Optimal boundary control problems for a hierarchical age-structured two-species model

This paper discusses the optimal control problems for a nonlinear age-structured two-species model, where elder individuals are more competitive than younger ones, and each species is described by a nonlinear integropartial differential equation with a global feedback boundary condition. First, we establish the existence of a unique nonnegative bounded solution by means of frozen coefficients and the fixed-point theorem. More importantly, we discuss the least deviation-cost problem and the most benefit-cost problem. For the least deviation-cost problem, the existence of an optimal strategy is established by means of Ekeland’s variational principle, and the minimum principle is obtained via an adjoint system. Meanwhile, the corresponding results for the most benefit-cost problem are given. In addition, some numerical experiment results are presented to examine the effects of parameters on the optimal policies and indexes.


Introduction
It is well known that populations consist of individuals with many structural differences, which include age, size, gender, and status. Structured population models distinguish individuals from one another according to these structural differences, to determine the birth, growth, and death rates, interaction with each other and with the environment, infectivity, etc. (see [1]). During the past one hundred years, age-and size-structured firstorder partial differential equations have provided a main tool for modeling population systems and have been recently employed in economics. To name a few, see [2][3][4][5][6][7][8][9][10][11][12] on the well-posedness, asymptotic behaviors, and optimal control of population models with age structure, while [1,[13][14][15][16][17][18][19][20][21][22] discussed the related problems of population systems with size structure.
Ecological studies show that there exist dominance ranks of individuals in many species, such as nonmammalian, primates, rodents, etc., (see [23]). Moreover, [24] pointed out that for many species the competition for resources that determine individual fertility and mortality is based on some hierarchy in the population that is related to individuals age or any other physiological structure. They also referred to this competitive phenomenon as "contest competition" (see [25]). Hierarchical structuring in a biotic population is the ranking of individuals one above the other based on their age, body size, height, or any other possible structuring variable that can affect their vital rates (see [26]). Further, Lomnicki [27] showed that the rank of an individual not only influences its access to some factors such as food resource, mates, shelter, and nesting, but also has some influence on the individual's vital rates. Thus, the existence of hierarchy has an impact on the dynamic behaviors of the population. Just as Gurney and Nisbet [26] said, the hierarchy of ranks in a population is one of the significant forces to maintain its ecological stability and species persistence. During the past decade, both discrete-time and continuous hierarchical structured population models have been studied by several researchers (see [28][29][30][31][32][33][34][35][36][37]). The hierarchical model is considered as a generalization of the age-and size-structure model. The key is to consider the so-called "internal environment" in the modeling. For example, Ackleh and Deng [29] discussed the following nonlinear hierarchical age-structured population model (a,t) ∂t + ∂p(a,t) ∂a = -μ(a, t, Q(p)(a, t))p(a, t), 0 < a < ∞, t > 0, p(0, t) = ∞ 0 β(a, t, Q(p)(a, t))p(a, t) da, t > 0, p(a, 0) = p 0 (a), 0 ≤ a < ∞, where the environment Q(p)(a, t) is a function of the density p, which is given by Parameter α is used to measure the degree of hierarchy in the population structure, and is called the hierarchical coefficient. More precisely, α is the weight of the lower ranks (i.e., age smaller than a) in the competition for resources. From [38], α = 0 (i.e., "contest competition") implies an absolute hierarchical structure, whereas values of α tending to 1 mean that, when competing for resources, the effect of higher ranks is similar to the effect of lower ranks. Moreover, the limit case α = 1 (i.e., "scramble competition") corresponds to the environment given by the total size of population at time t, i.e., without any hierarchical structure.
Most studies on the hierarchical population model mainly discuss the existence and uniqueness of solutions (see [28,30,36,37]), numerical approximation of solutions (see [28,29,32,33,37]), and the asymptotic behavior of solutions (such as stability and persistence, see [31,34,35,38]). However, studies on optimal control problems of hierarchical species models are rather rare. As far as we know, so far only [39,40] have investigated optimal control problems in hierarchical species. However, there is no investigation on the optimal boundary control of hierarchical multispecies model. The purpose of this paper is to make some contribution in this direction. In this paper, we will discuss the optimal control problems in a hierarchical system composed of two interacting species with age structure. The control variables describe the inflow rate of the baby individual from an external source (such as artificially stocking fry or artificially planting saplings), which appear in the boundary conditions.

Description of the problem
Our study is inspired by that of Ackleh and Deng [29]. To build our model, we assume that there are age structures in two species and they have the same life expectancy (i.e., maximal age) and is A ∈ (0, +∞). Moreover, we assume that the mortality and fertility of species depend mainly on the number of elderly individuals in addition to the age of individuals. From [29], this phenomenon can be described as that the vital rates of an individual depend not only on its age, but also on an "internal environment". Let p i (a, t) be the density of species i with age a at time t (i = 1, 2). In a similar way as we developed (1.2), the "internal environment" E(p i ) of species i can be defined as Here, constant α i is the degree of hierarchy. This means that younger individuals are less competitive than older ones. Moreover, when two species interact, we assume that the fertility of a species is only related to its own "internal environment", and the mortality of one species is not only related to its own "internal environment", but also dependent on the "internal environment" of the another species. Thus, we can assume that the fertility and mortality of species i are β i (a, E(p i )(a, t)) and μ i (a, E(p 1 )(a, t), E(p 2 )(a, t)), respectively. Hence, we propose the following hierarchical age-structured two-species model with boundary control (i = 1, 2) where Q = (0, A) × (0, T) and T ∈ (0, +∞) represents the control horizon. p 0 i (a) is the initial age distribution of species i. The control variable u i (t) stands for the number of baby individuals influx into species i at time t. Moreover, we assume that the control variable u i (t) belongs to The positive constant U i represents the maximum number of baby individuals influx into t)) be the solution of (2.1) and (2.2) with the control variable u.
Compared with known closely related ones, our model has the following features. It is clear that our model includes some existing two species models with age structure. If we take μ i (a, E(p 1 )(a, t), E(p 2 )(a, t)) = μ i (a, t)λ i (a, t)P j and β i (a, E(p i )(a, t)) = β i (t)m i (a, t) (i, j = 1, 2; i = j), then our model can be reduced to the model discussed in [9]. Moreover, if there is only one species and u i (t) ≡ 0, our system can be reduced to the system (1.1) and (1.2). In [29], Ackleh and Deng assumed that the vital signs β and μ in (1.2) are, respectively, decreasing and increasing with respect to "internal environment" Q (i.e., -∞ < β Q ≤ 0 and 0 ≤ μ Q < ∞). Based on these, the authors proved that the system has a unique solution by establishing a comparison principle and constructing monotone sequences. Moreover, they gave conditions for extinction and persistence of the population. Undoubtedly, the above works are important and helpful to understand the evolution of hierarchical species. However, it is also an important topic for human beings to use or transform nature (such as renewable-resource development, pest control, etc.) to serve human survival and development on the basis of understanding of nature. The research in this field corresponds to the control problem of the model mathematically. It is widely known that investigations on optimal control problems in hierarchical populations are rather rare. The purpose of this paper is to make some contribution in this direction.
On the one hand, if the system describes the interaction between pests or annoying animals (such as rodents), we always hope that each species can be reduced to an ideal distribution level that does not affect crop growth and will not become extinct. Thus, in this paper, we first consider the following optimization problem Here, the first integral represents the deviation between final state p u i (a, T) and ideal distributionp i (a), while the second one stands for the costs of control with factor σ i . Thus, an optimal policy for (2.3) is to manipulate the system with any given initial distribution as close as possible to an ideal distribution with least control cost. Hence, control problem (2.3) can be called the least deviation-cost problem.
On the other hand, if species in the system are of renewable resources (such as forest resources or fish resources), optimal harvesting is always one of the most fascinating topics. Thus, in this paper, we also consider the following optimization problem Here, function g i (a) represents the price of individuals with age a in species i. The motivation for (2.4) is as follows: How to input baby individuals during the period [0, T], such that one can obtain the most benefit by exploiting the final distributions of species with the least control cost? Hence, one can say optimization problem (2.4) is the most benefit-cost problem. Similar to the hypothesis of model (2.1) in [2, p.30], here, we make the following assumptions (i = 1, 2). Denote R + = [0, +∞).
This is reasonable because the birth rate of any species is a bounded function. Moreover, for any M > 0, there is L 1 (M) > 0 such that This means that the derivative of the birth rate of the species with respect to its "internal environment" is a bounded function. (A2) μ i (a, x, y) > 0 for any (a, x, y) ∈ (0, A) × R + × R + . For any (x, y) ∈ R + × R + , μ i (·, x, y) ∈ L 1 loc [0, A] and A 0 μ i (a, x, y) da = +∞. This condition will guarantee that there are no individuals with age beyond A for each species. Moreover, for any M > 0, there is L 2 (M) > 0 such that for a ∈ [0, A) and x 1 , This ensures that the initial distribution of each species is bounded.

Well-posedness of the state system
The purpose of this section is to discuss the well-posedness of (2.1) and (2.2). Here, we assume T > A and omit the proof for the case T ≤ A (which is analogous, but easier). Let Further, define the solution space as follows Note that X is a nonempty closed subset in X. Hence, (X , · * ) is also a Banach space. Moreover, for any w = (w 1 , w 2 ) ∈ X, we denote First, let q(a, t) = (q 1 (a, t), q 2 (a, t)) ∈ X be arbitrary but fixed, so is the function Consider the following linear system (3.2) Clearly, system (3.1) and (3.2) has a unique nonnegative solution (p 1 (a, t; q), p 2 (a, t; q)).
Using the characteristic lines (see [2]), the solution has the form (i = 1, 2): Proof It is easy to show that (3.5) has a unique nonnegative solution. Moreover, using (3.5)-(3.8) and Gronwall's inequality, we can derive that b i (t; q) ≤ B i with positive constant Now, we prove inequality (3.9). From (A1)-(A3), it follows that Thus, from (3.7) and (3.11)-(3.13), we have where Similarly, we also have Next, we prove inequality (3.10). From (A1)-(A3) and (3.5), it follows that Then, using Gronwall's inequality, we can obtain where Now, for any q = (q 1 , q 2 ) ∈ X , we define the mapping A : X → X by where A i (q 1 , q 2 ) (i = 1, 2) is defined by the right-hand sides of (3.3). Moreover, for any q ∈ X , it is easy to show that Aq ∈ X . Thus, if A is a contractive mapping on Banach space (X , · * ), then system (2.1) and (2.2) has a unique solution. Proof For any q i = (q i 1 , q i 2 ) ∈ X (i = 1, 2), it follows from Lemma 3.1 that Thus, . Hence, for any q i = (q i 1 , q i 2 ) ∈ X (i = 1, 2), it follows from (3.14) that Choose λ such that λ > M 4 . Then, A is a contraction on Banach space (X , · * ). By the Banach fixed-point theorem, A has a unique fixed point, which is the solution of system (2.1) and (2.2). Moreover, from the definition of solution space X , it follows that the solution is nonnegative and bounded.
is the solution of system (2.1) and (2.2) corresponding to u i .
Proof For any u = (u 1 , u 2 ) ∈ U , the solution of system (2.1) and (2.2) has the form a, t, a), a < t, Here, , t, a), (a, t) ∈ Q; 0, otherwise, is a solution of system (2.1) and (2.2) corresponding to u i ∈ U (i = 1, 2). It follows from (3.17) that where M 5 = (p * 1 A + B 1 T)L 1 (M * ) + (p * 1 + B 1 )ATβ * 1 L 2 (M * ). Using Gronwall's inequality, we can derive that Further, for a ≥ t, from (3.12), (3.15), and (3.16), it follows that For a < t, from (3.12), (3.15), and (3.16), it follows that Thus, from (3.21)-(3.23), it follows that Similarly, there exists a positive constant M 7 such that Thus, we have Remark 3.4 In this section, we first prove that the model has a unique nonnegative bounded solution by constructing a suitable solution space and equivalent norm, and using the freezing-coefficient method and fixed-point theory. The boundedness of the solution ensures that the species described in the system will not grow indefinitely. Then, using the basic inequality theory, we obtain the continuous dependence of solutions on the control variable, which will play a key role in the following discussions.

Least deviation-cost problem
In this section, we will discuss the optimization problem (2.3). Here and after, we denote by T U i (u i ) and N U i (u i ) the tangent cone and the normal cone of U i at u i , respectively. Moreover, from [41], we have the following result.

Minimum principle
In this subsection, we will establish first-order necessary conditions of optimality in the form of an Euler-Lagrange system.

Theorem 4.2
Let u * = (u * 1 , u * 2 ) and p * = (p * 1 , p * 2 ). If (u * , p * ) is an optimal pair for the optimization problem (2.3), then optimal strategies u * i must be structured as where q = (q 1 , q 2 ) is the solution of the adjoint system in which μ i2 and β i2 are partial derivatives of μ i and β i with respect to the second variable, respectively; μ i3 is a partial derivative of μ i with respect to the third variable.
Proof The existence of a unique bounded solution to (4.2) can be treated in the same manner as that for (2.1) and (2.2). For any v i ∈ T U i (u * i ) (see [41]), we have u ε i . = u * i + εv i ∈ U i (i = 1, 2) for sufficiently small ε > 0. Let p ε = (p ε 1 , p ε 2 ) be a solution of (2.1) and (2.2) corresponding to u = u ε . Then, the optimality of u * implies J(u * ) ≤ J(u ε ). Hence, Using the Theorem 3.3 and passing to the limit ε → 0 + , we can derive that in which z i (a, t) = lim ε→0 + p ε i (a,t)-p * i (a,t) ε (i = 1, 2). Via a similar discussion to that in [2], z 1 (a, t) and z 2 (a, t) do make sense. Further, it follows from (2.1) and (2.2) that (z 1 , z 2 ) satisfies Multiplying the first equation in (4.5) by q i (a, t) and integrating on Q, we have Hence, we can obtain This, together with (4.2), yields It follows from (4.4) and (4. = 1, 2), which implies the conclusion of this theorem. The proof is complete.

Existence of a unique optimal policy
In this subsection, we study the existence and uniqueness of the optimal control. Define the embedding mapping ϕ :
In a similar manner to that in Theorem 3.3, we can obtain the following result.
That is to showJ(ũ 1 ,ũ 2 ) = inf{J(u 1 , u 2 ) :ũ 1 ,ũ 2 ∈ U}. Note that θ i L ∞ (0,T) ≤ 1. Thus, from (4.11) and (4.12), it follows that Note thatũ = (ũ 1 ,ũ 2 ) ∈ U is the unique fixed point for the mapping P. Thus, . Hence, we have u ε →ũ as ε → 0 + . From Lemma 4.3 and (4.9), one hasJ(ũ) = inf{J(u) : u ∈ U}. Thus, u ∈ U is the optimal policy. Remark 4.6 In this section, using Ekeland's variational principle and fixed-point reasoning, we show the existence and uniqueness of optimal policies for the least deviation-cost problem, which gives us a solid theoretical ground for a practical application. As for the structure of the optimal policy, we have presented a feedback strategy, in Theorem 4.2, via the normal cone technique and adjoint system. Due to the high nonlinearities in our problem, one cannot expect an explicit optimal controller. However, the feedback strategy would be helpful for a numerical computation of the optimal policy (see Sect. 6).

Most benefit-cost problem
Here, we give the corresponding results for optimization problem (2.4). A similar discussion as that in Theorem 4.5, we can show that the most benefit-cost problem (2.4) has a unique solution. Further, similar to the proof of Theorem 4.2, we have the following result.
Theorem 5.1 (Maximum principle) Let u * = (u * 1 , u * 2 ) and p * = (p * 1 , p * 2 ). If (u * , p * ) is an optimal pair for the optimization problem (2.4), then where q = (q 1 , q 2 ) is the solution of the following adjoint system in which μ i2 and β i2 are partial derivatives of μ i and β i with respect to the second variable, respectively; μ i3 is a partial derivative of μ i with respect to the third variable.

Numerical simulations
In this section, we provide some examples to verify the effectiveness of the obtained results and find other dynamic properties of the system. From Theorem 4.2, optimal strategy u * i (i = 1, 2) is a fixed point of mapping F i . Clearly, the fixed point can be approximated by an iteration scheme. First, we discuss the effects of ideal distributionp(a) = (p 1 (a),p 2 (a)) on optimal policy u * (t) = (u * 1 (t), u * 2 (t)) and then on optimal functional J(u * ) for optimization problem (2.3).
Calculating in MATLAB with the above parameters, we obtain the numerical results for the optimal policy u * and optimal state p * corresponding to s = 2 (see Fig. 1). Moreover, the indexes J(u * ) = j × 10 5 corresponding to s = 2, 3, 4, . . . , 106 are listed in Table 1. Figure 1 shows the optimal strategy u * (t) and the optimal population density p * (a, t) with s = 2 within the ideal distributionp(a). Moreover, as can be seen from Table 1, optimal objective functional J(u * ) is increasing with respect to the coefficient s. This means that as the ideal distribution increases, the more baby individuals (i.e., the greater the control cost) should be input to make the final distribution of the species sufficiently close to the ideal distribution. Now, we investigate the effects of price function g(a) = (g 1 (a), g 2 (a)) on optimal policy u * (t) = (u * 1 (t), u * 2 (t)) and then on optimal functional J(u * ) for optimization problem (2.4) in the numerical approach.
Finally, we will discuss the effects of the costs factor σ i on the optimal functional J(u * ) for optimization problem (2.4). The other parameters are the same as those in Example 2. Table 3 demonstrates the changes of optimal benefits J(u * ) with respect to the control cost factor σ i . From the data in Table 3 alone, the objective optimal benefit J(u * ) decreases as the cost factor σ i increases. This means that as the cost factor increases, the total net economic benefit yielded from harvesting the species in the final state will decrease.

Conclusion and discussion
This paper is concerned with the optimal boundary control problems for a hierarchical age-structured two-species model. In the previous sections, we have established the wellposedness of the state system provided the control variable (the influx rate of baby for each species) is put in. Meanwhile, the continuous dependence of density of each species on the control parameters is established. More importantly, the least deviation-cost problem and the most benefit-cost problem are discussed. Here, the least deviation-cost problem means that the system with any initial distribution is manipulated to be as close to an ideal distribution as possible at the least cost. The most benefit-cost problem refers to maximizing the benefit achieved by developing the final population distribution with the least cost of control (i.e., the cost of inputting baby individuals). For the least deviation-cost problem, Theorem 4.2 establishes the minimum principle by using an adjoint system, and Theorem 4.5 shows the existence of a unique optimal strategy by means of Ekeland's variational principle. Meanwhile, the corresponding results, the existence of a unique optimal control policy and the maximum principle (see Theorem 5.1), for the most benefit-cost problem are also given. From Theorems 4.2 and 5.1, the optimal strategies can be regarded as fixed points of contraction mappings about the adjoint variables. Thus, we can approximate the optimal policies by the standard iterative paradigm. Moreover, some examples and numerical results are presented to verify the effectiveness of the obtained results and find other dynamic properties of the system. Although the phenomena observed in Tables 1, 2, and  3 have certain biological significance, they are intuitive in the general sense and do not have universal significance. However, the theoretical analysis of these phenomena is very difficult or even impossible to deal with.