Time-delay effect on a diffusive predator\u2013prey model with habitat complexity

Yanfeng Li,Haicheng Liu,Ruizhi Yang

doi:10.1186/s13662-021-03473-y

Abstract

Based on the predator–prey system with a Holling type functional response function, a diffusive predator–prey system with digest delay and habitat complexity is proposed. Firstly, the stability of the equilibrium of diffusion system without delay is studied. Secondly, under the Neumann boundary conditions, taking time delay as the bifurcation parameter, by analyzing the eigenvalues of linearized operator of the system and using the normal form theory and center manifold method of partial functional differential equations, the effect of time delay on the stability of the system is studied and the conditions under which Hopf bifurcation occurs are given. In addition, the calculation formulas of the bifurcation direction and the stability of bifurcating periodic solutions are derived. Finally, the accuracy of theoretical analysis results is verified by numerical simulations and the biological explanation is given for the analysis results.

Highlights

1.1 Development of the population model In the ecosystem, the functional response function can reflect the optimal feeding behavior of the predator with the maximum energy intake per unit time in order to achieve the maximum growth capacity of the population
Scholars have intensively studied on predator–prey models with Beddington– DeAngelis functional response function [1], a ratio-dependent functional response function [2], the Ivelev functional response function [3] and the Crowley–Martin functional response function [4]
More and more biological effects are applied to the predator–prey systems when studying the stability of the equilibrium, such as Allee effect [5], prey refuge effect [6,7,8,9,10,11], habitat complexity effect [12,13,14], and harvesting effect [28]

Summary

Introduction

1.1 Development of the population model In the ecosystem, the functional response function can reflect the optimal feeding behavior of the predator with the maximum energy intake per unit time in order to achieve the maximum growth capacity of the population. The biological significance is expressed as follows: r is the intrinsic growth rate of prey population; K is the maximum environmental capacity of prey population; c(1–β )xα 1+ch(1–β )xα represents Holling type functional response function, in which, α ≥ 1 and represents a kind of aggregation efficiency, when α = 1, it becomes Holling II type functional response, when α = 2, it becomes Holling III type functional response; c represents the attack rate of the predator on the prey; h is the handing time; e (0 < e < 1) is the conversion efficiency; d represents per capita death rate of predators; β (0 < β < 1) represents the intensity of the habitat complexity effect. Considering that the habitat is heterogeneous, we introduce the diffusion term in the model (1.1), and obtain the reaction–diffusion system with homogeneous Neumann boundary conditions, the model is as follows:.

Stability of boundary equilibria

Existence of Hopf bifurcation induced by time delay

Direction and periodic solutions of Hopf bifurcation