Discrete Optimal Control Method Based on the Optimal Strategy of Fishing

Considerationwas given to the discrete optimal controlmethod for the optimal fishing strategy. Ourmethod is new and efficient for discrete optimal control problem, which is different from the other optimal methods such as the traditional variational method, the Pontryagin principle of maximum, and the dynamic programming. The basic construction of the model is the traditional logistic function relating to the growth of fry. The discrete optimal control method for optimal fishing strategy was used to construct the optimal rate of each fishing strategy; the main focus of our work is on the rigorous mathematical analysis of the optimal control problem. The analysis allows one to obtain the optimal initial investment amount of the fry and the optimal size of the total catch. Furthermore, when the initial investment amount of the fry is below or above the optimal value and the intrinsic growth rate of fish R is too small, we derive that fishing operations should not be started in the last few years to make the overall fishing amount optimal. At last, several typical examples are given to illustrate the obtained results.


Introduction
Optimal control problem has been studied for many years, and in recent years, the optimal harvesting strategy of some species has been studied extensively by many authors [1][2][3].An optimal harvesting policy is given using Pontryagin's maximum principle by many authors [1,2].They consider the following fishery model: where  is biomass of fish population, () is the biological net growth rate,  is catch ability coefficient, and  is fishing effort.This model was originally developed by Schaefer [4] as a management tool for the Eastern Tropical Pacific Tuna Fishery.In the original Schaefer model, () was specified in "logistic" form where  and  are positive parameters called the "intrinsic growth rate" and the "carrying capacity, " respectively.
The term ℎ =  represents the rate of mortality imposed by the fishery, that is, the rate of catch, corresponding to a given input of fishing "effort" .Their objective is to determine max ℎ = . ( The study of predator-prey models with harvesting has also attracted the attention of researchers; see, for example, Brauer and Soudack [5][6][7], Beddington and Cooke [8], Dai and Tang [9], Hogarth et al. [10], Myerscough et al. [11], and Goh [12].
Extensive and unregulated harvesting of fishes can even lead to the dropping of the overall number of fishing.Some associated documents have shown that the optimal initial investment amount of the fry /2 is the optimal value to guarantee the optimal overall fishing amount [2], but they did not give the optimal rate of each fishing strategy and more important, however, is that when the initial investment amount of the fry is not /2, how can we decide the optimal fishing strategy ?
In this paper, we will give the discrete optimal control method of the optimal fishing strategy.This method is new and efficient for discrete optimal control problem, which is different from the other optimal methods such as the traditional variational method, the Pontryagin principle of maximum, and the discrete dynamic programming.The basic construction of the model is the traditional logistic function relating to the growth of fry.The discrete optimal control method for optimal fishing strategy was used to construct the optimal rate of each fishing strategy; the main focus of our work is on the rigorous mathematical analysis of the optimal control problem.The analysis allows one to obtain the optimal initial investment amount of the fry and the overall optimal fishing amount.Furthermore, when the initial investment amount of the fry is below or above the optimal value and the intrinsic growth rate of fish  is too small, we derive that fishing operations should not be started in the last few years to make the overall fishing amount optimal.However, the other optimal methods such as the traditional variational method require the controlled variable   not to be restricted; hence the variational method is not available.The discrete dynamic programming method is not convenient in the calculation for the logistic model is complex and not linear.The maximum principle in the calculation of the discrete control optimization is not convenient too.Hence, we pointed out that our method is new and efficient for discrete optimal control problem.Finally, several typical examples are given to illustrate the obtained results.
The aim of this paper is to undertake the mathematical analysis of the optimal control problem introduced above, namely, to maximize (7) subject to (4)- (6).We also provide some numerical results that illustrate the theoretical results by several typical examples.We remark that there are few optimal control studies in the literature for discrete optimal control problem.
An outline of the paper is as follows.In Section 2, discrete optimal control method of the optimal fishing strategy is given.This method is used to construct the optimal rate of each fishing strategy and then obtain the optimal initial investment amount of the fry and the overall optimal fishing amount.In Section 3, we discuss that when the initial investment amount of the fry is not the optimal value and the intrinsic growth rate of fish  is too small, fishing operations should not be started in the last few years to make the overall fishing amount optimal, and then, several typical examples are given to illustrate the obtained results.In Section 4, we give a detailed table to show some figures to illustrate the obtained results.Finally, in Section 5 we draw a conclusion.

Discrete Optimal Control Method to Construct the Optimal Fishing Rate
Let   denote the original amount of fry, which is the density of fish in the th years.Let be the natural growth rate of fish, where  > 0 is the intrinsic growth rate of fish,  > 0 is the maximum carrying capacity of environment, which is assumed to be fixed within three years, and  = 1, 2, 3 is the year under consideration.Let be the remaining amount of fish after  years, where   ∈ [0, 1] is the degree of each fishing strategy in the th years; choose the control set  ⊂   , defined as We define the objective functional where   (  ) represents the amount of fishing in the th years.Our goal is to maximize (  ), that is, to find a   such that where   satisfies conditions (4)-( 6).
Proof.The way we consider this problem is similar to the dynamic programming; to make the overall amount of fishing the optimal value, we let  * 3 = 1, which means that we fish all in the last year: It is easy to see that  3 is a quadratic function about  3 , when  * 3 = /2,  3 to be the maximum  * 3 = /4, we have completed the third fishing progress.
From condition (5), we have and then we can derive that and when  2 = /2, the inequality could be equality.Adjoining this inequality with (10) gives and this relation yields where  * 2 = /2; we have completed the second fishing progress.
Similarly, from condition (5), we have We can derive that Derive  1 ,   1 = (/2)/(1 −  1 ) 2 > 0, where  1 is an increasing function on  1 .Since then We are able to obtain that  1 ≤ 1 − 2/.When  * 1 = 1 − 2/, then where  * 1 = /2; we have completed the first fishing progress.To sum up, we can determine the optimal rate of each fishing strategy as and the overall amount of fishing to be the optimal value as The optimal initial investment amount of the fry is Corollary 2. Assume that  = ,  ≥ 2, and conditions ( 4)-( 6) hold; then there exists the optimal initial investment amount of the fry  * 1 = /2, and the optimal degree of each fishing strategy is making the overall amount of fishing the optimal value Note 1.In Theorem 1 we take  to be fixed, since the capacity of environment cannot change greatly in three years.When time is more than three years or when it has been a long time, we can consider that  is possible to be changed, and recounting is necessary.
Note 2. In front of the proof, we say that the way we consider this problem is similar to the discrete dynamic programming.Actually, this problem cannot be solved by other optimal methods such as the traditional variational method, the Pontryagin principle of maximum, and the dynamic programming.This is because we need a hypothesis in discrete optimal control problem: () and () are linearly independent, which can be assumed for discrete optimal control problem of free terminal, but for discrete optimal control problem of fixed terminal, this assumption is not established for the state equation ( + 1) = ((), (), ).
From Theorem 1 we can derive that the optimal initial investment amount of the fry is  * 1 = /2, and the optimal rate of each fishing strategy is  * 1 =  * 2 = 1 − 2/,  * 3 = 1, and then the overall amount of fishing is the optimal value  * = 3/4 − .However, in realistic application the initial investment amount of the fry may be above or below the optimal value  * 1 = /2 and  ≥ 2 may not hold; then how can we decide the optimal degree of each fishing strategy   to make the overall amount of fishing the optimal value?These questions are investigated below.
(1) When  ∈ (0,  1 ), one does not fish in the two previous kinds of fishing progress.
In every kind of fishing progress, one can determine the optimal fishing degree   and obtain the overall optimal amount of fishing .
Step 1.In the first fishing process, we divided the situation into two circumstances.
Step 2. In the second fishing process, we divided the situation into two circumstances too.
We get and then To sum up, when  ∈ [1/2, +∞), we have Step 3. In the third fishing process, we divided the situation into two circumstances.
(1) When  ∈ (0,  1 ) ( 1 < 1/2), we have It means that when  ∈ (0,  1 ) ( 1 < 1/2), we do not fish in the two previous kinds of fishing progress and fish all in the last fishing progress.This has no sense from a practical point of view.Therefore, we consider that not going in for fish farming is the better strategy.
(3) When  ∈ [1/2, +∞), we have It means that when  ∈ [1/2, +∞), we fish in each kind of progress.In every kind of fishing progress, we can determine the optimal fishing degree   and obtain the overall optimal amount of fishing , among which it is not difficult to verify Example 4. Consider Theorem 3  = 1/2 and  1 = ; determine the optimal strategy of fishing using the discrete optimal control method.
Step 1.In the first fishing process, we divided the situation into two circumstances.
(1) When  < 2, we have We get and then Step 2. In the second fishing process, we divided the situation into two circumstances too.
(1) When  ∈ (0, 2), we have It means that when  ∈ (0, 2), we do not fish in the two previous kinds of fishing progress.It is easy to see that It means that the maximum harvesting amount  * is less than the initial optimal investment amount  * 1 = /2 within three years; this has no sense from a practical point of view, let alone  1 ̸ = /2.Therefore, we consider that when the intrinsic growth rate of fish  < 2, not going in for fish farming is the better strategy.On the contrary, when  ≥ 2, we consider that setting  * 1 = /2 is the best policy decision and then determine   and   .
(2) When  ∈ [2, +∞), we have ( It means that when  ∈ [2, +∞), we fish in each kind of progress.In every kind of fishing progress, we can determine the optimal fishing degree   and obtain the overall optimal amount of fishing   .
Note.When  ∈ [2, +∞),  = 1/2, the result of Example 4 is just the very thing of Theorem 1.It means that Theorem 1 is a special case of Theorem 3.
Example 5. Consider Theorem 3  1 = ; we set  = 1/3 and determine the optimal strategy of fishing using the discrete optimal control method.
(1) When  ∈ (0, 2.02), we have It means that when  ∈ (0, 2.02), we do not fish in the two previous kinds of fishing progress.It is easy to see that where  * may be greater than /2 =  1 only when  ∈ [2, 2.02).It means that the maximum harvesting amount  * 1 +  * 2 is less than the initial optimal investment amount  1 = /2 within two years, and the maximum harvesting amount  * is less likely to be more than the initial optimal investment amount  1 = /2 within three years; this has no sense from a practical point of view.Therefore, we consider that when the intrinsic growth rate of fish  < 2.02, not going in for fish farming is the better strategy.
(2) When  ∈ [2.02, 2.25), we have It means that when  ∈ [2.02, 2.25), we do not fish in the first fishing progress.It is easy to see that the maximum harvesting amount  * 1 is less than the initial optimal investment amount  1 = /2 within one year; this has no sense from a practical point of view.
(3) When  ∈ [2.25, +∞), we have It means that when  ∈ [2.25, +∞), we fish in each kind of progress.In every kind of fishing progress, we can determine the optimal fishing degree   and obtain the overall optimal amount of fishing   .Corollary 7. Some people mistakenly think that "more is better" when they give the initial investment amount, but in fact this is not the case.Only when  1 = /2 can one obtain the optimal overall fishing amount.

Discussion
From Sections 2 and 3 and Examples 4 and 5, we set  = 3 and obtain the following conclusions.
When the initial investment amount of fry  1 = /2,  ≥ 2, we can determine   ( = 1, 2, 3) to make the overall fishing amount optimal.It is easy to see that when  < 2, we cannot guarantee /4 ≥ /2.It means that the maximum harvesting amount   (  ) = /4 is less than the initial optimal investment amount /2, which has no sense from a practical point of view, let alone  1 ̸ = /2.Therefore, we consider that when the intrinsic growth rate of fish  < 2, not going in for fish farming is the better strategy.On the contrary, when  ≥ 2, setting  * 1 = /2 is the best policy decision.Assume that  1 ̸ = /2; if  is large enough, we only have  1 <  * 1 , and the other   is possible to reach  *  ( = 2, 3); we do not cause great losses except for the first year.
In Section 3, we set  = 3 and obtain a series of conclusions.If  > 3, we can also gain some similar conclusions.When the initial investment amount of the fry is below or above the optimal value and the intrinsic growth rate of fish  is too small, we derive that fishing operations should not be started in the last few years to make the overall fishing amount optimal.To illustrate our conclusions clearly, we list the optimal total fishing amount  as follows.
From Table 1 we see that the overall optimal amount  of fishing is different when the initial investment amount of fry  1 is different; only when initial investment amount of fry is  1 = /2, is the overall fishing amount optimal.On the other hand, in every kind of fishing progress, we can determine the optimal fishing degree   and obtain the overall optimal amount of fishing .It is easy to see that when  <  1 , we do not fish in the two previous kinds of fishing progress; when  ∈ [ 1 , 1/2) ( 1 < 1/2), we do not fish in the first fishing progress; when  ∈ [1/2, +∞), we fish in each kind of progress.

Conclusion
From the rigorous mathematical analysis of the optimal control problem, we obtain the optimal initial investment amount of the fry and the optimal size of the total catch.When the initial investment amount of fry attains the optimal value  1 = /2, we can determine   ( = 1, 2, 3) to make the overall fishing amount optimal.Furthermore, when the initial investment amount of the fry is below or above the optimal value  1 = /2 and the intrinsic growth rate of fish  is too small, we derive that fishing operations should not be started in the last few years to make the overall fishing amount optimal.