Optimal Stationary Solution in Forest Management Model by Accounting Intra-Species Competition

Abstract

We consider a model of exploitation of a size-structured population when the birth, growth and mortality rates depend on the individual size and interspecies competition, while the exploitation intensity is a function of the size only. For a given exploitation intensity and under natural assumptions on the rates, we establish existence and uniqueness of a nontrivial stationary state of the population. In addition, we prove existence of an exploitation intensity which maximizes a selected profit functional of exploitation.

On a stationary state in the dynamics of a population with hierarchical competition

“…In accordance with this formula, the development of individuals of a given size is only affected by the individuals of the same size or larger. This distinguishes our paper from [1]- [3], where the integral was taken over the whole of [0, L] and the level of competition was the same for all sizes of individuals. The boundary condition has the form…”

Optimization of size-structured population with interacting species

“…The dynamics of size-structured population with n interacting species is governed by ∂x i (t, l) ∂t + ∂ g i (l, E(t))x i (t, l) ∂l = − μ i (l, E(t)) + u i (l) x i (t, l), (1) where x i (t, l) is the density of individuals of size l in the ith species, i = 1, 2, . .…”

“…. n, at time t, g i and μ i are intensity exponents of the growth and mortality rates for this species respectively, and the control u i determines the exploitation intensity [1]- [3]. The function E characterizes the total level of interspecies competition and is represented by the formula…”