Qualitative Analysis of a Leslie-Gower Predator-Prey System with Nonlinear Harvesting in Predator

Manoj Kumar Singh,Brajesh Kumar Singh,B S Bhadauria

doi:10.1155/2016/2741891

Abstract

This paper deals with the study of the stability and the bifurcation analysis of a Leslie-Gower predator-prey model with Michaelis-Menten type predator harvesting. It is shown that the proposed model exhibits the bistability for certain parametric conditions. Dulac’s criterion has been adopted to obtain the sufficient conditions for the global stability of the model. Moreover, the model exhibits different kinds of bifurcations (e.g., the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcations, Bogdanov-Takens bifurcation, and the homoclinic bifurcation) whenever the values of parameters of the model vary. The analytical findings and numerical simulations reveal far richer and complex dynamics in comparison to the models with no harvesting and with constant-yield predator harvesting.

Highlights

Marine life is a renewable natural resource that provides food to a large population of humans and is involved in the regulation of the Earth’s ecosystem
In Theorem 5, we have proved that the interior equilibrium point E3 is a cusp of codimension 2 whenever((x3 −2βy3 −βc)/(c+y3))− x3 = 0 and η4η5 ≠ 0, which implies that there may exist the Bogdanov-Takens bifurcation in system (5)
A Leslie-Gower predator-prey model has been analyzed in the presence of nonlinear predator harvesting

Summary

Introduction

Marine life is a renewable natural resource that provides food to a large population of humans and is involved in the regulation of the Earth’s ecosystem. The growing human needs for more food and more energy have led to increased exploitation of these resources which affects the Earth’s ecosystem. There are mainly three types of harvesting according to Gupta et al [3]: (i) h(x) = h, constant rate harvesting (see [4,5,6,7]), (ii) h(x) = qEx, proportionate harvesting (see [8, 9]), and (iii) h(x) = qEx/(m1E+m2x) (Holling type II), nonlinear harvesting (see [10,11,12,13]). Nonlinear harvesting is more realistic and exhibits saturation effects with respect to both the stock abundance and the effort level. Notice that h(x) → qE/m2 as x → ∞ and h(x) → qE/m1 as E → ∞

Objectives

Results

Conclusion