International Journal of Engineering Mathematics

Volume 2016 (2016), Article ID 2741891, 15 pages

http://dx.doi.org/10.1155/2016/2741891

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

^{1}Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow 226025, India^{2}Department of Mathematics, Faculty of Sciences, Banaras Hindu University, Varanasi 221605, India

Received 8 June 2016; Accepted 22 August 2016

Academic Editor: Krishnan Balachandran

Copyright © 2016 Manoj Kumar Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

Marine life is a renewable natural resource that not only provides food to a large population of humans but also 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. Thus, it is imperative to design harvesting strategies which aim at maximizing economic gains giving due consideration to the ecological health of the concerned Earth’s ecological system. Mathematical modeling in harvesting of species was started by Clark [1, 2]. There are mainly three types of harvesting according to Gupta et al. [3]: (i) , constant rate harvesting (see [4–7]), (ii) , proportionate harvesting (see [8, 9]), and (iii) (Holling type II), nonlinear harvesting (see [10–13]). Nonlinear harvesting is more realistic and exhibits saturation effects with respect to both the stock abundance and the effort level. Notice that as and as .

The objective of the present work is to study dynamical behaviors of a Leslie-Gower predator-prey model in the presence of nonlinear harvesting in predators depending on parameters of the model. There have been many papers on the Leslie-Gower predator-prey system with harvesting. For example, May et al. [14] proposed a Leslie-Gower predator-prey model in which two different kinds of constant-yield harvesting applied on both prey and predator species have been considered and this model was studied by Beddington and Cooke [15]. Beddington and May [16] proposed and studied Leslie-Gower predator-prey model when both the prey and predators were harvested with constant effort. Beddington and Cooke [15] studied a Leslie-Gower predator-prey model in which the preys are harvested at a constant-yield rate and predators are harvested with constant-effort rate. Zhu and Lan [17] studied a Leslie-Gower predator-prey model with constant-yield prey harvesting. Gong and Huang [18] studied the Bogdanov-Takens bifurcation for the model and showed that for different parameter values the model has a limit cycle or a homoclinic loop. Gupta et al. [3] discussed the bifurcation analysis of a Leslie-Gower prey-predator model in the presence of nonlinear harvesting in prey. Huang et al. [19] studied the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type model and showed that the model has Bogdanov-Takens (BT) singularity of codimension or a weak focus of multiplicity two for some parameter values. They have shown that as the parameters change, the model exhibits saddle-node bifurcation, repelling and attracting B-T bifurcations, and supercritical, subcritical, and degenerate Hopf bifurcations.

This article is organized as follows. In Section 2, we describe the mathematical model in detail. In Section 3, we obtain the number and location of equilibrium points and the local and global stability of the equilibria are investigated. In Section 4, the existence of saddle-node, Hopf, and Bogdanov-Takens bifurcations is shown. In Section 5, we present several numerical simulations that support our theoretical results. Finally, we present some ramification of our results in Section 6.

#### 2. Model Equations

##### 2.1. Leslie-Gower Model

In general, the Leslie-Gower predator-prey model [20] is given as follows:where and are the prey density and predator density at time , respectively; , , , , and are positive parameters and represent the intrinsic growth rate of prey, intrinsic growth rate of predator, carrying capacity of prey in the absence of predator, maximal predator per capita consumption rate, and measure of the food quality that the prey provides for conversion into predator births. Model (1) has been studied by Hsu and Huang [21]. They showed that the unique positive interior equilibrium point of system (1) is globally asymptotically stable under all biologically admissible parameters.

##### 2.2. Harvested Model

We assumed that only predator species is economically important and the nonlinear harvesting rate is considered. Model (1) reduces to the following:where the parameters and are positive and represent catchability coefficient and effort applied to harvest the individuals, respectively, and are suitable positive constants.

Model (2) is not well defined at . In order to define system (2) at , we improve the model as given in [22]; system (2) reduces toFor nondimensionalizing system (3), we introduce the following substitutions:System (3) reduces towhere , , , and . For the existence of the biological meaning of the variables in model, system (5) is studied in the closed first quadrant, , in -plane defined by .

#### 3. Equilibria and Their Local Stability

The equilibrium points of system (5) are the nonnegative real solutions of the system . It is obvious that system (5) has the trivial equilibrium point and the axial equilibrium point and the abscissa of the positive interior equilibrium points are the roots of the quadratic equation:and the ordinance of the positive interior equilibrium points is given by , .

The quadratic equation (6) has two positive roots and whenever (say), a double positive root whenever , and no positive root whenever . The number and location of the equilibrium points of system (5) lying in the set are given in the following lemma.

Lemma 1. *System (5) has *(a)*four equilibrium points, trivial equilibrium point , axial equilibrium point , and two interior equilibrium points and whenever , where , , , , and ;*(b)*three equilibrium points, trivial equilibrium point , axial equilibrium point , and a double interior equilibrium point whenever , where and ;*(c)*three equilibrium points, trivial equilibrium point , axial equilibrium point , and an interior equilibrium point whenever and , where and ;*(d)*three equilibrium points, trivial equilibrium point , axial equilibrium point , and an interior equilibrium point whenever , where and ;*(e)*two equilibrium points, trivial equilibrium point , and axial equilibrium point whenever and .*

*The number and location of interior equilibrium points have been depicted in Figure 1.*