Problems of optimal periodic resource harvesting for population models described by difference equations

We consider models of homogeneous or structured (by type, age, or other charac\-te\-ris\-tic) populations, the dynamics of which, in the absence of exploitation, is given by a system of difference equations
$x(k+1) = F\big(k, x(k)\big),$
where $x(k) = \big(x_1(k), \ldots, x_n(k)\big),$ $x_i(k),$\linebreak $i=1,\ldots,n$ is the amount of the $i$-th type or age class of the population at a time\linebreak $k=0,1,2,\ldots;$ $F(k,x)=\bigl(F_1(k,x), \ldots, F_n(k,x)\bigr),$ $F_i(k,x)$ are real functions that are defined and continuous on the set$\mathbb{R}^n_+ \doteq\big\{x\in\mathbb{R}^n : x_1\geqslant0, \ldots, x_n\geqslant0\big\}.$

It is assumed that at times $k=1, 2, \ldots$ the population is exposed to harvesting $u(k)=(u_1(k),\ldots,u_n(k))\in[0, 1]^n.$ Then the model of the exploited population is investigated, given by a system of difference equations
$<$
X(k+1) = F\bigl(k,(1-u(k))X(k)\bigr), \quad k=1, 2, \ldots,

where $X(k)=\big(X_1(k),\ldots,X_n(k)\big),$ $(1-u(k))X(k)=\big((1-u_1(k))X_1(k),\ldots,(1-u_n(k))X_n(k)\big),$ $X_i(k)$ and
$(1-u_i(k))X_i(k)$ is the amount of the resource of the $i$ type before and after harvesting at the time $k$ respectively, $i=1,\ldots,n.$

The problem of optimal harvesting of a renewable resources for an unlimited period of time under periodic operation mode, in which the highest values of collection characteristics are achieved, is investigated. The first of these characteristics is the average time profit given by the limit at $k\to\infty$ of the arithmetic mean of the cost of the resource over $k$ harvesting. Another characteristic is the harvesting effciency equal to the limit at $k\to\infty$ of the ratio of the cost of the resource gathered in $k$ harvestings to the amount of applied control (collection efforts). The results of the work are illustrated by examples of a homogeneous exploited population, given by a discrete logistic equation, and a structured population of two species.