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

Abstract

We shall show that for a given control u there exists a non-trivial stationary state in the dynamics of a size-structured population with hierarchical competition. More precisely, we assume that the dynamics is described by the equationwhere x(t, l) is the density of individuals of size l at time t, the functions g and µ characterize the growth and mortality of these individuals for level of competition E, and the control u characterizes the exploitation of the population. The competition function E is defined bywhere χ is a continuous function which is positive for positive values of the argument, and the interval [0, L] with L > 0 is the range of sizes on which the population is exploited. 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 formand it is responsible for the influx of individuals of size l = 0: this influx is a combination of natural reproduction and industrial regeneration p0 = const 0. The function r and the exponent β ∈ (0, 1) reflect the reproductiveness of the biomass and the non-linear dependence of the reproduction on the density, respectively. The model (1)-(3) is similar to well-known models (see, for instance, [4], [5]). We assume that all the functions involved, apart from u, are continuous and satisfy the following natural conditions: (a) for each l ∈ [0, L] the functions g and r are non-increasing with respect to the level of competition E, the function g is positive everywhere, and r is non-negative everywhere and positive on some non-empty subinterval of [0, L];(b) µ is a positive function which is non-decreasing with respect to E for each l ∈ [0, L]; (c) for 0 l1 < l2 L the ratio g(l1, · )/g(l2, · ) is non-increasing with respect to E. The conditions (a)-(c) express that the development of the population does not improve with an increase of competition and that this increase does not affect smaller-sized individuals any less than larger-sized ones.By an admissible measurable control u we mean one satisfying the condition 0 u1 u u2 on [0, L], where u1 and u2 are some piecewise continuous functions.