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Optimal harvesting control for an age-dependent competing population with diffusion
Boundary Value Problems volume 2014, Article number: 145 (2014)
This paper is mainly concerned with optimal harvesting policy for competing species with age dependence and diffusion. The existence and uniqueness of solutions for the system are proved by using the Banach fixed point theorem. The existence and uniqueness of optimal control are discussed by Ekeland’s variational principle. The maximum principle is obtained.
MSC: 35B10, 49K05, 65L12, 92B05.
The optimal harvesting control of age-structured and size-structured single species have been widely studied in the literature –. The control problems of multi-species have been investigated in the literature –. The objective function represents the total harvesting, respectively, in –. The diffusion factor has not been considered in –. It is well known that the profit due to harvesting is a quite important problem for optimal harvesting. In this paper, our purpose is to consider the optimal control of the profit functional for diffusion population. Specifically, we deal with the following optimal harvesting problem:
for all , where the corresponding state variable satisfies the state system
where represents the density of the th population. is the diffusion rate of the th population. We assume that the populations have the same life expectancy , . is the average morality of the th population, and describes the average fertility of the th population. represents the interaction coefficients (, ). () is a bounded domain with a smooth enough boundary , , , , , and is the outward unit normal. The function gives the initial density distribution of the population, and represents the harvesting effort function, which is the control variable in the model and satisfies
where , , . The integral
represents the profit due to harvesting.
Throughout this paper, we assume that:
(A1): , , where is a constant.
(A2): , a.e. in , , where , .
(A3): , , where is a positive constant (, ).
(A4): , , , , where is a constant, .
The rest of this paper is organized as follows. In Section 2, we prove that under the assumptions listed above, the system has a unique non-negative solution. In Section 3, the necessary conditions of optimality for the control problems is given. In the final section, we prove the existence and uniqueness of the optimal control.
2 The existence and uniqueness of solution for system (1.2)
For the sake of convenience, we introduce the following definitions of the solution.
Given the solution of system (1.2), the function , , which belongs to for almost any characteristic line of the equation
where , .
Then we rewrite the characteristic line as
We introduce the following notations:
For any given, system (1.2) has a unique non-negative solutionsuch that
, , , where is a positive constant;
is continuous in .
For any given , , we define
The given system
by Theorem 4.1.3 in , we know that the above system has a unique non-negative solution,
From the comparison principle of linear system , it follows that
where , and this is the solution of the following system:
For any , , let the corresponding state be (), . It follows from (1.2) that
Multiplying the th equation in system (2.3) by (), integrating on , and according to the Hölder inequality, we have
where . By using the Gronwall inequality, we have
where . Set
Define the mapping
Denote the norm on by
Define the following norm for :
Obviously the norm is equivalent to the norm . Using (2.5), we get
It is obvious that and is a contraction on , so there is a unique fixed point, which is the solution of system (1.2).
Since , let , we have , a.e. . In the following, we study the continuity of the solution of (1.2) for the control variable . Let , , . From (1.2), it follows that
In a similar manner as that in (2.4), we deduce that
Let , we have
Adding up (2.7) from to and using the Gronwall inequality, we get
where . Multiplying (2.8) by and integrating on , we have
Inequality (2.9) implies that is continuous with respect to . This completes the proof. □
3 The necessary conditions
Before stating our main results, we prove the following lemma, which is useful in proving our results.
Letbe an optimal pair for Problem (1.1), for anyand for any, , , where, then the following limit holds:
where, , andis the solution of
The existence and uniqueness of the solution of (3.1) follow in the same manner as by Theorem 2.1, the solution , and , where is a constant. We introduce the following notations:
It is straightforward that is the solution of
where , , .
By (2.9), we have
So we can claim that in , as . By (3.1) and (3.2), we see that is the solution of the following equation:
where , , .
Multiplying the first equation in (3.4) by , integrating on , using the Gronwall inequality, in the same manner as in (2.5), and taking in , we deduce
Consequently, the proof is completed. By the same argument as in (2.9), the following lemma holds. □
In order to give the necessary conditions, we introduce the adjoint equations of (3.1).
If, are the solutions of the following system corresponding to the control, , respectively:
where, , , , , are constant and independent of, , , .
We are now able to state the main result in this section.
Suppose that (A1)-(A5) hold, ifis an optimal pair for Problem (1.1), andis the solution of (3.5), which is corresponding to the control, then we have the following necessary conditions:
The existence and uniqueness of the solution to (3.5) can be proved by Theorem 2.1. For any and any , . Since is an optimal control for (1.1), we get
By Lemma 3.1 and inequality (3.3), passing to the limit as in (3.6), we conclude
Multiplying (3.5) by , then integrating over , we deduce that
Combining (3.7) with (3.8), we have
This completes the proof. □
4 The existence and uniqueness of optimal control
In this section, we prove the existence and uniqueness of optimal control. From , we give the following lemma.
Letbe a reflexive Banach space and letbe a lower semi-continuous convex function. Ifis a closed, convex, and bounded subset of, thenattains its infimum on. In other words, then there issuch that
The main conclusion is presented as follows:
Ifis small enough, then Problem (1.1) has a unique optimal control in.
For any , , , , , we define
we shall prove that is strictly monotone. Indeed, denote by and the state corresponding to controls and , respectively.
By (1.3), we have
where is the solution of Problem (3.1) corresponding to the control . By the same argument as given in (3.8), we have
Combining (4.1) and (4.2), we obtain
where is the solution of Problem (3.5) corresponding to the control ().
Given , and , and the norm being equivalent to the norm , combining Theorem 2.1 and Lemma 3.1, we have
where is small enough. If , then we have
Hence, is strictly monotone. Consequently, is strictly concave in .
Define the function :
It is clear that is concave in . It follows that the nonlinear function is upper weakly semi-continuous. Since is a closed, convex, and bounded subset in and is strictly convex in , by Lemma 4.1, attains the unique maximum in , which implies Problem (1.1) has a unique optimal control in . □
Brokate M: Pontryagin’s principle for control problem in age-dependent population dynamics. J. Math. Biol. 1985, 23: 75-101. 10.1007/BF00276559
Murphy LF, Smith SJ: Maximum sustainable yield of a nonlinear population model with continuous age structure. Math. Biosci. 1991, 104: 259-270. 10.1016/0025-5564(91)90064-P
Murphy LF, Smith SJ: Optimal harvesting of age-structured population. J. Math. Biol. 1990, 29: 77-90. 10.1007/BF00173910
Anita S, Iannelli M, Kim MY, Park EJ: Optimal harvesting for periodic age-dependent population dynamics. SIAM J. Appl. Math. 1998, 58: 1648-1666. 10.1137/S0036139996301180
Anita S: Optimal harvesting for a nonlinear age-dependent population dynamics. J. Math. Anal. Appl. 1998, 226: 6-22. 10.1006/jmaa.1998.6064
Ainseba B, Anita S, Langlais M: On the optimal harvesting of a nonlinear age-structured population dynamics. Abstr. Appl. Anal. 2001, 6: 357-368. 10.1155/S108533750100063X
Busoni G, Matucci S: A problem of optimal harvesting policy in two-stage age-dependent populations. Math. Biosci. 1997, 143: 1-33. 10.1016/S0025-5564(97)00011-4
Luo Z, Wang M: Optimal harvesting control problem for linear age-dependent population dynamics. Appl. Math. J. Chin. Univ. Ser. B 2003, 18: 412-420. 10.1007/s11766-003-0068-4
Gurtin ME, Murphy LF: On the optimal harvesting of age-structured populations: some simple models. Math. Biosci. 1981, 55: 115-136. 10.1016/0025-5564(81)90015-8
Gurtin ME, Murphy LF: On the optimal harvesting of persistent age-structured populations. J. Math. Biol. 1981, 13: 131-148. 10.1007/BF00275209
Liu Y, Cheng XL, He ZR: On the optimal harvesting of size-structured population dynamics. Appl. Math. J. Chin. Univ. 2013, 28(2):173-186. 10.1007/s11766-013-2965-5
Zhao C, Wang M, Zhao P: Optimal harvesting problems for age-dependent interacting species with diffusion. Appl. Math. Comput. 2005, 163: 117-129. 10.1016/j.amc.2004.01.030
Luo Z:Optimal harvesting problem for an age-dependent -dimensional food chain diffusion model. Appl. Math. Comput. 2007, 186: 1742-1752. 10.1016/j.amc.2006.08.168
Luo Z:Optimal harvesting control problem for an age-dependent competing system of species. Appl. Math. Comput. 2006, 183: 119-127. 10.1016/j.amc.2006.05.180
He ZR: Optimal harvesting of two competing species with age dependence. Nonlinear Anal., Real World Appl. 2006, 7: 769-788. 10.1016/j.nonrwa.2005.04.005
He ZR, Liu R, Liu LL: Optimal harvesting for populations system with age-dependent predator. Adv. Math. 2012, 42(5):691-700.
Luo Z, Xing T, Li X: Optimal birth control of free horizon problems for predator-prey system with age-structure. J. Appl. Math. Comput. 2010, 34: 19-37. 10.1007/s12190-009-0302-1
He ZR, Liu LL, Luo ZX: Global positive periodic solutions of age-dependent competing systems. Appl. Math. J. Chin. Univ. 2011, 26(1):38-46. 10.1007/s11766-011-2503-2
Luo Z, Fan X: Optimal control for an age-dependent competitive species model in a polluted environment. J. Appl. Math. Comput. 2014, 228: 91-101. 10.1016/j.amc.2013.11.069
Luo ZX, Yu XD, Ba ZG:Overtaking optimal control problem for an age-dependent competition system of species. Appl. Math. Comput. 2013, 218: 8561-8569. 10.1016/j.amc.2012.02.019
Anita S: Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic, Dordrecht; 2000.
The authors would like to thank the anonymous referees for their valuable comments on and suggestions regarding the original manuscript. The project is supported by Graduate Student Innovation of Jilin Normal University (201114) and the Department of Education of Jilin Province (2013445).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and approved the final version.
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Fu, J., Wu, X. & Zhu, H. Optimal harvesting control for an age-dependent competing population with diffusion. Bound Value Probl 2014, 145 (2014). https://doi.org/10.1186/s13661-014-0145-z
- optimal control
- Ekeland’s principle
- maximum principle