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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.

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 [

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. [

This article is organized as follows. In Section

In general, the Leslie-Gower predator-prey model [

We assumed that only predator species is economically important and the nonlinear harvesting rate is considered. Model (

Model (

The equilibrium points of system (

The quadratic equation (

System (

four equilibrium points, trivial equilibrium point

three equilibrium points, trivial equilibrium point

three equilibrium points, trivial equilibrium point

three equilibrium points, trivial equilibrium point

two equilibrium points, trivial equilibrium point

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

This diagram shows the number and location of the positive interior equilibrium points of system (

Now, we shall discuss the stability of the equilibrium points. The Jacobian matrix of system (

The trivial equilibrium point

The Jacobian matrix of system (

(a) The equilibrium point

(b) The equilibrium point

(c) The equilibrium point

(d) The equilibrium point

(e) The equilibrium point

(f) The equilibrium point

(a) The eigenvalues of the Jacobian matrix

(b) The determinant

(c) The determinant

(d) The determinant

(e) The determinant

(f) The determinant

From Lemma

The equilibrium points

From Lemma

After simplification, we obtain

In Theorem

The equilibrium point

a saddle-node whenever

a cusp of codimension

First, we shall shift the equilibrium point

In Section

In Theorem

System (

the equilibrium point

system (

A critical magnitude of the bifurcation parameter

Let

From the discussion above, the equilibrium point

In Section

System (

It has been shown that if

In Theorem

System (

We consider that the parameters

Making the affine transformations

Consider the

Consider the

Applying the Malgrange Preparation theorem [

The sign of

Applying the parameter dependent affine transformations

By means of

The system above is topologically equivalent to the normal form of the Bogdanov-Takens bifurcation which is given by

saddle-node bifurcation curve:

Hopf bifurcation curve:

Homoclinic bifurcation curve:

In this section, we will present the numerical simulation results which will support our analytical findings.

(a)

For ((a) and (b))

(b)

(a)

(c)

(d)

Translating the equilibrium point

Making the affine transformations

Performing the

Applying the Malgrange Preparation theorem, we have

We have

Applying the parameter dependent affine transformations

Consider the

The local representations of the bifurcation curves are as follows:

In this paper, a Leslie-Gower predator-prey model has been analyzed in the presence of nonlinear predator harvesting. It is shown that the system has at most four equilibrium points in

It is shown that the system can have zero, one, or two interior equilibrium points through saddle-node bifurcation as the bifurcation parameter

The dynamical analysis of model (

The authors declare that they have no competing interests.