About existence of the limit to the average time profit in stochastic models of harvesting a renewable resource

We investigate population dynamics models given by difference equations with stochastic parameters. In the absence of harvesting, the development of the population at time points $k=1,2,\ldots$ is given by the equation $X(k+1)=f\big(X(k)\big),$ where $X(k)$ is amount of renewable resource, $f(x)$ is a real differentiable function. It is assumed that at times $k=1,2,\ldots$ a random fraction $\omega\in[0,1]$ of the population is harvested. The harvesting process stops when at the moment $k$ the share of the collected resource becomes  greater than a certain value $u(k)\in[0,1),$ in order to save a part of the population for reproduction and to increase the size of the next harvest. In this case, the share of the extracted resource is equal to $\ell(k)=\min\big\{\omega(k),u(k)\big\}, k=1,2,\ldots.$ Then the model of the exploited population has the form
$X(k+1)=f\left(\right(1-l\left(k\right) \left)X\right(k) ),k=1,2,...,$
where $x(0)$ is the initial population size.
For the stochastic population model, we study the problem of choosing a control $\overline{u}=(u(1),\ldots,u(k),\ldots)$ that limits at each time moment $k$ the share
of the extracted resource and under which the limit of the average
time profit function
$H(\overline{l},x(0\left)\right)\doteq \underset{n\to \infty }{lim}\sum _{k=1}^{n}X\left(k\right)l\left(k\right), где \overline{l}\doteq \left(l\right(1),\dots ,l(k),\dots )$
exists and can be estimated from below with probability one
by as a large number as possible.
If the equation $X(k+1)=f\big(X(k)\big)$ has a solution of the form $X(k)\equiv x^*,$
then this solution is called the equilibrium position of the equation.
For any $k=1,2,\ldots,$ we consider random variables $A(k+1,x)=f\bigl((1-\ell(k))A(k,x)\bigr),$ $B(k+1,x^*)=f\bigl((1-\ell(k))B(k,x^*)\bigr)$; here $A(1,x)=f(x),$  $B(1,x^*)=x^*.$
It is shown that when certain conditions are met, there exists a control $\overline{u}$
under which there holds the estimate of the average time profit
$\frac{1}{m}\sum _{k=1}^{m}M\left(A\right(k,x\left)l\right(k\left)\right)\le H(\overline{l},x(0\left)\right)\le \frac{1}{m}\sum _{k=1}^{m}M\left(B\right(k,x^{*}\left)l\right(k\left)\right),$
where $M$ denotes the mathematical expectation.
In addition, the conditions for the existence of control $\overline{u}$ are obtained
under which there exists, with probability one, a positive limit to the
average time profit equal to
$H(\overline{\ell},x(0)) =$
\lim\limits_{k\to\infty} MA(k,x)\ell(k) =
\lim\limits_{k\to\infty} MB(k,x^*)\ell(k).