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

Abstract

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,... is given by the equation X(k+1)=f(X(k)), where X(k) is amount of renewable resource, f(x) is a real differentiable function. It is assumed that at times k=1,2,... a random fraction ω∈[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)∈[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 l(k)=min{ω(k), u(k)}, k=1,2,.... Then the model of the exploited population has the form X(k+1)=f((1-l(k))X(k)), k=1,2,..., where X(1)=f(x(0)). For the stochastic population model, we study the problem of choosing a control u=(u(1), …, u(k),…), 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((l)̅, x(0))≐lim┬(n→∞)⁡〖∑^n_k=1▒〖X(k)l(k), где 〗〗(l)̅≐(l(1),…, l(k),…) exists and can be estimated from below with probability one by as a large number as possible. If the equation X(k+1)=f(X(k)) has a solution of the form X(k)≡x^*, then this solution is called the equilibrium position of the equation. For any k=1,2,..., we consider random variables A(k+1,x)=f((1-l(k))A(k,x)), B(k+1,x^*)=f((1-l(k))B(k,x^*)); here A(1,x)=f(x), B(1,x^*)=x^*. It is shown that when certain conditions are met, there exists a control u under which there holds the estimate of the average time profit 1/m∑^m_k=1▒〖M(A(k,x)l(k))≤H((l)̅, x(0))≤1/m∑^m_k=1▒〖M(B(k,x^*)l(k)),〗〗 where M denotes the mathematical expectation. In addition, the conditions for the existence of control u are obtained under which there exists, with probability one, a positive limit to the average time profit equal to H((l)̅, x(0))=lim┬(k→∞)⁡〖MA(k,x)l(k)=〗lim┬(k→∞)MB(k,x^*)l(k).