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# Dynamics of consumer-resource systems with consumer's dispersal between patches

The second author is supported by NSF grant of China (12071495, 11571382)

• This paper considers consumer-resource systems with Holling II functional response. In the system, the consumer can move between a source and a sink patch. By applying dynamical systems theory, we give a rigorous analysis on persistence of the system. Then we show local/global stability of equilibria and prove Hopf bifurcation by the Kuznetsov Theorem. It is shown that dispersal in the system could lead to results reversing those without dispersal. Varying a dispersal rate can change species' interaction outcomes from coexistence in periodic oscillation, to persistence at a steady state, to extinction of the predator, and even to extinction of both species. By explicit expressions of stable equilibria, we prove that dispersal can make the consumer reach overall abundance larger than if non-dispersing, and there exists an optimal dispersal rate that maximizes the abundance. Asymmetry in dispersal can also lead to those results. It is proven that the overall abundance is a ridge-like function (surface) of dispersal rates, which extends both previous theory and experimental observation. These results are biologically important in protecting endangered species.

Mathematics Subject Classification: 34C37, 92D25, 37N25.

 Citation: • • Figure 1.  Effect of dispersal $D_1$ on dynamics of system (1). Fix $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 0.9, a_{21} = 0.5, b = 1, K = 4, D_2 = 0.1$. (a-b) Let $D_1 = 0.03$ and $D_1 = 0.15$, respectively. The resource and consumer coexist in periodic oscillations, while the amplitude decreases with the increase of $D_1$. (c) Let $D_1 = 0.28$. The resource and consumer coexist at a steady state $P^*( 2.1250, 1.6272, 2.2787)$. (d) Let $D_1 = 0.9$. The consumer goes to extinction even though the resource species persists

Figure 2.  Dynamics of system (1). Let $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 0.9, a_{21} = 0.8, b = 1, K = 4, D_1 = 0.3, D_2 = 0.1$. Numerical simulations display that all positive solutions (except $P^*$) of systems (1) converge to the unique limit cycle and exhibit periodic oscillations

Figure 3.  Comparison of $T_1(D_1, D_2)$ and $T_0$. The red and black lines represent $T_1$ and $T_0$, respectively. Fix $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 2, a_{21} = 0.8, b = 1, K = 2$. (a) Fix $D_2 = 0.1$ but let $D_1$ vary. $T_1$ approaches its maximum $T_{1max} = 1.5089$ at $\bar{D_1} = 0.4276$, and the curve is hump-shaped. We have $T_1>T_0$ if $D_1<0.6305$; $T_1<T_0$ if $D_1>0.6305$ as shown in Proposition 4(ii). (b) Fix $D_1 = 0.86$ but let $D_2$ vary. $T_1$ approaches its maximum $T_{1max} = 1.5089$ at $\bar{D_2} = 0.3023$. The curve is hump-shaped if $D_2<0.6014$ but is convex if $D_2>0.6014$. We have $T_1>T_0$ if $D_2>\hat{D_2} = 0.1728$; $T_1<T_0$ if $D_2<0.1728$

Figure 4.  The surface of $T_1(D_1, D_2)$ when both $D_1$ and $D_2$ vary. Fix $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 2, a_{21} = 0.8, b = 1, K = 2$. Then $T_1$ approaches its maximum $T_{1max} = 1.5089$ at a line $D_1 = 0.213+2.137D_2$, as shown in proposition 5(iii). This figure provides an intuition of the surface of $T_1 = T_1(D_1, D_2)$, which is a combination of Figs. 3a-b, i.e., when $D_2$ is fixed, the surface becomes Fig. 3a; when $D_1$ is fixed, the surface becomes Fig. 3b

Figure 5.  Dynamics of system (1). Fix $r = 1, r_1 = 0.2, r_2 = 0.1, a_{12} = 0.9, b = 1, K = 4, D_1 = 0.28, D_2 = 0.1$. (a) Let $a_{21} = 0.4$. The consumer goes to extinction even though the resource species persists. (b) Let $a_{21} = 0.5$. The resource and consumer coexist at a steady state $P^*(2.1250, 1.6272, 2.2787)$. (c-d) Let $a_{21} = 0.65$ and $a_{21} = 0.8$, respectively. The resource and consumer coexist in periodic oscillation, while the amplitude increases with the increase of $a_{21}$

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