Estudo analítico da equação de Fisher linearizada: determinação de tamanhos mínimos de fragmentos populacionais

Abstract

The study of population dynamics is a frequent topic in mathematical biology. The most important points to consider in these type of simulations are the growth behaviour of a specific population over time and the description of the spatial aspects. The addressed problem in this work is to describe a population moving in space subject to a growth rate contained by a saturation. A dynamic one-dimensional Fickian diffusion model was adopted. In order to introduce discontinuity in the model, described by the Fisher-Komolgorov-Petroviski-Piskunov (FKPP) equation, some spatial heterogeneities were added. In biological terms, this discontinuity can be understood as a space fragmentation. The aim of this work is to determine the minimum critical size that a fragment must have so it can be prosperous in systems of one or two fragments, all of the same size. It was found that the minimum critical size depends on the number of fragments, the degree of life favorability, as well as the distance between the fragments. It was also found that the higher the isolation from the external environment , the higher the critical size of each fragment in case of isolation.