OPTIMAL HARVESTING FOR AGE-STRUCTURED POPULATION DYNAMICS WITH SIZE-DEPENDENT CONTROL

. We investigate two optimal harvesting problems related to age-dependent population dynamics; namely we consider two problems of max- imizing the proﬁt for age-structured population dynamics with respect to a size-dependent harvesting eﬀort. We evaluate the directional derivatives for the cost functionals. The structure of the harvesting eﬀort is uniquely determined by its intensity (magnitude) and by its area of action/distribution. We derive an iterative algorithm to increase at each iteration the proﬁt by changing the intensity of the harvesting eﬀort and its distribution area. Some numerical tests are given to illustrate the eﬀectiveness of the theoretical results for the ﬁrst optimal harvesting problem.


1.
Setting of the problems. There is an extensive mathematical literature devoted to the optimal harvesting of age-structured population dynamics. Various methods are used to search an optimal harvesting effort (control) in a certain space of functions depending on time and age, to derive its structure or to approximate it. A quite general harvested age-structured population dynamics is described by the following McKendrick model ∂ t p(a, t) + ∂ a p(a, t) + µ(a, t)p(a, t)

+M(
A 0 p(a, t)da)p(a, t) = −u(a, t)p(a, t), (a, t) ∈ (0, A) × (0, T ) p(0, t) = A 0 β(a, t)p(a, t)da, t ∈ (0, T ) p(a, 0) = p 0 (a), a ∈ (0, A), (1) where A ∈ (0, +∞) is the maximal age for the population species and T ∈ (0, +∞) is the final moment of the harvesting process. Here β(a, t) is the fertility rate, µ(a, t) is the mortality rate for individuals of age a at the moment t, and p 0 (a) is the initial density of the population of age a. M(

M(
A 0 p(a, t)da) being an additional mortality rate, and is due to the competition for resources. Function u is a control and plays the role of an additional mortality rate, and is called "harvesting effort"; the quantity T 0 A 0 u(a, t)p(a, t)da dt represents the total harvest. When trying to implement a given harvesting effort u(a, t) we need to be able to establish the age of individuals. We, however are trying to do this by investigating the size of individuals. Actually, we notice that the size of an individual is a nondecreasing function of age (see Figure 1), strictly increasing on an age-interval [0, a 0 ] and constant on [a 0 , A] (a 0 ∈ (0, A)). It means that when the size of an individual is less than the size corresponding to age a 0 (the maximal size) we may identify its age uniquely, and when the size is equal to the size corresponding to a 0 , then we only may say that its age is between a 0 and A. Hence, for individuals of age between a 0 and A the harvesting effort has to be the same (because we cannot distinguish their age). What we can vary (in time) is the intensity of the harvesting effort.
Assume now that we are actually interested in the industrial fishing (fish harvesting). We may vary at any moment t the intensity of the harvesting w(t). For any given moment t we harvest at the same rate the individuals with a size superior to a certain value (which obviously corresponds to ages superior to a certain value α(t) ∈ [0, a 0 ]). Hence, here is the form of a realistic harvesting effort where H is the Heaviside function. Obviously, α : [0, T ] → [0, a 0 ] is a function (whose graph looks as in Figure 2) to be determined, and the intensity of the harvest is a function w : [0, T ] → [0, L] (where L ∈ (0, +∞) is the maximal affordable intensity of the harvesting effort) and has to be determined as well.
Our goal is to maximize the total harvest/yield. Hence, we have to determine the optimal intensity of the harvesting effort w * , and the optimal α * (or the optimal area where the control should act; see Figure 2). We may view this as a shape optimization problem. Here is the first optimal harvesting problem to be investigated where p is the solution to (1) corresponding to u given by (2). Here are the hypotheses to be used We denote by (the cost functional at (w, α) ∈ W × A, where p w,α is the solution to (1), with u given by (2)). We shall say indistinctly that (w, α) or u(a, t) = w(t)H(a − α(t)) is the control. For any (w, α) ∈ W × A there exists a unique solution p w,α to (1), where u(a, t) = w(t)H(a − α(t)). By a solution to (1) we mean a function p ∈ L ∞ ((0, A) × (0, T )), absolutely continuous along almost any characteristic line a.e. a ∈ (0, A).
Here Dp is given by Dp(a, t) = lim ε→0 p(a + ε, t + ε) − p(a, t) ε (Dp is a directional derivative). Since the solution to (1) is absolutely continuous along almost any characteristic line, the conditions a.e. t ∈ (0, T ) and lim a.e. a ∈ (0, A), are meaningful. It is important to notice that the solution p satisfies the first equation in (1) along the characteristic lines. In order to prove the existence and uniqueness of a solution to (1) we integrate along the characteristic lines and use the Banach fixed point result. For the definition of a solution to (1) and for other age-dependent systems, and for the existence, uniqueness, and other basic properties of such solutions we refer to Chapter 2 in [2]. Sometimes, we modify as time increases the value of α (recalibrate the harvesting devices) and we have to pay a certain cost which is proportional to the length of Γ α , where Γ α = {(t, α(t)); t ∈ [0, T ]}. If we assume that α is smooth, then the cost to be paid for controlling/acting in the hashed area (see Figure 2) is where k ∈ (0, +∞) is a constant. In this situation a reasonable optimal harvesting problem is the following one where p is the solution to (1) corresponding to u given by (2).
For both optimal harvesting problems we shall evaluate the directional derivatives. For the first optimal harvesting problem we will derive an iterative algorithm to improve at each step the intensity of the harvesting effort w and the area where the control acts (the hashed region in Figure 2).
Our present paper is organized as follows. Section 2 is devoted to the evaluation of the directional derivative of the cost functional for problem (OH). A numerical iterative algorithm to improve at each step the harvesting effort is derived in section 3. Some numerical tests are given as well. A separable case for (OH) is treated in section 4. The evaluation of the directional derivative of the cost functional for problem (OH2) is the main goal of section 5. Some final comments are made.
2. The directional derivative of the cost functional for problem (OH). The maing goal of this section is to calculate the directional derivative of J. Consider some arbitrary v ∈ L ∞ (0, T ) and ζ ∈ L ∞ (0, T ) such that for any θ > 0, sufficiently small, as θ → 0 (since p w,α is absolutely continuous along almost any characteristic line, Here δ(a − ρ) denotes the Dirac mass at ρ. By a solution to (3) we mean a function z ∈ L ∞ ((0, A) × (0, T )), that satisfies the first equation in (3) along almost any characteristic line (of equation a − t = const.), and such that a.e. a ∈ (0, A).
Notice that the time intervals when w(t) = 0 correspond to prohibition periods for fishing. (7) and Theorem 2.1, we develop a conceptual iterative algorithm to improve at each step the control (w, α), i.e. the intensity of the harvesting effort and the region where the control acts (in order to obtain a higher value for J).

In Step 2 and
Step 7, ε 1 > 0 and ε 2 > 0 are prescribed convergence parameters, and · is the L 2 -norm. The algorithm we have used is a gradient-type one. For details about the gradient methods see [6, §2.3]. System (1) (Step 1 and Step 4) is approximated by an Euler-type scheme, ascending with respect to time levels. Using the same idea we approximate system (5) (Step 1 and Step 4), but descending with respect to time levels. The integrals from Step 1, Step 3, and Step 5 are numerically computed using the trapezoidal formula corresponding to the discrete grid.
In Table 1 and in Figure 4 below, it can be seen that the algorithm provides a higher value for J at each iteration.
The control u corresponding to the last iteration is given in Figure 5(a). The section of the control u corresponding to the time level t = 0.6 can be seen in Figure  5(b).
We take another value for the constant B for the function β, B = 75, w (0) = 1, and α (0) = 0.2. The algorithm ends when the condition in Step 7 is fulfilled.  control u corresponding to the last iteration can be seen in Figure 6(a) and the section for the time level t = 0.1 in Figure 6(b). In this case, the highest value of J is 0.7143. It can be observed that the harvesting effort takes the maximum value L when the fertility rate β is approximatively 0. In the first test, at the beginning of the time interval, the harvest effort is 0. In second test, at the beginning of the time interval, the harvest effort is 0, for a ≤ a 0 , and L, for a > a 0 . A similar behaviour can be seen at the end of the time interval for the both tests: u is 0, for a ≤ 0.2, and L, for a > 0.2 (0.2 is the initialization value for α).

4.
A separable case for (OH). An important particular case is obtained when we consider that α ≡ 0, meaning that the harvesting effort is w(t) (the intensity of fishing) and this shows that for w(t) = 0 we have prohibition for fishing and when w(t) > 0 the intensity of fishing does not depend on age (is the same for all ages). Remark that the control is u(a, t) = w(t) and the unique solution to (1) is separable and may be written as where g is the unique solution to and h is the unique solution to where G(t) = A 0 g(a, t)da, for any t ∈ [0, T ].

SEBASTIAN ANIŢ A AND ANA-MARIA MOŞNEAGU
We rediscover an optimal harvesting problem investigated in [4]: Here h w is the solution to (11). The cost functional Φ is defined on W by Let us evaluate the directional derivative for this function. Consider an arbitrary v ∈ L ∞ (0, T ) such that for sufficiently small θ > 0 we have w(t) + θv(t) ∈ [0, L] a.e. t ∈ (0, T ). This yields Let r be the solution to the following problem    r (t) = M(G(t)h w (t))r(t) + M (G(t)h w (t))G(t)r(t)h w (t) +w(t)(r(t) + G(t)), t ∈ (0, T ) r(T ) = 0.
If we multiply the first equation in (13) by y(t) and integrate over [0, T ] we get after an easy calculation (and using (12)) that Using the form of dΦ(w)(v) we obtain that The existence of an optimal control w * for (OHs) follows in a standard manner.
Here ψ = ∂Ψ is the subdifferential of Ψ, where Remark α satisfies a Signorini boundary condition. This allows to derive a conceptual algorithm to improve at any step w and α in order to obtain a bigger value for J 2 (as in [5]).