A Stage-Structured Predator-Prey Model in a Patchy Environment

Xuejuan Lu,Shengqiang Liu,Yuming Chen

doi:10.1155/2020/6028019

Abstract

In this paper, we propose a stage-structured predator-prey model with migrations among patches in an n-patch environment. The net reproduction number for each patch in isolation is obtained along with the net reproduction number of the system of patches, ℛ0. Inequalities describing the relationship among these numbers are also given. Furthermore, threshold dynamics determined by ℛ0 is established: the predator dies out if ℛ0<1 while the predator persists if ℛ0>1. Focusing on the case with two patches, we obtain that the dispersal decreases the net reproduction number ℛ0. By numerical simulations, we find that the dispersal may be a good thing or a bad thing because the dispersal could make the predator population thrive or extinct, and hence we might seek steady state in the ecological environment by controlling parameters related to the prey and the predator.

Highlights

We focus on a model with prey and predator dispersal in an n-patch environment
In order to obtain a positive equilibrium for system (12), we assume that s􏼐diag􏼐ri − dNi 􏼑 + MN􏼑 > 0, (13)
We prove the global attractivity of the predator-free equilibria (PFE) as follows

Summary

The Model

We formulate a predator-prey model in n patches by taking into consideration diffusion among patches and a stage structure in the predator. There are prey individuals, juvenile predator individuals, and adult predator individuals, denoted by Ni, PJi , and PAi , respectively. We assume that the predation function fi(Ni) satisfies the following basic assumptions for Ni ∈ (0, ∞). (3) f′i(Ni) ≥ 0, i ∈ Nn. e following three types of predation functions f(N) in [21] satisfy the above assumptions. We assume that each component of A0 is nonnegative with the following initial conditions:. To show the existence of predator-free equilibria (PFE), we let PJi PAi 0, i ∈ Nn in equation (1) to get dNi Ni + 􏽘 mNij Nj j∈Nn (12). In order to obtain a positive equilibrium for system (12), we assume that s􏼐diag􏼐ri − dNi 􏼑 + MN􏼑 > 0,.

Threshold Dynamics

Persistence and Existence of Positive Equilibria

Simulations

Discussion