Εφαρμογές της θεωρίας δυναμικών συστημάτων στη δυναμική βιολογικών πληθυσμών

Abstract

The purpose of this thesis is the application of the theory of dynamical systems to biological populations which obey the Allee e®ect and their dy- namics can be described by discrete maps. In chapter 1 we give some elements of the theory of dynamical systems, i.e. the de?nition and the classi?cation of ?xed points of a map, the bifurcations points and some types of bifurcations. In chapter 2 we present some elements of population dynamics. Starting from the de?nition of the population and the basic equation for a biological species, we refer to the exponential model and the logistic map. Next, we show the necessity of existence of the Allee e®ect at some populations. Then, we study models of two populations, e.g. the Lotka { Volterra model and metapopulation models, e.g. the Levins model. In chapter 3 we consider a chain of identical sites appropriate for colo- nization of a biological species and we use a map of third degree, where the points 0 and 1 are asymptotically stable ?xed points. We observe three dif- ferent behaviors, the source { sink e®ect, the rescue e®ect and the extinction of the metapopulation. Also, we study the wavefront of the invasion of a biological species to new territory and the e®ect of the heterogeneity of sites. In chapter 4 we consider once again a chain of identical sites and a map of ?fth degree where the point 0 is a asymptotically stable ?xed point, while the behavior for larger values of the population varies with respect to a the dispersal probability. In chapter 5, in order to study the combined in°uence of predation and the Allee e®ect, we construct a model of a specialist predator (which dynamics is of the logistic type) and its prey (which obeys the Allee e®ect). We observe the paradox of enrichment and also the fact that predation excludes the appearance of the rescue e®ect.