Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems.∗

Abstract

Logistic equations have been widely used in the mathematical biology literature in the modeling of the spread, over a given spatial domain, of all sorts of biological quantities (e.g. population species densities, genes), [11, 12], [5], [13]. Their success in this field is due to the ability of these equations in describing natural features of biological systems as self–production of species (such as birth and death rates) and medium limitations, or competition for resources. At the same time, spatial heterogeneities of biological characteristics of each model can be easily taken into account. A typical basic model includes diffusion, boundary conditions and a nonlinear space-dependent reaction term. At the next level of complexity, when considering seasonal or environmental influences, one must include time as an explicit variable in the problem. In other words the equations become non autonomous. In this context a widely used assumption, which may be taken as realistic in many real situations, is that of periodic time dependence in the equations; see [11] and [8]. However, non periodic (e.g. almost periodic) situations appear as relevant in realistic applications. In each of these situations a crucial question becomes that of determining ranges of parameters and typical models in which some important solutions exist. Namely, those reflecting the coexistence or, on the other hand, extinction of some of the species involved. Of course the space-time structures described by these important solutions are of capital importance in the applications. An important remark is that, due to the intrinsic nature of the quantities modeled, (population densities for example), in the applications one is only interested in considering nonnegative solutions of the equations. It is remarkable however that the dynamics of positive solutions is typically much simpler than these of arbitrary solutions, [2]. For example, in the case of autonomous problems, conditions are known implying the existence of a unique globally asymptotically positive steady state which completely describes the

Cite this paper

@inproceedings{RodguezBernal2006ExistenceUA, title={Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems.∗}, author={Ańıbal Rodŕıguez-Bernal and Alejandro Vidal-L{\'o}pez}, year={2006} }