Polynomial profits in renewable resources management
Introduction
A biological resource is grown to provide an economical profit. Up to a fixed age , this population consists of juveniles whose density at time and age satisfies the usual renewal equation [1, Chapter 3] and 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 , each individual of the 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 and the populations whose evolution is described by the renewal equations with obvious meaning for the functions . Here, the selection procedure is described by a parameter , varying in , which quantifies the percentage of the population directed to the market, so that The overall dynamics is completed by the description of reproduction, which we obtain here through the usual age dependent fertility function using the following nonlocal boundary condition 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 population at given stages of its development. More precisely, we assume there are fixed ages , with , where the fractions of the population are kept, while the portions are sold. A very natural choice is to set , meaning that nothing is left unsold after age . The dynamics of the whole system has then to be completed introducing the selection that takes place at the age , for .
Summarizing, the dynamics of the structured population is thus described by the following nonlocal system of balance laws, see also Fig. 1 : where we inserted the initial data .
Our key result is the proof that for all and all , the quantities , and are polynomial in the values attained by the control parameters and .
We now pass to the introduction of the expressions of cost and income. To this aim, we first fix a time horizon , with . Then, a reasonable expression for the income is The latter term above is the sum of the incomes due to the selling of the individuals at the ages . Typically, each value function 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 , accounts for the total amount of the population at time and it can also be seen as the capital consisting of the biological resource at time . Neglecting this term obviously leads to optimal strategies that leave no juveniles at the final time . The value function is also assumed to be polynomial, see Section 3.3.
To model the various costs, we use a general integral functional of the form The cost functions , for , are assumed to be polynomial in , for all and . In the simplest case of linear cost and income, (1.2) and (1.3) reduce to Here, is the unit value of juveniles of age , while is the price at time per each individual of the population sold at maturity . Similarly, the quantity , for , is the unit cost related to the keeping of individuals of the population , of age , at time .
Below, we provide the essential tools to establish effective numerical procedures able to actually compute the profit as a function of the (open loop) control parameters and . In particular, this also allows to find choices of the time dependent control parameters and that allow to maximize . Moreover, the procedures presented below provide an alternative to the use of bang –bang 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 , while is the usual characteristic function of the set , so that if and only if , whereas vanishes outside . The positive integers and are fixed throughout, as also the positive strictly increasing real numbers , . It is also of use to introduce the real intervals , , and .
Below, for a real valued function defined on an interval , we call its total variation, while 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 all the population is sold; this corresponds to the case .
Technical details
As in [1], [2], [3], we recall that the initial – boundary value problem for the renewal equation admits a unique solution that can be explicitly computed integrating along characteristics as where the maps and , with and , are defined as
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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