Population dynamics from a topological point of view

Résumé

Mathematical models are important in many different disciplines since from the knowledge of an initial data it is possible to determine how the system varies in the time. In the last decades, an infinity of models have appeared to describe different biological situations. This growing interest has generated a recent discipline, Math-Biology. Math-Biology has a character multidisciplinary involving mathematicians, physicists, engineers, biologists... The importance of this topic is reflected on hundreds of analytical, numerical and experimental papers that can be consulted in the bibliographies of the monographs \cite{kot} and \cite{caswell}. Throughout this thesis we focus our attention on population dynamics. Roughly speaking, the main aim is to use mathematical models in order to study the interaction of different species sharing the same environment. More precisely, given an initial data, i.e. the number of individuous of each species at time $t=0$, our purpose is to determine the evolution of the number of individuous in the time. It would be very desirable to determine, in a explicit way, this evolution. However, in most of models, it is impossible to obtain such an analytic expression from the initial data.