Estimating the Average Temporal Benefit in Probabilistic Environmental–Economic Models

Abstract

This paper considers environmental–economic models of optimal harvesting, given by impulsive differential equations, which depend on random parameters. We assume that lengths of intervals \({{\theta }_{k}}\) between the moments of impulses \({{\tau }_{k}}\) are random variables and the sizes of impulse influence depend on random parameters \({{{v}}_{k}},\)\(k = 1,2, \ldots \) One example of such objects is an impulsive equation that simulates dynamics of the population subject to harvesting. In the absence of harvesting, the population development is described by the differential equation \(\dot {x} = g(x),\) and in time points \({{\tau }_{k}}\) some random share of resource \({{{v}}_{k}},\)\(k = 1,2, \ldots \) is taken from the population. We can control the gathering process so that to stop harvesting when its share will appear big enough to save the biggest possible remaining resource to increase the size of the next gathering. Let the equation \(\dot {x} = g(x)\) have an asymptotic stable solution \(\varphi (t) \equiv K,\) whose attraction region is the interval \(({{K}_{1}},{{K}_{2}}),\) where \(0 \leqslant {{K}_{1}} < K < {{K}_{2}}.\) We construct the control \(u = ({{u}_{1}}, \ldots ,{{u}_{k}}, \ldots ),\) which limits a share of the harvested resource at each moment of time \({{\tau }_{k}}\) so that the quantity of the remaining resource, since some moment \({{\tau }_{{{{k}_{0}}}}},\) would be not less than the given value \(x \in ({{K}_{1}},K).\) For any \(x \in ({{K}_{1}},K)\) the estimates of the average temporal benefit, valid with the probability of one, are obtained. It is shown that there is a unique \(x* \in ({{K}_{1}},K),\) at which the lower estimate reaches the greatest value. Thus, we described the way of population control at which the value of the average time profit can be lower estimated with the probability of one by the greatest number whenever possible.