Optimal contraception control for a nonlinear population model with size structure and a separable mortality

This paper is concerned with the problem of optimal contraception control 
for a nonlinear population model with size structure. 
First, the existence of separable solutions is established, which is crucial in obtaining 
the optimal control strategy. Moreover, it is shown that the population density 
depends continuously on control parameters. 
Then, the existence of an optimal control strategy is proved via 
compactness and extremal sequence. Finally, the conditions of the 
optimal strategy are derived by means of normal cones and adjoint systems.

1. Introduction. Wild animals are not only valuable natural resources but also important components of a healthy ecosystem. However, relatively high densities of small rodents (called vermin) can result in a considerable threat to crops and can even destroy the ecological balance. This makes it necessary for us to intervene the growth of the vermin. Usually, chemical drugs are applied to poison the vermin, which will pollute the environment and destroy the ecological system. Recently, decreasing the reproductive rate instead of increasing mortality has been suggested as a promising way for managing the impact of overabundant species (see, for example, [16]). Often, female sterilant is used to achieve this purpose. See [21,28] on contraception control for models without stage structures and see [8,9,10,14,25] for optimal birth control on models with age structures.
Note that age is only a special kind of size and size is one of the most natural and important variables to describe population dynamics. Here by size we mean some indices displaying the physiological or statistical characteristics of population individuals. Sizes can be mass, length, diameter, volume, maturity, and so on. For instance, for some animals and most trees, their metabolic capacity is related to their surface area [27]. Further studies show that the amount of food obtained by individuals is proportional to their surface area and the cost of the metabolism is proportional to their volume [29]. Moreover, the size of the individual can determine it's prey object and niche, thus it can determine the ability of intra-competition and inter-competition [5]. As a result, modelling population dynamics with size structure has been an active and fruitful theme in mathematical biology (to name a few, see [2,4,6,7,11,12,13,15,17,18,19,20,22,23,24,26,31]). Unfortunately, only a few papers deal with the control problem [2,11,12,13,15,18,19,23,31]. Moreover, all focus on optimal harvesting except Araneda et al. [2] on optimal harvesting time and He et al. [11] on optimal birth control.
To the best of our knowledge, so far there is no investigation on the optimal contraception control of size-structured population models. The purpose of this paper is to make some contribution in this direction. The remaining part of this paper is organized as follows. First, we propose the model in Section 2, followed by the existence of separable solutions in Section 3. The last two sections are devoted to the optimal control policy. The existence of a unique optimal policy is proved via compactness and extremal sequence in Section 4 while optimality conditions are derived in Section 5.

2.
The model formulation. Our study is inspired by that of Kato [19], where the author investigated the optimal harvesting problem for the following nonlinear size-structured population model x ∈ [0, l), with the objective functional is a maximal size and T is a given time; V (x, t) represents the growth rate depending on the individual's size x and time t; β(x, t, J(t)) is the fertility rate depending on size x, time t and the total population J(t) weighted by b(x); µ(x, t) is the natural mortality rate and Φ(I(t)) stands for an external mortality rate which depends on the total population I(t) weighted by m(x) possibly due to the intra-competition such as the limitation of the habitat; α(t) is a harvesting rate depending only on time t and f (x, t) stands for a certain inflow such as immigration rate. To build our model, we assume that the female sterilant applied at any time is eaten completely by the vermin (including the male vermin) and at any time the vermin individual with the same size eats the same amount of the female sterilant. In a similar way as to develop (1), we propose the following model on contraception OPTIMAL CONTROL OF A SIZE-STRUCTURED POPULATION 3605 control for nonlinear size-structured population dynamics, In the coming discussion, we make the following assumptions.
Let u α be the solution of (2) corresponding to α ∈ Ω. Similar to [13], we investigate the following optimization problem, whereū(x) ∈ L ∞ (0, l) is a given ideal distribution of the vermin population, that is, the maximal amount of the vermin which will not affect the crop's growth. The function g(u α (x, t) −ū(x)) represents the proximity of the controlled variable to the ideal distribution while the function h(t)u α (x, t)α(x, t) represents the costs of the contraception control, which include the costs of the female sterilant and the costs of the related labor in applying the sterilant. Therefore, an optimal control policy is one that the density of the vermin falls to as close as possible to the ideal distribution and the control cost is as low as possible. Though (2) has a unique solution for each u 0 , this will not help us much in dealing with the optimal control problem. As in [19], we shall seek separable solutions to (2).
3. The separable solutions. In this section, we provide some properties of the solutions, which include boundedness and the continuous dependence of the population density on the control parameter. As in [12], we first introduce some definitions.
Definition 3.2. The derivative of the function u(x, t) at (x, t) along the characteristic curve ϕ is given by For an arbitrary point (x, t) in the first quadrant of the x-t plane such that Utilizing the characteristic curve technique as in [19], we can define the solution of (2) as follows.
is said to be a solution of the subsystems (5) and (6) if it satisfies the following two equations whereĨ hold. Then, for any α ∈ Ω, subsystems (5) and (6) have a unique solution (ũ y (x, t), y(t)), which is non-negative.
The result is established by applying the Banach fixed point theorem. Firstly, by (6), we have y(t) = exp{− t 0 Φ(Ĩ(s)y(s))ds} ≥ θ, which means that y(t) ∈ A. By [19, Theorem 4.1], we have that, for fixed y(t) ∈ A, subsystem (5) has a unique non-negative solutionũ y (x, t) ∈ L ∞ (Q), which satisfies for h ∈ A. It's easy to see that θ ≤ [Ah](t) ≤ 1, so A is a map from A to A. Moreover, by (8) and (A 5 ), we can get Then, for any h 1 , h 2 ∈ A, we have Thus, choosing λ > C Φ (r 1 )r 1 yields that A is a contraction on the space (A, · λ ). By the Banach fixed point theorem, A owns a unique fixed point, or equivalently, there exists a uniqueỹ ∈ A such that Thirdly, by [19,Lemma 2.4], for any y 1 , y 2 ∈ A, there exists a constant M > 0 such that Define a map B : A → A by (By)(t) =ỹ(t) for y ∈ A.
In summary, we have proved that subsystems (5) and (6) have a unique solution (u y (x, t), y(t)), which is non-negative and uniformly bounded.
Combining Theorem 3.5 with the fact that (2) has a unique solution for each u 0 immediately produces the following result.
To conclude this section, we prove that the population density depends continuously on the control parameter α.
Theorem 3.7. Assume that (A 1 )-(A 7 ) hold. Then the solution u α (x, t) of (2) is continuous with respect to the control variable α, that is, there exists a positive constant B such that, for any t ∈ [0, T ], α 1 , α 2 ∈ Ω, we have where u 1 and u 2 are the solutions of (2) corresponding to α 1 and α 2 ∈ Ω, respectively.
The required result follows immediately from (13)- (15) and hence the proof is complete. 4. Existence of optimal control policy. The purpose of this section is to prove the existence of the optimal control policy.
is uniformly bounded about the control variable α ∈ Ω. Note that dI α (t) By the assumptions and Theorem 3.6, we know that I 1 is uniformly bounded about the control variable α ∈ Ω. For I 2 , by the second equation of (2), we obtain Using the assumptions, we know that I 2 is also uniformly bounded about the control variable α ∈ Ω. In summary, we have proved that dI α (t) dt is uniformly bounded about the control variable α ∈ Ω.
Next, we use Fréchet-Kolmogorov guidelines (see, for example, [30]) to prove that {I α (t) : α ∈ Ω} is a relatively compact set in L 2 (0, T ). For convenience, we denote I α (t) = 0 if t < 0 or t > T . Then I α (t) is continuous on R. To apply Fréchet-Kolmogorov guidelines, we need to verify the following three things. 1 • Uniform boundedness of I α (t) about α. This is easy to see since  Since {u αn } is uniformly bounded about the control variable α ∈ Ω, there exists a subsequence of {α n }, still denoted by {α n }, such that u αn u * weakly in L 2 (Q) as n → ∞ for some u * ∈ L 2 (Q). For {u αn }, it follows from the Mazur Theorem (see [1]) that there exists the following convex combination of {u αn }, such thatũ n → u * in L 2 (Q) as n → ∞. It follows from Lemma 4.1 that there exists a subsequence of {α n }, still denoted by {α n }, such that I αn → I * as n → ∞ and I αn (t) → I * (t) for almost every t ∈ (0, T ). Consequently, I * (t) = l 0 b(x)u * (x, t)dx. Define the function sequence of the control variable as For any n + 1 ≤ i ≤ k n , 0 ≤ α i (x, t) ≤ L since α i (x, t) ∈ Ω. Combining this with λ n i ≥ 0 and u αi (x, t) ≥ 0, we have which implies thatα n ∈ Ω. From the boundedness of {α n } and the weak compactness of the bounded sequence, we obtain that there exists a subsequence of {α n }, still denoted by {α n }, such thatα n α * weakly in L 2 (Q) as n → ∞.
Next, we show that the control α * ∈ Ω is an optimal policy. On the one hand, On the other hand, by the definition of J(·), (16), and (17), we have Therefore, J(α * ) = d = inf α∈Ω J(α), which means that α * is an optimal control policy for the control problem (2)-(3). This completes the proof.
5. Optimality conditions. The purpose of this section is to derive the first-order necessary conditions of optimality in the form of an Euler-Lagrange system.
Treating (21) in the same manner as that done in Theorem 3.5-Theorem 3.7, we can get the following result with the proof being omitted.
Now we are ready to present the optimality conditions. where ξ(x, t) is the solution of the adjoint system (21).