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1 Introduction

Ecological researches show that the outcomes resulting from interactions between species can vary with context-dependent factors (see [6]). The outcome of the interaction is that the density of each species may end up above, equal to, or below its carrying capacity in isolation from the other species (see [29]). Thus, the interaction outcomes of a two species system are classified into positive , neutral or negative effects (see [11]). Wang and Wu [30] pointed that the interaction outcomes between two species are not fixed but vary with biotic and/or abiotic factors. Moreover, Wang and DeAngelis [29] divided the outcomes between two species into six forms: mutualism , commensalism , predation/parasitism , amensalism , competition and neutralism .

Holland and DeAngelis [11] established the consumer–resource theory by using dif- ferential equations, which provides a way to deal quantitatively with the problem. The consumerresource system is a system that discusses the process of energy or nutrient transfer between a consumer organism and a resource in which a resource is defined as a biotic or abiotic factor, which could increase the consumers’ growth, while a consumer will utilize the supply of resource and then reduces its growth rate (see [26]). The one way and two-way flow of energy or matter between species makes that the consumer resource system can be divided into unidirectional and bidirectional system (see [29]). In nature, there are many bidirectional consumer resource interactions (such as lichens and plant mycorrhizal fungi) and unidirectional consumer resource interactions (such as insect pollinator and host plant).

The traditional consumer resource interaction is simulated by the type relationship in which the consumer obtains some material benefits at the cost of the resource, such as the classic predator–prey models or parasite-host models. The unidirectional consumer–resource mutualisms are consistent with the traditional consumer–resource interaction (see [10, 27, 28]). Wang and DeAngelis [29] considered the following unidirec- tional consumer–resource system:

Here and , respectively, represent the densities of the resource species and the consumer species at time .

However, populations consist of individuals with many structural differences, such as age, size, location, status, movement, etc. According to these structural characteristics, structured population models distinguish individuals from one another to determine the birth, growth and death rates, interaction with each other and with environment, etc. (see [22]). In the last century, structured population models have played a significant role in the mathematical analysis and control of populations in biology and demography. Especially, age structured first-order partial differential equations (PDEs) provide a main tool for modeling population systems and are recently employed in economics (see [2, 14, 20, 21, 23, 24, 31]). In [20], the authors investigated the oscillation theory of the following unidirectional consumer–resource model with age-structure in the consumer:

where is the density of the resource species at time , and I the density ofhe consumer species at time with age . is the number of matured (reproducing) consumers with the age-dependent maturation function and are the intrinsic growth rate and logistic type limitation of resource species, respectively. is the death rate of the consumer species describes the positive effect on the growth of the resource species due to mutualistic interactions with the consumer species, where denotes the saturation level of the functional response of the consumer species, and denotes the half-saturation density of resource species represents the consumption level of resource species by the matured consumer in the boundary condition denotes the new born individual of the consumer species depending on resource supplied by , where is the interaction strength, and is the half-saturation constant.

Note that age is only a special kind of size and size of an individual has a strong influence upon dynamical processes like its feeding, growth and reproduction, which in turn affect the dynamics of the population as a whole [5]. Here sizes can be mass, length, diameter, surface area, volume, maturity, and so on. For some animals, the amount of food obtained by individuals is proportional to their surface area, and the cost of the metabolism is proportional to their volume (see [25]). As a result, modeling population dynamics, it is natural to assume that the vital rates, such as fertility, mortality, and growth rates of individuals, depend on their body size and time (see [3, 7–9, 15, 17–19, 32, 33]).

To the best of our knowledge, so far there is no investigation on the optimal control of size-structured population models of consumer–resource mutualisms. The purpose of this paper is to make some contribution in this direction. To build the model, we assume that is the density of the resource at time and represents the density of the consumer at time with size . Let . Here is the maximum size of any individual in the consumer species, and is a given time. Similar to [20], let

be the number of matured consumers with the size-dependent maturation function In a similar way as to develop (2), we propose the following unidirectional consumer resource mutualisms system with size structure in the consumer to study the optimal harvest problem

All meanings of the parameters are exact to or similar as those for system (2) except the following. Here is the growth rate of individual’s size, that is,. The control variables and are the harvesting efforts for the resource species and the consumer species, respectively, which belong to

Here and are positive constans. Let be solution of (4) corresponding to . As done in [16], in this paper, we discuss the optimization problem as follows:

where

Here and are, respectively, the economic values of the individual of the resource and the consumer at time is the weight factor of the costs for implementing the controls. Thus, the optimization problem represents the total net economic benefit yielded from harvesting the resource and the consumer during a time of .

Denote . We make the following assumptions throughout this paper.

1. (A1) is a bounded continuous function; is of -class with respect to for each uniformly for . Further, there is a Lipschitz constant such that

2. (A2) is a positive constant. Moreover, for any , we assume that .

3. (A3) , and there is a positive constant such that

2 Well-posedness of the state system

This section is devoted to the well-posedness of system (4). As in [16], we first introduce the definition of characteristic curve.

Definition 1. (See [16, Def. 1].) The unique solution of the initialvalued problem is said to be a characteristic curve. Let be the characteristic curve through

For any point in the first quadrant of such that that is , define initial time. It is clear that . Using characteristic curve technique as in [1], the solution of system (4) can be defined as follows.

Definition 2. A pair of functions is said to be a solution of system (4) if it satisfies

where are given by

for and

To discuss the well-posedness of (4), let and define a new norm in by

for some . Obviously, is equivalent to the usual norm on . Thus, is a Banach space with the norm . Further, denote

and define the space

and

It is clear that is a nonempty closed subset in . Define by

where are defined by the right-hand sides of (6) and (7), respectively. Clearly, if is a fixed point of , then it must be a solution of (4) and vice versa.

Next, we show that is a contraction mapping on . To do this, we first introduce the following lemmas.

Lemma 1. (See [13, Lemma 3.3].) For any is continuous, decreasing and onto, and hence has the inverse , which is continuous from .

Lemma 2. (See [13, Lemma 3.4].) Let . Then is differentiable with respect to , and

and is differentiable with respect to , and

Theorem 1. Assume that (A1).(A3) hold. Then, for , system (4)has a unique solution.

Proof. First, we show that the mapping maps into. For any , . It is clear that . From (A2) it follows that . Thus, for,

we have

Let . By Definition 1. Moreover, from it follows that . Then from Lemma 2,. Thus, we have

This, together with (10), yields Gronwall’s inequality it follows that

For , inequality (12) still holds, and the proof is more simple.

For any , we consider .

From (6) it is easy to see that

Further, from (7), (11) and (12), for , we can see that

For , the above inequality still holds, and the proof is more simple. Thus, maps into itself.

Next, we discuss the compressibility of . For any and , from (6) it follows that

where , we have

For . By Definition 1 it follows that . Then from Lemma 2 it follows that . Thus,

Further, we can obtain

For , as in [12], using Fubini’s theorem and Lemma 1, we have

From Lemma 1 it follows that if . Further, . By Definition 1. Moreover, it follows from that . Thus, from Lemma 2 we have .Thus, we have

Hence, from (14)–(16) we obtain

where , the above inequality still holds, and the proof is more simple.

It follows from (13) and (17) that

Choose such that . Thus, is a contraction mapping on the Banach space . Hence, A owns a unique fixed point, which is the solution of (4).

To conclude this section, we will discuss the continuous dependence of solutions n the control variable. Let.

Theorem 2. For any be solutions of (4)corresponding to , respectively. If is small enough, then there are positive constants and such that

and

Proof. We only prove the first estimate as the proof for the second one is similar. From (6) it follows that

where .Further, from (7) we have

By a similar discussion as that in (changing variable ) we have

By a similar discussion as that in (changing variable ) we also have

Hence, we can obtain

where . The result follows immediately from above analysis.

3 The adjoint system

In this section, we will derive adjoint system of (4). Here and below we denote by and the tangent cone and normal cone of , respectively.

Lemma 3. (See [4, Prop. 5.3].) Suppose that satisfies

Then there

Lemma 4. Let be solution of (4)corresponding to . For each such that for sufficiently small , we have

as is the solution of (4)corresponding to is the solution of the following system:

Proof. The existence and uniqueness of the solution to (18) can be established in a similar way as in Theorem 1. By [1, Lemma 3.1.3], make sense. Note that are solutions of system (4) corresponding to , respectively. For simplicity, we denote ecccc. It follows from Theorem 2 that

as . Thus, must be solution of

It follows from Theorem 2 that

as Taking in (19) and using the above results yield system (18).

The adjoint system corresponding to control and state is

Methods similar to Theorem 1 can be used to prove the existence of the solution to system (20). Moreover, for (20), by similar discussion as in Theorem 2 one has the following result.

Theorem 3. For each , the adjoint system (20)has a unique bounded solution. Moreover, for sufficiently small, there is a positive constant such that

where and are the solutions of system (20)corresponding to and , respectively.

4 Optimality conditions

In this section, we will give first-order necessary conditions of optimality in the form of an Euler–Lagrange system.

Theorem 4. Let be an optimal harvest policy for the optimization problem (5), let and be the corresponding optimal state of system (4). Then

where the truncated mappings are given by

where is the solution of the following system:

Proof. For any , one has for sufficiently small be the solution of system (4) corresponding . From the optimality of it follows that . Thus, from Lemma 4 it follows that

Here is the solution of (18) with replaced by , respectively. Next, we show that

In fact, multiplying the first equation in (23) by and integrating on , we obtain

Multiplying the second equation in (23) by and integrating on , we obtain

It follows from (26) and (27) that

Similarly, multiplying the first equation of (18) by and multiplying the second equation of (18) by , with replaced by in (18), respectively, we also have

By (28) and (29) we obtain that equality (26) is true. Substituting (26) into (24), for each , we have

Hence, we have . This implies the conclusion of this theorem.

5 Existence of a unique optimal harvesting

The purpose of this section is to show that the optimization problem (5) has a unique solution by means of Ekeland’s variational principle. First, we embed the functional in the space by defining

Lemma 5. The functional is upper semicontinuous with respect to in

Proof. Assume that . From Riesz theorem there is a subsequence of still denoted by such that

as . Then, using the Lebesgue’s dominated convergence theorem, we have

Let be the solutions of system correspondingto and , respectively. From Theorem 2 it follows that

Thus, we obtain

Similarly, we also have

From Fatou’s lemma it follows that . This means that is upper semicontinuous.

Theorem 5. If is small enough, there is a unique optimal harvesting policy , which is in feedback form and determined by (21).(22).

Proof. Define the mapping by

where are, respectively, the solutions of (4) and (20) corresponding to . Now, we show owns a unique fixed point, which maximizes .

From Lemma 5 and Ekeland’s variational principle it follows that for each , there exists such that

where

It is clear that perturbed functional attains its supremum at . In the same manner as that in the proof of Theorem 4, we obtain

where are solutions of (4) and (20) corresponding to ,

Step 1. We show that the mapping has only one fixed point.

(i) For any , from (22) it follows that . Thus, maps into itself.

(ii) From Theorems 2 and 3 we know that are continuous about the control variable Thus, for any , we have

where is a constant. Obviously, is a contraction if is small enough. Thus, has a unique fixed point. In addition, Theorem 4 shows that if the optimal policy exists, it must be the fixed point of . Thus, the uniqueness holds.

Step 2. We show that is the optimal policy. That is, as

Note that

and

Hence, we have

If is sufficiently small such that , then

Thus, . From Lemma 5 it follows that

This means that is the optimal policy.

6 Numerical tests

In this section, we provide some examples to illustrate the effectiveness of the obtained results. Note that our problem is highly nonlinear, and one cannot expect an explicit optimal controller. In the following examples, we do not consider the interaction between resource and consumer species and do not consider the costs of controls. We take , .

Example 1. (see Figs. 1–3).

Example 2. (see Fig. 4).

Example 3. (see Figs. 5–7).

From the numerical simulations given in Figs. 1 and 4 we can see that the optimal harvesting strategies for both the resource species and the consumer species basically have a bang-bang structure. Further, by comparing Fig. 1 and Fig. 4 it can be seen that given other parameters, the optimal harvesting strategy for the resource species has nothing to do with its initial value, and the optimal harvesting strategy for the consumer species has nothing to do with its initial size distribution. Thus, it leads to the conclusion that the bang-bang structure of optimal policies is much more common in optimal population management. In this paper, we assume that the maximum harvesting efforts for the resource species and the consumer species are, respectively, positive constants. However, from the numerical simulations in Example 3 it can be seen that if the maximum harvesting effort for the resource species is a bounded function with respect to time ., and the maximum harvest effort for the consumer species is a bounded function with respect to time and individual size , the optimal harvesting strategies for both the resource species and the consumer species basically have a bang-bang structure. From the right part of Figs. 1, 4 and 5 it can be seen that for consumer species, harvesting individuals with larger sizes is conducive to obtaining more economic benefits. This has obvious biological significance because we assume that individuals with larger size have greater economic value.

7 Conclusion

This paper is concerned with the harvesting problem for a size-structured model of unidi- rectional consumer–resource mutualisms in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive ef- fect on the consumer. In the previous sections, we have established the well-posedness of the system by constructing a suitable solution space and equivalent norm. Then the continuous dependence of solutions on the control variable and the adjoint system of the state system are investigated. More important result is the existence of a unique optimal harvesting policy, which provides a theoretical basis for practical application. As for the structure of the optimal policy, in Theorem 4, we have presented a feedback strategy.

Let us make some comments on the difference of our results and methods with those of closely related works. For the optimal control problems of size-structured population models, the authors in [15,17,18] proved that the optimization problems admit at least one solution but paid no attention to the uniqueness. In addition, the structure of the optimal strategy did not considered in [15].

In our paper, we show that there is a unique optimal harvesting policy, and the struc- ture of the optimal policy is given in the form of feedback. As far as we know, most of optimal control problems for population systems are naturally formed in an infinite time horizon. However, in this paper, we consider the optimal harvest problem with a fixed horizon , where. To our knowledge, even for the population model of ordinary differential equations, the infinite-horizon optimal control problems are still challenging. For example, it is difficult to establish a suitable transversality condition so that one can choose the correct solution of adjoint system for which Pontryagin maximum principle is applicable. For more details of the infinite-horizon optimal control (including age-structured systems and size-structured systems), please refer to [24]. We leave these for our future work. Moreover, as done in [20], we can investigate the existence and stability of positive equilibrium and the existence of nontrivial periodic solution of the system.

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