About existence of the limit to the average time profit in stochastic models of harvesting a renewable resource
- 1 January 2022
- journal article
- Published by Tambov State University - G.R. Derzhavin in Russian Universities Reports. Mathematics
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).This publication has 17 references indexed in Scilit:
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