On Estimation of an Average Time Profit in Probabilistic Environmental and Economic Models

Abstract

We consider environmental-economical models of optimal harvesting, given by the differential equations with impulse action, which depend on random parameters. We assume, that lengths of intervals θ_k between the moments of impulses τ_k are random variables and the sizes of impulse influence depend on random parameters v_k, k = 1, 2, . . . One example of such objects is an equation with impulses, modelling dynamics of the population subject to harvesting. In the absence of harvesting, the population development is described by the differential equation ˙x = g(x) and in time moments τk some random share of resource v_k, k = 1, 2, . . . is taken from population. We can control gathering process so that to stop harvesting when its share will appear big enough to keep possible biggest the rest of a resource to increase the size of the following gathering. Let the equation ˙x = g(x) have an asymptotic stable solution ϕ(t) ≡ K and the interval (K₁, K₂) is the attraction area of the given solution (here 0 ≤ K₁ < K < K₂). We construct the control u = (u₁, . . . , u_k, . . .), limiting a share of harvesting resource at each moment of time τk, so that the quantity of the remained resource, since some moment τk0 , would be not less than the given value x ∈ (K₁, K). For any x ∈ (K₁, K) the estimations of average time profit, valid with probability one, are received. It is shown, that there is a unique x^∗ ∈ (K₁, K), at which the lower estimation reaches the greatest value. Thus, we described the way of population control at which the value of average time profit can be lower estimated with probability 1 by the greatest number whenever possible.