Dynamic Behaviors of a Single Species Stage Structure Model with Michaelis–Menten-TypeJuvenile Population Harvesting

Xiangqin Yu,Fengde Chen,Zhenliang Zhu

doi:10.3390/math8081281

Abstract

A single species stage structure model with Michaelis–Menten-type juvenile population harvesting is proposed and investigated. The existence and local stability of the model equilibria are studied. It shows that for the model, two cases of bistability may exist. Some conditions for the global asymptotic stability of the boundary equilibrium are derived by constructing some suitable Lyapunov functions. After that, based on the Bendixson–Dulac discriminant, we obtain the sufficient conditions for the global asymptotic stability of the internal equilibrium. Our study shows that nonlinear harvesting can make the dynamics of the system more complex than linear harvesting; for example, the system may admit the bistable stability property. Numeric simulations support our theoretical results.

Highlights

Up to now, no scholars have considered the dynamic behaviors of the stage structure model of single species with Michaelis–Menten-type juvenile population harvesting
Our results indicate that the Michaelis–Menten-type harvesting is more sensitive to the impact of model dynamics
The first type of bistability phenomenon shows that if b < c(e − a), ∆(b) < 0, the model is persistent when the initial value lies in the first quadrant

Summary

Introduction

It is well known that many species go through two stages, juvenile and adult, and that there are significant differences in the characteristics of species at different stages, so the stage structure model can better show the real-world phenomena. Yu, Zhu, Lai, et al [33] proposed the single species stage structure system with Michaelis–Menten-type adult species harvesting They obtained sufficient conditions for the global asymptotic stability of the unique boundary equilibrium and positive equilibrium of the model. Their results indicated that the system has bifurcation phenomena; within the proper harvesting range, the two species can coexist; over harvesting will lead to species extinction. Up to now, no scholars have considered the dynamic behaviors of the stage structure model of single species with Michaelis–Menten-type juvenile population harvesting. The paper ends with some numeric simulations and a brief discussion

Existence and The Types of Equilibria

The Boundary Equilibrium

The Internal Equilibria

Global Stability of Equilibria

Numerical Simulations

Conclusions