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This paper tries to highlight a delayed prey-predator model with Holling type III functional response and harvesting to predator species. In this context, we have discussed local stability of the equilibria, and the occurrence of Hopf bifurcation of the system is examined by considering the harvesting effort as bifurcation parameter along with the influences of harvesting effort of the system when time delay is zero. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by applying the normal form theory and the center manifold theorem. Lastly some numerical simulations are carried out to draw for the validity of the theoretical results.

Differential equation models for interactions between species are one of the classical applications of mathematics to biology, dating back to the first half of this century. The development and use of analytical techniques and the growth of computer power have progressively improved our understanding of these types of models.

The study of population dynamics with harvesting is a subject of mathematical bioeconomics, which in turn related to the optimal management of renewable resources, Clark [

Regulation of exploitation of biological resources has become a problem of major concern nowadays in view of the dwindling resource stocks and the deteriorating environment. Exploitation reduces the biomass of the concerned species, exhibits oscillation, and even causes extinction of some other species. Rosenweig-MacAurtho model experiences oscillation under selective effort of Kar and Ghosh [

It has been noticed over the years by many authors that time delays have come to play an important role in almost all branches of science, for example, ecology and biology. The time delay is considered in the population dynamics when the rate of change of the population is not only a function of the present population but also depends on the past population. Delay is frequently used in a predator-prey model to represent the biological process more accurately.

Wangersky and Cunningham [

The purpose of the work is to illustrate the combined effects of harvesting and delay on the dynamics of predator-prey system.

In this paper, we consider the following prey-predator system:

For

Our paper is organized in the following way: existence of equilibria of the system (

We now study the existence and nature of the steady states. Particularly we are interested in the interior equilibrium of the system. To begin with, we list all possible steady states of the system (

For the stability analysis of the equilibria, let the variational matrix of the system (

At

At

The characteristic equation of the variational matrix

Therefore, interior equilibrium

If

It is known that, for prey-predator systems, existence and stability of a limit cycle are related to the existence and stability of a positive equilibrium. If the limit cycles do not exist, in this case the equilibrium is globally asymptotically stable. On the other hand if the positive equilibrium exists and is unstable, there must occur at least one limit cycle.

Let us consider system (

Now we consider the following Theorem 4.2 of Kuang and Freedman [

Suppose for system (

Following Theorem

The characteristic equation (

Let the interior equilibrium of the system (

Also

To construct Figure

Variation of

We will investigate the dynamics of delay system (

Let

It is known that the steady state is asymptotically stable if all roots of the characteristic equation (

Equation (

We want to determine if the real part of some root increases to reach zero and eventually becomes positive as

From (

If

On the other hand, if

Substituting

Thus

If

For

This will signify that there exists at least one eigenvalue with positive real part for

From (

Therefore, the transversality condition holds, and hence, Hopf bifurcation occurs at

Let

If conditions

Without loss of generality, denote the critical values

Then system (

For

For

By the discussion in Section

By direct computation, we can obtain that

Using the same algorithms and similar computation process as in Hassard et al. [

Based on the discussion above, we can obtain the following results.

The sign of

As the problem is not a case study, the real-world data are not available for this model. We, therefore, take here some hypothetical data with the sole purpose of illustrating the results that we have established in the previous sections.

(i) Let us consider the parameters of the system (

The system is asymptotically stable for

Phase plane trajectories corresponding to different initial levels. The figure clearly indicates that the equilibrium

The system is unstable for

There is a stable limit cycle surrounding

(ii) For the values

Both the prey and predator populations converge to their equilibrium values.

The equilibrium

Phase plane trajectories corresponding to different initial levels. The figure clearly indicates that the equilibrium

Bifurcating periodic solutions from

Bifurcating periodic solutions from

This paper deals with a delayed prey-predator model with Holling type III functional response and harvesting of predator species. Oscillatory behavior and existence of limit cycles in harvested predator-prey system are common in nature. In the absence of delay, we have proved that exactly one stable limit cycle occurs when positive equilibrium is unstable. Since harvesting is associated with economic interests, our analysis shows that the harvesting of predator species plays an important role in the shaping of the dynamical behavior of the system. From Figure

Taking delay as bifurcation parameters, we see that a Hopf bifurcation occurs, whenever delay increases a critical value. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and center manifold theorem.

The authors declare that there is no conflict of interests regarding the publication of this paper.