Polynomial profits in renewable resources management

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Abstract

A system of renewal equations on a graph provides a framework to describe the exploitation of a biological resource. In this context, we formulate an optimal control problem, prove the existence of an optimal control and ensure that the target cost function is polynomial in the control. In specific situations, further information about the form of this dependence is obtained. As a consequence, in some cases the optimal control is proved to be necessarily bang–bang, in other cases the computations necessary to find the optimal control are significantly reduced.

Introduction

A biological resource is grown to provide an economical profit. Up to a fixed age ā, this population consists of juveniles whose density J(t,a) at time t and age a satisfies the usual renewal equation [1, Chapter 3] tJ+agJ(t,a)J=dJ(t,a)Ja[0,ā], gJ and dJ being, respectively, the usual growth and mortality functions, see also [2], [3], [4]. For further structured population models, we refer for instance to [5], [6], [7], [8], [9].

At age ā, each individual of the J population is selected and directed either to the market to be sold or to provide new juveniles through reproduction. Correspondingly, we are thus lead to consider the S and the R populations whose evolution is described by the renewal equations tS+agS(t,a)S=dS(t,a)StR+agR(t,a)R=dR(t,a)Raā,with obvious meaning for the functions gS,gR,dS,dR. Here, the selection procedure is described by a parameter η, varying in [0,1], which quantifies the percentage of the J population directed to the market, so that gS(t,ā)S(t,ā)=ηgJ(t,ā)J(t,ā)gR(t,ā)R(t,ā)=(1η)gJ(t,ā)J(t,ā).The overall dynamics is completed by the description of reproduction, which we obtain here through the usual age dependent fertility function w=w(a) using the following nonlocal boundary condition gJ(t,0)J(t,0)=ā+w(α)R(t,α)dα.In this connection, we recall the related results [10], [11], [12] in structured populations that take into consideration a juvenile–adult dynamics.

Once the biological evolution is defined, we introduce the income and cost functionals as follows. The income is related to the withdrawal of portions of the S population at given stages of its development. More precisely, we assume there are fixed ages ā1,,āN, with ā<ā1<ā2<<āN, where the fractions ϑ1,,ϑN of the S population are kept, while the portions (1ϑ1),,(1ϑN) are sold. A very natural choice is to set ϑ̄N0, meaning that nothing is left unsold after age āN. The dynamics of the whole system has then to be completed introducing the selection S(t,āi+)=ϑiS(t,āi)that takes place at the age āi, for i=1,,N.

Summarizing, the dynamics of the structured (J,S,R) population is thus described by the following nonlocal system of balance laws, see also Fig. 1 : tJ+agJ(t,a)J=dJ(t,a)J(t,a)R+×[0,ā]tS+agS(t,a)S=dS(t,a)S(t,a)R+×[ā,+[{ā1,,āN}tR+agR(t,a)R=dR(t,a)R(t,a)R+×[ā,+[gS(t,ā)S(t,ā)=ηgJ(t,ā)J(t,ā)tR+gR(t,ā)R(t,ā)=(1η)gJ(t,ā)J(t,ā)tR+gJ(t,0)J(t,0)=ā+w(α)R(t,α)dαtR+S(t,āi+)=ϑiS(t,āi)tR+,i=1,,NJ(0,a)=Jo(a)a[0,ā]S(0,a)=So(a)a[ā,+[R(0,a)=Ro(a)a[ā,+[,where we inserted the initial data (Jo,So,Ro).

Our key result is the proof that for all t and all a, the quantities J(t,a), S(t,a) and R(t,a) are polynomial in the values attained by the control parameters η and ϑ.

We now pass to the introduction of the expressions of cost and income. To this aim, we first fix a time horizon T, with T>0. Then, a reasonable expression for the income is I(η,ϑ;T)=0āPa,J(T,a)da+i=1N0TPit,1ϑi(t)S(t,āi)dt.The latter term above is the sum of the incomes due to the selling of the S individuals at the ages ā1,,āN. Typically, each value function sPi(t,s) can be chosen linear in its second argument, but the present framework applies also to the more general polynomial case. The former term in the right hand side of (1.2), namely 0āPa,J(T,a)da, accounts for the total amount of the J population at time T and it can also be seen as the capital consisting of the biological resource at time T. Neglecting this term obviously leads to optimal strategies that leave no juveniles at the final time T. The value function jP(t,j) is also assumed to be polynomial, see Section 3.3.

To model the various costs, we use a general integral functional of the form C(η,ϑ;T)=0T0āCJt,a,J(t,a)dadt+0Tā+CSt,a,S(t,a)dadt+0Tā+CRt,a,R(t,a)dadt.The cost functions wCu(t,a,w), for uJ,S,R, are assumed to be polynomial in w, for all a and t. In the simplest case of linear cost and income, (1.2) and (1.3) reduce to I(η,ϑ;T)=0āp(a)J(T,a)da+i=1N0Tpi(t)1ϑi(t)S(t,āi)dt.C(η,ϑ;T)=0T0ācJ(t,a)J(t,a)dadt+0Tā+cS(t,a)S(t,a)dadt+0Tā+cR(t,a)R(t,a)dadt.Here, p(a) is the unit value of juveniles of age a, while pi(t) is the price at time t per each individual of the population S sold at maturity āi. Similarly, the quantity cu(t,a), for u{J,S,R}, is the unit cost related to the keeping of individuals of the population u, of age a, at time t.

Below, we provide the essential tools to establish effective numerical procedures able to actually compute the profit P(η,ϑ;T)=I(η,ϑ;T)C(η,ϑ;T)as a function of the (open loop) control parameters η and ϑ. In particular, this also allows to find choices of the time dependent control parameters η and ϑ that allow to maximize P. Moreover, the procedures presented below provide an alternative to the use of bangbang controls. For a comparison between the two techniques we refer to Section 3.3.

The next section presents the main results of this note, while specific examples are deferred to paragraphs 3.1, 3 Examples, 3 Examples. All analytic proofs are in Section 4.

Section snippets

Main results

Throughout we denote R+=0,+, while χA is the usual characteristic function of the set A, so that χA(x)=1 if and only if xA, whereas χA vanishes outside A. The positive integers κ,m and N are fixed throughout, as also the positive strictly increasing real numbers ā, ā1,,āN. It is also of use to introduce the real intervals IJ=[0,ā], IS=IR=ā,+, and IT=[0,T].

Below, for a real valued function u defined on an interval I, we call TV(u) its total variation, while BV(I;R) is the set of real

Examples

The examples in paragraphs 3 Examples, 3 Examples rely on several numerical integrations of (1.1). They were accomplished using the explicit formula (4.2). To compute the gains and the costs (1.2) –(1.3), we used the standard trapezoidal rule.

For simplicity, we assume throughout that at age āN all the population S(t,āN) is sold; this corresponds to the case ϑN0.

Technical details

As in [1], [2], [3], we recall that the initial – boundary value problem for the renewal equation tu+agu(t,a)u=du(t,a)uu(0,a)=uo(a)gu(t,au)u(t,au+)=b(t)t0aauadmits a unique solution that can be explicitly computed integrating along characteristics as u(t,a)=uoAu(0;t,a)ψu(0,t,a)aAu(t;0,au)bTu(au;t,a)guTu(au;t,a),auψuTu(au;t,a),t,aa<Au(t;0,au),where the maps tAu(t,to,ao) and aTu(a;to,ao), with tR+ and a,aoIu, are defined as tAu(t;to,ao) is the solution toȧ=gu(t,a)a(to)=ao andaTu(a;to,a

Acknowledgments

This work was partially supported by the 2015–INDAM–GNAMPA project Balance Laws in the Modeling of Physical, Biological and Industrial Processes. The first author acknowledges the support of the CaRiPLo foundation, project 2013-0893.

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