Dynamical Analysis for the Hybrid Network Model of Delayed Predator-Prey Gompertz Systems with Impulsive Diffusion between Two Patches

Li Xiang,Dan Zhang,Zhichun Yang,Lingzhi Huang,Yurong Zhang,Jinliang Wang

doi:10.1155/2020/5041761

Abstract

In this paper, we consider a hybrid network model of delayed predator-prey Gompertz systems with impulsive diffusion between two patches, in which the patches represent nodes of the network such that the prey population interacts locally in each patch and diffusion occurs along the edges connecting the nodes. Using the discrete dynamical system determined by the stroboscopic map which has a globally stable positive fixed point, we obtain the global attractive condition of predator-extinction periodic solution for the network system. Furthermore, by employing the theory of delay functional and impulsive differential equation, we obtain sufficient condition with time delay for the permanence of the network.

Highlights

Along with the continuous development of the network science, the mathematical models organized as networks have received considerable attention [1]–[3]
The dynamical behaviors of the predator-prey model defined on the network have enjoyed remarkable progress [4,5,6,7,8]
In [7], Chang studied instability induced by time delay for a predator-prey model on complex networks and instability conditions were obtained via linear stability analysis of network organized systems

Summary

Introduction

Along with the continuous development of the network science, the mathematical models organized as networks have received considerable attention [1]–[3]. In [7], Chang studied instability induced by time delay for a predator-prey model on complex networks and instability conditions were obtained via linear stability analysis of network organized systems. Many stage-structured predator-prey models with time delay and impulsive diffusive were investigated [25,26,27,28]. Jiao et al [26] and Dhar and Jatav [27] investigated a delayed predator-prey model with impulsive diffusion and sufficient conditions of the global attractiveness of the predator-extinction periodic solution and the permanence were derived. Motivated by the above discussion, in this paper, we shall organize the patches into networks to investigate a delayed stage-structured functional response predator-prey Gomportz model with impulsive diffusion between two predators territories.

Hybrid Network Model-Organized PredatorPrey System

Extinction of the Predator

Permanence

Illustrative Example and Discussion