#### Abstract

We study an optimal harvesting for a nonlinear age-spatial-structured population dynamic model, where the dynamic system contains an external mortality rate depending on the total population size. The total mortality consists of two types: the natural, and external mortality and the external mortality reflects the effects of external environmental causes. We prove the existence and uniqueness of solutions for the population dynamic model. We also derive a sufficient condition for optimal harvesting and some necessary conditions for optimality in an optimal control problem relating to the population dynamic model. The results may be applied to an optimal harvesting for some realistic biological models.

#### 1. Introduction

Optimal control problems for the age-structured systems are of interest for many areas of application, as harvesting, cost control, birth control, and epidemic disease control [1–5]. Many authors studied some optimal harvesting problems for an age-dependent population dynamic system ([2, 6–8] and references therein). One of the aims of such optimal control problems is to find some conditions of optimality for some objective functionals.

Aniţa [2] considered the optimal harvesting problem for the following nonlinear age-dependent population dynamic model. Let be the population density of age at time , and let and be the natural death rate of individuals of age and the harvesting rate, respectively. The evolution of an age-structured population subject to harvesting is described as a partial differential equation: where and stand for the total population and the external mortality rate, respectively, is the natural fertility-rate, and is the maximal age of the individual.

Now let us make the previous model a generalized model considering the location and the external environmental cause. Let be a bounded domain in with a smooth boundary , , . We denote by the distribution of individuals of age at time and location in . Let the natural fertility rate and be the natural death rate of individuals of age at time and location and density , where we note that the death rate depends on the density but most of study for models of the age structured population has been done with the death rate dependnt on the time and age only [2, 4]. For a more realistic situation, it is natural to assume that the death rate depends on the density as well as the time and age. Moreover, we set that the total mortality consists of a natural mortality and an external mortality and that the total population has a special weight at age and location : where is the maximal age of the individual. In this total population , the weight function gives the effects differently on each age under the external environmental causes: the virus, the climate change, the earthquake, and storm waves. So the external mortality rate reflects the long-term or short-term effects of external environments such as the virus, the climate change, and the earthquake. We also assume that the flux of population, as emigration, takes the form with , where is the gradient vector with respect to the location variable .

Now we consider the following nonlinear age-spatial-structured population dynamics model with external mortality: where is a bounded domain in with smooth boundary , is a harvesting rate, and is a positive function in .

We study an optimal control problem relating to the dynamic system (1.3) as follows: where , is a given bounded weight function is the solution of the dynamic control system (1.3), and is the set of controllers given by for some , , a.e., in . This problem is called the primal problem. The objective functional in represents the profit from harvesting, that is, the profit term is the proportion of the species harvested multiplied by the selling price dependent on age at time and location . In a biological system, we may apply the dynamic system (1.3) to the fish, animal, and plant dynamic models.

The purpose of this paper is to prove the existence and compactness of solutions for the dynamic system (1.3) and to investigate an optimal harvesting problem for a nonlinear age-spatial-structured population dynamic model with external mortality. The optimal approach introduced in this work may be applicable in the realistic biological models with field data beyond the theoretical model.

The paper is organized as follows. In Section 2, we obtain the existence, uniqueness, and compactness of solutions for the dynamic system (1.3). In Section 3, we derive a sufficiently condition for the optimal control problem . Finally, a necessary condition for the optimal control problem is given in Section 4.

#### 2. Existence, Uniqueness, and Compactness of Solutions

In this work, we assume the following: The fertility rate , a.e., . The mortality rate and is increasing and Lipschitz continuous with respect to the variable . is bounded and Lipschitz continuous, that is, there exists a constant such that and is continuously differentiable. a.e., . a.e., . is a nonnegative bounded and measurable function in with for all .

The existence of a solution to the dynamic system (1.3) is given by the following lemma (see also [1]). Here we assume that a function belongs to , for almost any characteristic line ; , . In addition, we assume that or , which may be a natural biological condition for population dynamics.

Lemma 2.1. * Let the assumptions – hold. For any , the dynamic system (1.3) admits a unique and nonnegative solution which belongs to . *

*Proof. * We will use the Banach fixed-point theorem for proof. Let . Denote by the mapping , where is the solution of
Then, the mapping is well defined form to (see Lemma 2 of [9]). For any , we denote , with and . By definition of , we get the following equation:
where . Using the conditions and , we get after some calculations that
where is a positive constant. For sufficiently small , we get the existence of a unique fixed point for . Since the solution satisfies
and is the solution of the dynamic system (1.3) corresponding to , we complete the proof.

For , denote

Lemma 2.2. * The set is relatively compact in . *

*Proof. *For any small enough, we get that
is a solution of
Using the condition , we obtain
where we have used the fact that and are bounded in , is bounded in and is bounded in .

Therefore, is bounded in . By Aubin's compactness theorem that for any , the set is relatively compact in . On the other hand, we get also

Combining these two results, we conclude the relative compactness of in .

#### 3. Existence of the Optimal Solution

Now, we show the existence of the optimal solution for the primal problem .

Theorem 3.1. * Let the assumptions – hold. Then, the primal problem has at least one optimal pair. *

*Proof. * Let . Then, we have
where is the solution of the dynamic system (1.3) corresponding to and . Now let be a sequence such that
Since a.e., in , we conclude that there exists a subsequence such that
For a strong convergence to , we consider the sequence such that
where is an increasing sequence of integer numbers.

Let the totality , and we assume that . For any , suppose that for every . Then, the set is a convex neighborhood of of and for all .

Let be the Minkowski functional of . Note here that if we choose with and , then we get .

Consider a real linear subspace and put for . This real linear functional on satisfies on . Thus, by the Hahn-Banach extension theorem, there exists a real linear extension of defined on the real linear space such that on . is a neighborhood of , the Minkowski functional is continuous in . Hence, is a continuous real linear functional defined on the real linear normed space . Moreover, we have
This is contradiction to . Therefore, converges strongly to in . Consider now the sequence of controls:
This control is an element of the set . So we can take a subsequence, also denoted by such that
By Lemma 2.2, we obtain
and since weakly in , we get
Obviously, is a solution of
By conditions of and , we get
where and is the Lipschitz constant. Therefore, we have
By and , we obtain
Since in , we have
Passing to the limit in (3.10), we obtain that is the solution of the dynamic system (1.3) corresponding to . Therefore, we have

#### 4. Necessary Conditions for Optimality

In this section, we study a necessary condition of optimality for the primal problem .

Theorem 4.1. * Let the assumptions – hold. Suppose that is an optimal pair for the primal problem . If is the solution of
**
then we have
**
Here, are the given functions in the control set which is introduced in the introduction, is the derivative of with respect to , and is the derivative of with respect to . *

*Proof. *Since is an optimal pair for the primal problem we get
for all and for all such that
Let and be the solution of the dynamic system (1.3) corresponding to . Then the above equality implies
Let be the solution to
Since in as , after some simple calculations and passing to the limit in (4.9) then we obtain
for all and for all such that
Multiplying (4.1) by and integrating over , we get
for all and for all such that

By inequality (4.13), we get , where is the normal cone at in . Therefore, if , then
and if , then is any arbitrary value belonging to the interval . This completes the proof of Theorem 4.1.

From now on, let and be a bounded domain in with a smooth boundary . We consider an optimal control problem: find subject to where .

We note that this is a special case of the optimal control problem by the dynamic system (1.3) (). Then, we obtain the necessary condition for optimality.

Theorem 4.2. * Let the assumptions , and hold. Suppose that is an optimal pair for the problem . If is a solution of the adjoint system :
**
then we have
*

*Remark 4.3. *We may consider natural death rates
as examples which satisfy our hypotheses. Here is the increasing function of . It is natural to assume that the mortality rate depends on density of individuals as well as the total population. Also, we can consider the weight function which gives an effect on age as follows:

*Remark 4.4. *Let a functional and the set of controllers be defined by
respectively.

We introduce another control problem corresponding to the primal problem , which is called the dual problem:
subject to the adjoint system (4.17).

Then, we can establish a duality theorem saying that the primal problem is equal to the dual problem , which is the result in [10].

#### Acknowledgement

This work was supported by a 2-Year Research Grant of Pusan National University.