Formação De Padrões Espaciais Na Dinâmica De Populações

Abstract

Colombo, Eduardo H.; Anteneodo, Celia (advisor). Spatial pattern formation in population dynamics. Rio de Janeiro, 2014. 83p. Dissertação de Mestrado -Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro.Motivated by the richness of phenomena produced by living beings, this work aims to study the formation of spatial patterns in biological populations. From the mesoscopic point of view, we define the basic processes that may occur in the dynamics, building a partial differential equation for the evolution of the population distribution. This equation incorporates two generalizations of a pre-existing model for the dynamics of one species, which takes into account long-range (nonlocal) interactions. The first generalization is to consider that diffusion is nonlinear, i.e., it is affected by the local density such that the diffusion coefficient follows a power law. On the other hand, because of the high complexity involved in the nature of model parameters, we introduced as a second generalization time-fluctuating parameters. We idealize these fluctuations as Gaussian temporally uncorrelated (white) noises. To study the resulting model, we use an analytical and numerical approach. Analytical tools are based on the linearization of the evolution equation and are therefore approximate. However, as evidenced by numerical results, we draw important conclusions. The nonlocal feature of the interaction is the main mechanism which induces pattern formation. We show that the extent of these interactions is what characterizes the dominant mode. Thus, for parameter values above a critical threshold patterns emerge. Analytically, we also show that even below this threshold, fluctuations in the parameters can induce the appearance of spatial order. The effects of nonlinear diffusion are only superficially captured by the linear analysis. Numerically, we show that their presence modifies the patterns shape. We mainly observed the existence of a qualitative difference between the cases when diffusion is facilitated or not by high densities. In the first case, we note that the patterns become fragmented, that is, population becomes composed of disconnected clusters.