Coexistence and optimal control problems for a degenerate predator–prey model

Abstract

In this paper we present a predator-prey mathematical model for two biological populations which dislike crowding. The model consists of a system of two degenerate parabolic equations with nonlocal terms and drifts. We provide conditions on the system ensuring the periodic coexistence, namely the existence of two non-trivial non-negative periodic solutions representing the densities of the two populations. We assume that the predator species is harvested if its density exceeds a given threshold. A minimization problem for a cost functional associated with this process and with some other significant parameters of the model is also considered.

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

“…Similar topological methods are also employed to a great extent for the existence of non-negative periodic solutions of degenerate and doubly degenerate parabolic equations, see [3], [9], [20], [30], [31], [38], [42], [44], [45], [48], [55], [56], [58], [59], [60], [61], [62], [63], [64], [67], [68]. Nonlocal models to study aggregation in biological systems with degenerate diffusion are proposed in several papers, see the recent [12], [43] and the references therein.…”

“…[3], [5], [6], [7], [21], [22], [25], [26], [29], [31], [35], [42], [50], [56], [57], [60], [67], [68]. We also recall the related problems faced in [23] and [24] also for higher order operators, and in [19] for p = 2 and N = 1.…”

Coexistence periodic solutions of a doubly nonlinear parabolic system with Neumann boundary conditions

“…In recent years, periodic problems for degenerate parabolic equations with homogeneous Dirichlet boundary conditions have been the subject of intensive study, see for example [10][11][12][13][14][15][16] and references therein. In particular, authors of [16] proved the existence of the coexistence solutions of problem (1.1) with no-flux boundary conditions replaced by homogeneous Dirichlet boundary conditions under some conditions, which involves the first eigenvalue of the related eigenvalue problem.…”

“…As illustrated in [23], the presence of nonlocal terms seems to render upper-lower solution arguments difficult to apply. In this paper, we approach the existence of the coexistence periodic solutions of problem (1.1) by the Leray-Schauder degree, which has even been used in [12,15,20,23] for periodic problems with Dirichlet boundary conditions. It is noted that in our present situation, the standard regularization of problem (1.1) as in [12,15,23] is invalid.…”

“…In this paper, we approach the existence of the coexistence periodic solutions of problem (1.1) by the Leray-Schauder degree, which has even been used in [12,15,20,23] for periodic problems with Dirichlet boundary conditions. It is noted that in our present situation, the standard regularization of problem (1.1) as in [12,15,23] is invalid. In fact, for the well-posedness of the regularized problem under the Neumann boundary condition, it usually requires the regularized parabolic equations to have some kind of the coercive property.…”