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The present paper deals with a dynamic reaction model of a fishery. The dynamics of a fishery resource system in an aquatic environment consists of two zones: a free fishing zone and a reserve zone. To protect fish population from over exploitation, a control instrument tax is imposed. The existence of its steady states and their stability are studied. The optimal harvest policy is discussed next with the help of Pontryagin's maximum principle. Our theoretical results are confirmed by numerical simulation.

With the growing need of human for more food and energy, several resources have been increasingly exploited. It has caused wide public concern to protect the ecosystem. A scientific management of commercial exploitation of the biological resource like fisheries and forestry is necessitated. The first attempt of mathematical modelling of resource management problems was made in the article by Hotelling [

On the other hand, 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 of marine fisheries naturally involves the problems of law enforcement. Several governing instruments are suggested for the choice of a regulatory control variable. These are imposition of taxes and license fees, leasing of property rights, seasonal harvesting, direct control, and so forth. Various issues associated with the choice of an optimal governing instrument and its enforcement in fishery were discussed by Anderson and Lee [

In order to keep a sustainable fishing resource. We will take some actions in fishing areas to protect certain fish stocks by restricting the fishermen’s fishing action. Such restriction would be implemented in the form of taxation. Following [

From [

The aim of this paper is to find a proper taxation policy which would give the best possible benefit through harvesting to community while preventing the extinction of the fishing species. The structure of this paper is as follows. In the next section, we study the steady-state existence of positive equilibrium. In Section

We find the steady-states of (

From [

Please note that there may be many possibilities that the above equation has positive solutions. We assume that the following inequalities hold:

Knowing the value of

If inequalities (

Next, we will consider the existence of the positive equilibrium

From (

Thus we have following theorem.

If (

We first consider the local stability of equilibria. The variational matrix of the system (

At

In this cubic equation, one root is

The equilibrium

At

If inequalities (

To determine the local stability character of the interior equilibrium

If (

In this section, we will consider the global stability of the unique interior equilibrium of system (

If (

Define a Lyapunov function

Differentiating

The objective of the regulatory agency is to maximize the total discounted net revenues that the society drives due to the harvesting activity. Symbolically, this objective amounts to maximizing the present value

Our objective is to determine a tax policy

We apply Pontryagin’s maximum principle in Burghes and Graham [

Now the adjoint equations are

Substituting the value of

Now using the values of

The existence of an optimal equilibrium solution has been created, which satisfies the necessary conditions of the maximum principle. As stated by Clark [

From the above analysis carried out in this section, we observe the following.

From (

Considering the interior equilibrium, (

From (

This shows that an infinite discount rate results in the complete dissipation of economic revenue. For zero discount rate, it is indicated that the present value of continuous time stream gains its maximum value.

In this section, we use Matlab 7.0 to simulate a numerical example to illustrate our results.

Let

Then for the above values of the parameter, optimal tax becomes

Solution curves corresponding to the tax

Phase space trajectories corresponding to the optimal tax

In Figures

Variation of free fishing zone population with time for different tax levels.

Variation of reserve area population with time for different tax levels.

Variation of harvesting effort with time for different tax levels.

We take

From these results, it is clear that if the fishermen have to pay no tax, then they use a large amount of effort compared to the case when the fishermen have to pay the optimal tax. As a result, the steady-state values of the two species for the case of

The computer analyzed results for the time course display of the two species

By comparing Figures

The trend of population

In this paper, we study an optimal harvesting problem for fishery resource with prey dispersal in a two-patch environment: one is a free fishing zone and the other is a reserve zone, focusing attention on the use of taxation as an optimal governing instrument to control exploitation of the fishery. In Sections

This work was partially supported by the NNSF of China (10961018), the NSF of Gansu Province of China (1107RJZA088), the NSF for Distinguished Young Scholars of Gansu Province of China (2011GS03829), the Special Fund for the Basic Requirements in the Research of University of Gansu Province of China and the Development Program for HongLiu Distinguished Young Scholars in Lanzhou University of Technology.