Dynamics of a new delayed stage-structured predator-prey model with impulsive diffusion and releasing

Jianjun Jiao,Limei Li,Shaohong Cai

doi:10.1186/s13662-016-1038-5

Jianjun Jiao, Limei Li + Show 1 more

Open Access

https://doi.org/10.1186/s13662-016-1038-5

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Abstract

In this work, we propose a new delayed stage-structured predator-prey model with impulsive diffusion and releasing. By the stroboscopic map of the discrete dynamical system, we obtain a prey-extinction boundary periodic solution. Furthermore, we prove that the prey-extinction boundary periodic solution is globally attractive. We also prove that the investigated system is permanent by the theory on the delay and impulsive differential equations. Our results indicate that time delay, impulsive diffusion, and impulsive releasing have influence to the dynamical behaviors of the investigated system. The results of this paper also provide a tactical basis for pest management.

Highlights

Many authors [ – ] and papers [, ] have studied the predator-prey, competitive, and cooperative models
If the population dynamics with the effects of spatial heterogeneity is modeled by a diffusion process, most previous papers focused on the population dynamical system modeled by the ordinary differential equations
By the comparison theorem for impulsive differential equations [ ], we know that there exists a sufficient small ε > and t (> t + τ ) such that the inequality zi(t) ≤ zi (t) + ε (i =, ) holds for t ≥ t, zi(t) ≤ [zi∗ e–(wi –kiβiy∗i )lτ + zi∗ ∗e–(wi +kiβiy∗i )( –l)τ ] + ε for all t ≥ t

Summary

Introduction

Many authors [ – ] and papers [ , ] have studied the predator-prey, competitive, and cooperative models. The stage structure of a species has been considered very little. Almost all animals have the stage structure of being immature and mature. [ – ] studied the stage structure of species with or without time delays. Aiello et al [ ] considered a time delayed stage structure of being immature and mature of the population model. If the population dynamics with the effects of spatial heterogeneity is modeled by a diffusion process, most previous papers focused on the population dynamical system modeled by the ordinary differential equations. Theories of impulsive differential equations [ , ] have been introduced into population dynamics. Impulsive differential equations are found in most domains of applied science [ , , , , – ]. We introduce the model and background concepts.

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Discussion