Non-linear harvesting of prey with dynamically varying effort in a modified Leslie-Gower predator-prey system

Sunita Gakkhar ,Reenu Rani

doi:10.28919/cmbn/2498

Abstract

In this paper, a three dimensional dynamical system incorporating non- linear harvesting effort for prey is investigated. The Holling type-II functional response is considered for prey while predator is assumed to follow Modified Leslie-Gower type dynamics. The steady states of the system are obtained and the local dynamics is explored. The sufficient condition is derived for global stability of its positive interior equilibrium point. The conditions for bionomic equilibrium and uniform persistence of the system have been investigated. It is also observed that the system exhibits transcritical bifurcation for a threshold level of taxation. A taxation policy is discussed with the help of Pontryagin's Maximum Principle as an effective control instrument to preserve the prey species from extinction and maintain a sustainable fishery. https://doi.org/10.28919/cmbn/2498

Highlights

This paper is concerned with the study of a Modified Leslie- Gower type predator in a predator- prey system with nonlinear harvesting of prey population
The harvesting effort is taken as a dynamic variable and taxation as a control instrument
The conditions for existence of steady states and their stability behavior have been examined by using Eigen Value Method, Routh- Hurwitz criteria and Lyapunov method

Summary

Introduction

The classical ecological non-linear models of interacting population have been discussed extensively by many authors. Dubey analyzed a non-linear mathematical model to study a resource dependent fishery model with optimal harvesting policy by considering taxation as a control instrument[5]. They proved that the fishery resources can be protected from over-exploitation by increasing the tax and discounted rate. The present paper deals with a dynamic reaction model in the case of a predator- prey type fishery system, while the model considered here, is especially based on a modified version of the Leslie-Gower scheme, where only the prey species is subjected to non- linear harvesting. The main aim of this paper is to find the proper taxation policy which gives the best possible benefit through harvesting

The Mathematical Model And Its Qualitative Analysis

Persistence

Bionomic Equilibrium and Optimal Harvesting Policy

Numerical Simulations

Effort’E’

Conclusion