Stability and Bifurcation for a Single-Species Model with Delay Weak Kernel and Constant Rate Harvesting

Xiangrui Li,Shuibo Huang

doi:10.1155/2019/1810385

Abstract

In this paper, we consider the effect of constant rate harvesting on the dynamics of a single-species model with a delay weak kernel. By a simple transformation, the single-species model is transformed into a two-dimensional system. The existence and the stability of possible equilibria under different conditions are carried out by analysing the two-dimensional system. We show that there exists a critical harvesting value such that the population goes extinct in finite time if the constant rate harvesting u is greater than the critical value, and there exists a degenerate critical point or a saddle-node bifurcation when the constant rate harvesting u equals the critical value. When the constant rate harvesting u is less than the critical value, sufficient conditions about the existence of the Hopf bifurcation are derived by topological normal form for the Hopf bifurcation and calculating the first Lyapunov coefficient. The key results obtained in the present paper are illustrated using numerical simulations. These results indicate that it is important to select the appropriate constant rate harvesting u.

Highlights

Ecological population dynamics is an important research eld of mathematical biology. e single-species model is the cornerstone of research for mathematical biology
A single-species model with a delay weak kernel and a constant harvesting rate is established in this paper
We know that system (11) is unstable when constant rate harvesting u > Kr/4, that is, harvesting rates on population exceed the maximum sustained yield (MSY), which will reduce the population to extinction in finite time, since dx/dt < 0

Summary

Introduction

Ecological population dynamics is an important research eld of mathematical biology. e single-species model is the cornerstone of research for mathematical biology. Due to the size of the population at earlier times to determine the present effect on resource availability, motivated by the above work, introducing a constant rate harvesting u into the Volterra’s model [24], we consider a single-species model with a delay weak kernel and a constant rate harvesting: 1t rx(t)􏼠1 − 􏽚. Where x(t) represents the unit density of the species at time t, r > 0 represents the intrinsic growth rate of the species (reflecting the characteristics of the species itself ), K > 0 is carrying capacity, αe− α(t− s) is a common weak kernel function, and u stands for constant rate harvesting, here α and u are positive constants.

A Simple Analysis of System and Preliminaries

Degenerate Critical Point and SaddleNode Bifurcation

Hopf Bifurcation

Conclusions