On how to exploit a population given by a difference equation with random parameters

A. A. Rodin ; L. I. Rodina ; A. V. Chernikova

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 2, pp. 211-227

Résumé

We consider a model of an exploited homogeneous population given by a difference equation depending on random parameters. In the absence of exploitation, the development of the population is described by the equation $$ X(k+1)=f\bigl(X(k)\bigr), k=1,2,\ldots, $$ where $X(k)$ is the population size or the amount of bioresources at time $k,$ $f(x)$ is a real differentiable function defined on $I=[0,a]$ such that $f(I)\subseteq I.$ At moments $k=1,2,\ldots$, a random fraction of the resource $\omega(k)\in\omega\subseteq[0,1]$ is extracted from the population. The harvesting process can be stopped when the share of the harvested resource exceeds a certain value of $u(k)\in[0,1)$ to keep as much of the population as possible. Then the share of the extracted resource will be equal to $\ell(k)=\min (\omega(k),u(k)).$ The average temporary benefit $H_*$ from the extraction of the resource is equal to the limit of the arithmetic mean from the amount of extracted resource $X(k)\ell(k)$ at moments $1,2,\ldots,k$ when $k\to\infty.$ We solve the problem of choosing the control of the harvesting process, in which the value of $H_*$ can be estimated from below with probability one, as large a number as possible. Estimates of the average time benefit depend on the properties of the function $f(x)$, determining the dynamics of the population; these estimates are obtained for three classes of equations with $f(x)$, having certain properties. The results of the work are illustrated, by numerical examples using dynamic programming based on, that the process of population exploitation is a Markov decision process.

Détail

Keywords: difference equations, equations with random parameters, average time profit.
Mots-clés : optimal exploitation