Properties of the average time benefit for probabilistic models of exploited populations

A model of a homogeneous population given in the absence of exploitation by a differential equation $\dot x=g(x)$ is considered. At each moment of time $\tau_k=kd,$ where $d>0,$ $k=1,2,\ldots,$ some random share of the resource $\omega_k\in [0,1]$ is extracted from this population. We assume that it is possible to stop the harvesting if its share turns out to be greater than a certain value $u\in [0,1):$ then the share of the extracted resource will be $\ell_k=\ell(\omega_k,u)=\min(\omega_k,u),$ $k=1,2,\ldots.$ The average time benefit from resource extraction is investigated, it is equal to the lower limit of the arithmetic amount of the resource obtained in $n$ extractions as $n\to\infty$. It is shown that the properties of this characteristic are associated with the presence of a positive fixed point of the difference equation $X_{k+1}=\varphi\bigl(d,(1-u)X_{k}\bigr),$ $k=1,2,\ldots,$ where $\varphi(t,x)$ is a solution of the equation $\dot x=g(x)$ satisfying the initial condition $\varphi(0,x)=x.$ The conditions for the existence of the limit and the estimates of the average time benefit performed with probability one are obtained. The results of the work are illustrated by examples of exploited homogeneous populations depending on random parameters.