned}$$

From the comparison theorem (Hale 1969), we have

$$\begin{aligned} \limsup _{t\rightarrow \infty } y_1(t)+y_2(t)+\frac{a_{21} }{a_{12} } x(t) \le \frac{r_1 K_1 }{ {\bar{r}}}. \end{aligned}$$

Thus, the solution $(x(t),y_1(t), y_2(t))$ is bounded and system (1) is dissipative. $\square$

Proposition 5.2

• (i) If $D_1 < D_1^+,$ there is a unique positive equilibrium $E_{12}(y_1^+,y_2^+)$ of system (2), which is globally asymptotically stable in int$R_+^2$ as shown in Fig. 1.

• (ii) If $D_1 \ge D_1^+,$ the boundary equilibrium O(0, 0) of system (2) is globally asymptotically stable in int$R_+^2$.

Proposition 5.3

(Hofbauer and Sigmund 1998)

• (i) If $a_{12} > r/K_1$, there is a unique positive equilibrium ${\bar{E}}({\bar{x}},{\bar{y}}_1)$ of system (3), which is globally asymptotically stable in int$R_+^2$.

• (ii) If $a_{12} \le r/K_1$, the boundary equilibrium $E_1(0,K_1)$ of system (3) is globally asymptotically stable in int$R_+^2$.

Appendix B: Results for Sect. 3

Here, global dynamics of system (1) are shown by constructing Lyapunov functions. Every Lyapunov function is formed by combining the sub-Lyapunov function for each patch by the graph method (see Remark 5.5).

Theorem 5.4

• (i) If $D_1 \ge D_1^+$, the boundary equilibrium O(0, 0, 0) of system (1) is globally asymptotically stable in int$(R_+^3)$, and system (1) is not persistent.

• (ii) If $D_1^- \le D_1 < D_1^+$, the boundary equilibrium $P_{23}(0, y_1^+,y_2^+)$ of system (1) is globally asymptotically stable in int$(R_+^3)$, and system (1) is not persistent.

• (iii) If $D_1 < D_1^-$, the positive equilibrium $P^*(x^*,y_1^*,y_2^*)$ of system (1) is globally asymptotically stable in int$(R_+^3)$, and system (1) is uniformly persistent.

Proof

(i) Let

$$\begin{aligned} V(x, y_1,y_2) = y_1 + \frac{D_2 }{r_2+ D_2 } y_2. \end{aligned}$$

From $D_1 \ge D_1^+$, we have

$$\begin{aligned} \frac{\mathrm{d}V}{\mathrm{d}t}\bigg |_{(2.1)} = y_1 \left( r_1 - \dfrac{r_2D_1 }{r_2 +D_2} - \frac{r_1 }{K_1}y_1 - a_{21} x\right) \le 0. \end{aligned}$$

Let $\dfrac{\mathrm{d}V}{\mathrm{d}t}\big |_{(2.1)} =0$. Then, we obtain $y_1 = 0$, which means $x =0$ by the first equation of (1), and $y_2 =0$ by the second equation of (1). By the LaSalle invariance principle, equilibrium O is globally asymptotically stable in int$(R_+^3)$.

(ii) Let

$$\begin{aligned} U_1&= y_1- y_1^+ -y_1^+ \ln \frac{y_1}{y_1^+ },\quad U_2= \frac{ D_2 y_2^+}{D_1 y_1^+ } \left( y_2- y_2^+ -y_2^+ \ln \frac{y_2}{y_2^+ }\right) \\ V(x, y_1,y_2)&= \dfrac{a_{21}}{a_{12} } x + U_1 +U_2= \dfrac{a_{21}}{a_{12} } x + y_1- y_1^+ -y_1^ + \ln \frac{y_1}{y_1^+}\\&\quad + \frac{ D_2 y_2^+}{D_1 y_1^+ } \left( y_2- y_2^+ -y_2^+ \ln \frac{y_2}{y_2^+ }\right) . \end{aligned}$$

Since $D_1 \ge D_1^-$, we have $-\dfrac{r}{a_{12}} + y_1^+ \le 0$. Then,

$$\begin{aligned} \frac{\mathrm{d}V}{\mathrm{d}t}\bigg |_{(2.1)} = \left( -\frac{r}{a_{12}} + y_1^+\right) a_{21} x - \frac{r_1}{K_1 } (y_1- y_1^+ )^2 + D_2 y_2^+ \left( 2- \frac{y_2 y_1^+ }{y_2^+ y_1 } - \frac{y_2^+ y_1 }{y_2 y_1^+ } \right) \le 0 \end{aligned}$$

where we use the inequality $2- u - \dfrac{1}{u } \le 0$ if $u > 0$, and the equality holds if and only if $u =1$. Let $\dfrac{\mathrm{d}V}{\mathrm{d}t}|_{(2.1)}=0$. Then, we obtain $y_1 = y_1^+$, which means $y_2 = y_2^+$ by $2- \dfrac{y_2 y_1^+ }{y_2^+ y_1 } - \dfrac{y_2^+ y_1 }{y_2 y_1^+ }= 0$. Thus, we obtain $x = 0$ by the second equation of (2). By the LaSalle invariance principle, equilibrium $P_{23}$ is globally asymptotically stable in int$(R_+^3)$.

(iii) From $D_1 < D_1^-$, we have $D_1 < D_1^+$. Denote

$$\begin{aligned} V_1&= \dfrac{a_{21} }{a_{12}}\left( x- x^* -x^* \ln \frac{x}{x^* }\right) + y_1- y_1^* -y_1^* \ln \frac{y_1}{y_1^* },\\ V_2&= \frac{ D_2 y_2^*}{D_1 y_1^* } \left( y_2- y_2^* -y_2^* \ln \frac{y_2}{y_2^* }\right) . \end{aligned}$$

Let $V=V_1 +V_2$. Then,

$$\begin{aligned} \frac{\mathrm{d}V}{\mathrm{d}t }|_{(2.1) } = - \frac{r_1}{K_1 } a_{12} (y_1- y_1^* )^2 + D_2 y_2^* \left( 2- \frac{y_1 y_2^* }{y_1^* y_2 } - \frac{y_1^* y_2 }{y_1 y_2^* } \right) \le 0 \end{aligned}$$

where we use the inequality $2- u - \dfrac{1}{u } \le 0$ if $u > 0$. Let $\dfrac{\mathrm{d}V}{\mathrm{d}t}\big |_{(2.1)}=0$. Then, we obtain $y_1 = y_1^*$, which means x(t) is a constant by the first equation of (1), and then, $y_2(t)$ is a constant by the second equation of (1). By the uniqueness of positive equilibrium in (1), we obtain that $x = x^*$ and $y_2 = y_2^*$. By the LaSalle invariance principle, equilibrium $P^*$ is globally asymptotically stable in int$(R_+^3)$. $\square$

Remark 5.5

Construction of Lyapunov functions in the proof of Proposition 5.2 and Theorem 5.4. We focus on the Lyapunov function in the proof of Theorem 5.4(iii), while similar discussion can be given for others.

When there is no diffusion and the species coexist at $({\bar{x}},{\bar{y}}_1)$ in patch 1, there is a Lyapunov function for the $(x,y_1)$-subsystem (Hofbauer and Sigmund 1998):

$$\begin{aligned} {\bar{V}}_1= \dfrac{a_{21} }{a_{12}}\left( x- {\bar{x}} -{\bar{x}} \ln \frac{x}{{\bar{x}} }\right) + y_1- {\bar{y}}_1 -{\bar{y}}_1 \ln \frac{y_1}{{\bar{y}}_1 }. \end{aligned}$$

When there is diffusion and the prey can persist at $y_2={\bar{y}}_2$ in patch 2, a direct computation shows that there is a Lyapunov function for the $y_2$-subsystem:

$$\begin{aligned} {\bar{V}}_2= y_2- {\bar{y}}_2 -{\bar{y}}_2 \ln \frac{y_2}{{\bar{y}}_2 }. \end{aligned}$$

When there is diffusion and the species coexist at $P^*(x^*,y_1^*,y_2^*)$ in system (1), we obtain $V_1, V_2$ from ${\bar{V}}_1, {\bar{V}}_2$ by replacing ${\bar{x}},{\bar{y}}_1,{\bar{y}}_2$ with $x^*,y_1^*,y_2^*$, where the coefficient $\dfrac{ D_2 y_2^*}{D_1 y_1^* }$ of $V_2$ is obtained by the graph method (see Li and Shuai 2010). Thus, we obtain the Lyapunov function

$$\begin{aligned} V&=V_1 +V_2= \dfrac{a_{21} }{a_{12}}\left( x- x^* -x^* \ln \frac{x}{x^* }\right) + y_1- y_1^* -y_1^* \ln \frac{y_1}{y_1^* } \\&\quad + \frac{ D_2 y_2^*}{D_1 y_1^* } \left( y_2- y_2^* -y_2^* \ln \frac{y_2}{y_2^* }\right) . \end{aligned}$$