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

Abstract

A model of a homogeneous population given in the absence of exploitation by a differential equation x ̇=g(x) is considered. At each moment of time τ_k=kd, where d>0, k=1,2,..., some random share of the resource ω_k ∈ [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∈[0,1): then the share of the extracted resource will be l_k=l(ω_k,u)=min(ω_k,u), k=1,2,.... 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→∞. 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) = φ(d,(1-u)X_k), k=1,2,..., where φ(t,x) is a solution of the equation x ̇=g(x) satisfying the initial condition φ(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.