Global Asymptotic Stability for Discrete Single Species Population Models

A. Bilgin,M. R. S. Kulenović

doi:10.1155/2017/5963594

A. Bilgin, M. R. S. Kulenović

Open Access

https://doi.org/10.1155/2017/5963594

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Abstract

We present some basic discrete models in populations dynamics of single species with several age classes. Starting with the basic Beverton-Holt model that describes the change of single species we discuss its basic properties such as a convergence of all solutions to the equilibrium, oscillation of solutions about the equilibrium solutions, Allee’s effect, and Jillson’s effect. We consider the effect of the constant and periodic immigration and emigration on the global properties of Beverton-Holt model. We also consider the effect of the periodic environment on the global properties of Beverton-Holt model.

Highlights

The following difference equation is known as Beverton-Holt model: xn+1
Where a > 0 is a rate of change and xn is the size of population at nth generation. It was introduced by Beverton and Holt in 1957 and depicts density dependent recruitment of a population with limited resources in which resources are not shared
In the special case of three-generation model we find the precise basins of attraction of all locally stable equilibrium solutions and locally stable period-two solutions

Summary

Introduction

Where a > 0 is a rate of change (growth or decay) and xn is the size of population at nth generation. Where a > 0 is a rate of change (growth or decay), h > 0 is a constant immigration, and xn is the size of population at nth generation. The following difference equation is known as BevertonHolt model with periodic immigration or stocking: hn,. Where a > 0 is a rate of change (growth or decay), hn > 0 is a periodic immigration or stocking, and xn is the size of population at nth generation. The following difference equation is known as the Beverton-Holt model with periodic environment: Kn μKnxn , + (μ − 1) xn (13). Where μ > 1 is a rate of change (growth or decay), Kn > 0 is a periodic sequence of period p modeling periodicity of environment (periodic supply of food, energy, etc.), and xn is the size of population at nth generation. In the special case of three-generation model we find the precise basins of attraction of all locally stable equilibrium solutions and locally stable period-two solutions

Preliminaries

Single Species Two-Generation Models

Local and Global Dynamics of Several Generation Models