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
$$X(k+1)=f\big((1-\ell(k))X(k)\big), \ \ \, k=1,2,\ldots,$$
where $X(1)=f\big(x(0)\big),$ $x(0)$ 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
$$H\bigl(\overline{\ell},x(0)\bigr) \doteq\displaystyle{\lim_{n\to\infty}\,\dfrac{1}{n}\,\sum_{k=1}^{n}X(k)\ell(k)}, \ \ \, \text{where} \ \,\, \overline{\ell}\doteq(\ell(1),\ldots,\ell(k),\ldots),$$
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
$$\dfrac{1}{m}\sum\limits_{k=1}^{m} M\bigl(A(k,x)\ell(k)\bigr) \leqslant H(\overline{\ell},x(0)) \leqslant \dfrac{1}{m}\sum\limits_{k=1}^{m} M\bigl(B(k,x^*)\ell(k)\bigr),$$
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
$$H(\overline{\ell},x(0)) = \lim\limits_{k\to\infty} MA(k,x)\ell(k) = \lim\limits_{k\to\infty} MB(k,x^*)\ell(k).$$