Positive periodic solutions and optimal control for a distributed biological model of two interacting species

Abstract

The paper deals with the existence of positive periodic solutions to a system of degenerate parabolic equations with delayed nonlocal terms and Dirichlet boundary conditions. Taking in each equation a meaningful function as a control parameter, we show that for a suitable choice of a class of such controls we have, for each of them, a time-periodic response of the system under different assumptions on the kernels of the nonlocal terms. Finally, we consider the problem of the minimization of a cost functional on the set of pairs: controlperiodic response. The considered system may be regarded as a possible model for the coexistence problem of two biological populations, which dislike crowding and live in a common territory, under different kind of intra-and inter-specific interferences.

Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

“…In the very recent paper [60], the authors replace the nonlocal terms of (1.1) by Ω K i (ξ, t)u(ξ, t)dξ, for i = 1, 3, and Ω K i (ξ, t)v(ξ, t)dξ, for i = 2, 4. By means of local conditions, different from those proposed in [25,26], the authors obtain the coexistence of the two species via a similar topological approach when p, q ≥ 2, m, n ≥ 1 and, thus, only the slow and normal diffusion occurs, i.e. m(p − 1) ≥ 1, n(q − 1) ≥ 1.…”

“…The aim of this paper is to extend the results of [25] and [26], concerning the existence of non-negative, non-trivial periodic solutions, to a system of singular-degenerate parabolic equations. To the best of our knowledge, this is the first result for the case when 1 < p, q < 2, m > p and n > q, also in the case of a single equation.…”

“…To the best of our knowledge, this is the first result for the case when 1 < p, q < 2, m > p and n > q, also in the case of a single equation. We recall that the cases p, q > 2, m, n > 1 and p = q = 2, m, n > 1 were treated respectively in [25] and [26], see also [21] and [22] for a system of anisotropic (p(x), q(x))-Laplacian parabolic equations, with p(x), q(x) > 2 in Ω, and m = n = 1. In the very recent paper [60], the authors replace the nonlocal terms of (1.1) by Ω K i (ξ, t)u(ξ, t)dξ, for i = 1, 3, and Ω K i (ξ, t)v(ξ, t)dξ, for i = 2, 4.…”

“…m(p − 1) ≥ 1, n(q − 1) ≥ 1. More precisely, models for the interaction between two biological species sharing the same isolated territory, with the interactions represented by means of the kernels K i , i = 1, 2, 3, 4, were considered in related systems of doubly degenerate parabolic equations in [25], [60] and in systems of porous medium equations in [26]. On the other hand, some previous biological models found in the literature, see e.g.…”

“…[1], [2], [51], [52] involve the p-Laplacian with p > 1 (and m = 1). Furthermore, we observe that the equations of the system we consider treat all the possible types of diffusion: slow, normal and fast, while in [25], [60] and [26] only the slow and normal diffusions were presented. In fact, as it can be easily checked, if p ∈ 1,…”