# American Institute of Mathematical Sciences

## On predation effort allocation strategy over two patches

 1 Department of Applied Mathematics, National Pingtung University, Pingtung 90003, Taiwan 2 Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7

* Corresponding author: Xingfu Zou

Received  August 2020 Published  September 2020

Fund Project: This paper is dedicated to Professor Sze-Bi Hsu in celebrating his retirement and in honour of his great contributions to the fields of mathematical biology and dynamical systems. C.-Y. Cheng was partially supported by the Ministry of Science and Technology, Taiwan, R.O.C. (Grant No. MOST 108-2115-M-153-003); X. Zou was partially supported by NSERC of Canada (RGPIN-2016-04665)

In this paper, we formulate an ODE model to describe the population dynamics of one non-dispersing prey and two dispersing predators in a two-patch environment with spatial heterogeneity. The dispersals of the predators are implicitly reflected by the allocation of their presence (foraging time) in each patch. We analyze the dynamics of the model and discuss some biological implications of the theoretical results on the dynamics of the model. Particularly, we relate the results to the evolution of the allocation strategy and explore the impact of the spatial heterogeneity and the difference in fitness of the two predators on the allocation strategy. Under certain range of other parameters, we observe the existence of an evolutionarily stable strategy (ESS) while in some other ranges, the ESS disappears. We also discuss some possible extensions of the model. Particularly, when the model is modified to allow distinct preys in the two patches, we find that the heterogeneity in predation rates and biomass transfer rates in the two patches caused by such a modification may lead to otherwise impossible bi-stability for some pairs of equilibria.

Citation: Chang-Yuan Cheng, Xingfu Zou. On predation effort allocation strategy over two patches. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020281
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Different regions of $\mu$ to examine the criterion $F(\alpha_1, \alpha_2)>0$ (when $\xi>1$)
Strategies $\alpha_1$ vs $\alpha_2$ for $F(\alpha_1, \alpha_2)>0$ (with $\xi>1$): (A) for $\mu$ in I-1, (B) for $\mu$ in I-2 and I-3, (C) for $\mu$ in I-4
Strategies $\alpha_1$ vs $\alpha_2$ for $G(\alpha_1, \alpha_2)>0$ (with $\xi>1$). (A) for $\mu$ in the region I-1, (B) for $\mu$ in the regions I-2 and I-3, (C) for $\mu$ in the region I-4
$\tilde{G}(\alpha_1, \alpha_2)>0$ for $p = 0.5$, $e = 0.2$, $K_1 = 1$, $\xi = 2$ ($K_2 = 2$), $\mu_1 = 0.022$ and $\mu_2 = 0.02$
$\tilde{G}(\alpha_1, \alpha_2)>0$ for $p = 0.5$, $e = 0.2$, $K_1 = 1$, $\xi = 2$ ($K_2 = 2$), $\mu_1 = 0.018$ and $\mu_2 = 0.02$
Bi-stability in the system (1) with $r = 1$, $K_1 = K_2 = 1$, $p_1 = p_2 = 0.5$, $\mu_1 = \mu_2 = 0.1333$, $e_{11} = 0.2$, $e_{12} = 0.05$, $e_{21} = 0.05$ and $e_{22} = 0.2$, $\alpha_1 = \frac{1}{3}$ and $\alpha_2 = \frac{2}{3}$. A solution either converges to $E^1_1$ or $E^2_2$ depending on the initial condition
Bi-stability in the system (1) with $r = 1$, $K_1 = K_2 = 1$, $p = 0.5$, $\mu = 0.025$, $e_{11} = 0.5$, $e_{12} = 0.1$, $e_{21} = 0.3$ and $e_{22} = 0.3$, $\alpha_1 = 0.4$ and $\alpha_2 = 0.6$. A solution either converges to $E^1_1$ or $E^2_3$ depending on the initial condition
Bi-stability in the system (1) with $r = 1$, $K_1 = K_2 = 1$, $p = 0.5$, $\mu = 0.05$, $e_{11} = 0.2$, $e_{12} = 0.1$, $e_{21} = 0.1$ and $e_{22} = 0.2$, $\alpha_1 = 0.4$ and $\alpha_2 = 0.6$. A solution either converges to $E^1_3$ or $E^2_3$ depending on the initial condition
Existence and stability of boundary equilibria for system (1). Cond-1: $\frac{\mu_2(\mathcal{R}^2_0-1)}{p_2e_2(\alpha_1\alpha_2K_1+\beta_1\beta_2K_2)}<\frac{\mu_1(\mathcal{R}^1_0-1)}{p_1e_1(\alpha_1^2K_1+\beta_1^2K_2)}$; Cond-2: $\frac{\mu_1(\mathcal{R}^1_0-1)}{p_1e_1(\alpha_1\alpha_2K_1+\beta_1\beta_2K_2)}<\frac{\mu_2(\mathcal{R}^2_0-1)}{p_2e_2(\alpha_2^2K_1+\beta_2^2K_2)}$; LS: locally stable
 Equilibrium Existence Stability Condition for stability $E^0_0, E^0_1, E^0_2$ always exists unstable $E^0_3$ always exists LS $\mathcal{R}^1_0<1, \mathcal{R}^2_0<1$ $E^1_1$ $\mathcal{R}^1_1>1$ LS $\hat{\mathcal{R}}^1_1>1$, $\mathcal{R}_1^2<\mathcal{R}_1^1$ $E^1_2$ $\mathcal{R}^1_2>1$ LS $\hat{\mathcal{R}}^1_2>1$, $\mathcal{R}_2^2<\mathcal{R}_2^1$ $E^1_3$ $\mathcal{R}^1_0>1$, $\hat{\mathcal{R}}^1_1<1$, $\hat{\mathcal{R}}^1_2<1$ LS Cond-1 $E^2_1$ $\mathcal{R}^2_1>1$ LS $\hat{\mathcal{R}}^2_1>1$, $\mathcal{R}_1^1<\mathcal{R}_1^2$ $E^2_2$ $\mathcal{R}^2_2>1$ LS $\hat{\mathcal{R}}^2_2>1$, $\mathcal{R}_2^1<\mathcal{R}_2^2$ $E^2_3$ $\mathcal{R}^2_0>1$, $\hat{\mathcal{R}}^2_1<1$, $\hat{\mathcal{R}}^2_2<1$ LS Cond-2 $E^*$ Theorem 3.8
 Equilibrium Existence Stability Condition for stability $E^0_0, E^0_1, E^0_2$ always exists unstable $E^0_3$ always exists LS $\mathcal{R}^1_0<1, \mathcal{R}^2_0<1$ $E^1_1$ $\mathcal{R}^1_1>1$ LS $\hat{\mathcal{R}}^1_1>1$, $\mathcal{R}_1^2<\mathcal{R}_1^1$ $E^1_2$ $\mathcal{R}^1_2>1$ LS $\hat{\mathcal{R}}^1_2>1$, $\mathcal{R}_2^2<\mathcal{R}_2^1$ $E^1_3$ $\mathcal{R}^1_0>1$, $\hat{\mathcal{R}}^1_1<1$, $\hat{\mathcal{R}}^1_2<1$ LS Cond-1 $E^2_1$ $\mathcal{R}^2_1>1$ LS $\hat{\mathcal{R}}^2_1>1$, $\mathcal{R}_1^1<\mathcal{R}_1^2$ $E^2_2$ $\mathcal{R}^2_2>1$ LS $\hat{\mathcal{R}}^2_2>1$, $\mathcal{R}_2^1<\mathcal{R}_2^2$ $E^2_3$ $\mathcal{R}^2_0>1$, $\hat{\mathcal{R}}^2_1<1$, $\hat{\mathcal{R}}^2_2<1$ LS Cond-2 $E^*$ Theorem 3.8
Conditions for $F(\alpha_1, \alpha_2)>0$ in variant values of $\mu$
 Value of $\mu$ I-1 I-2 I-3 I-4 Conditions $\left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_2\alpha_2\\ \alpha_2>b/a \end{array} \right.$ or $\left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_2>b/a \end{array} \right.$ $\left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_2  Value of$ \mu $I-1 I-2 I-3 I-4 Conditions$ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_2\alpha_2\\ \alpha_2>b/a \end{array} \right. $or$ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_2>b/a \end{array} \right.  \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_2
Conditions for $G(\alpha_1, \alpha_2)>0$ in variant values of $\mu$
 Value of $\mu$ I-1 I-2 I-3 I-4 Conditions $\left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_1\alpha_2\\ \alpha_1>b/a \end{array} \right.$ or $\left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_1>b/a \end{array} \right.$ $\left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_1  Value of$ \mu $I-1 I-2 I-3 I-4 Conditions$ \left\{ \begin{array}{l} \alpha_1>\alpha_2\\ \alpha_1\alpha_2\\ \alpha_1>b/a \end{array} \right. $or$ \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_1>b/a \end{array} \right.  \left\{ \begin{array}{l} \alpha_1<\alpha_2\\ \alpha_1
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