Uniform Persistence, Periodicity and Extinction in a Delayed Biological System with Stage Structure and Density-Dependent Juvenile Birth Rate

Limin Zhang,Chaofeng Zhang

doi:10.4236/ajcm.2016.62014

Limin Zhang, Chaofeng Zhang

Open Access

https://doi.org/10.4236/ajcm.2016.62014

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Abstract

A delayed biological system of predator-prey interaction with stage structure and density dependent juvenile birth rate is investigated. It is assumed that the prey population has two stages: immature and mature. The growth of the immature prey is density dependent and is a function of the density of adult prey. Such phenomenon has been reported for beetles, tribolium, copepods, scorpions, several fish species and even crows. The growth of the predator is affected by the time delay due to gestation. By some Lemmas and methods of delay differential equation, the conditions for the uniform persistence and extinction of the system are obtained. Numerical simulations illustrate the feasibility of the main results and demonstrate that the density dependent coefficient has influence on the system populations’ densities though it has no effect on uniform persistence and extinction of the system.

Highlights

In the natural world, there are many species whose individual members have a life history that takes them through two stages: juvenile and adult
We investigate the extinction of the predator population in system (1.2) with initial conditions (1.3) under some condition
We propose a stage-structured predator-prey system with time delay and density-dependent juvewith the initial conditions (φ1 (θ ), φ2 (θ ),ψ (θ )) = (1,1,1) and β (t ) = 0.01, the other parameters given by equation (4.1), where t ∈[0,100] and θ ∈[−0.6,0]

Summary

Introduction

There are many species whose individual members have a life history that takes them through two stages: juvenile and adult. (2016) Uniform Persistence, Periodicity and Extinction in a Delayed Biological System with Stage Structure and Density-Dependent Juvenile Birth Rate. The past state as well as current conditions can influence biological dynamics and such interactions have motivated the introduction of time delay in stage-structured predator-prey systems [2]-[5] [8]-[13]. Juvenile population birth rate is a function of adults’ density and remains bounded when adults’ size is large due to limited resources [20]. This densitydependent regulator has been found in beetles, tribolium, copepods, scorpions, several fish species and even crows by Polis [21].

Uniform Persistence and Periodicity

Extinction

Examples

Conclusion