A spatial generalization of the Ricker model and the break of chaos with applications.

摘要

The Ricker model is a well-studied discrete model of population dynamics given through the map xn +1 = xner(1- xn). It is known that this map is chaotic for large r, but that the addition of a constant perturbation will induce a series of period-doubling reversals until at last there is a stable 2-cycle for large r. In this thesis we propose a spatial Ricker model on a discrete lattice. The interactions are such that each cell is influenced only by itself and its nearest neighbours at the previous stage. Under the influence of a constant perturbation we find that there is no chaos for sufficiently large r, and that in place of chaos there may be two kinds of points: points which have 2-cycle dynamics, and points which exhibit nearly stable dynamics. That is, while other points are in 2-cycle, these points only ever have values very close to the lower phase of this 2-cycle. We also consider the case of a negative perturbation, as well as a few others, one of which is of spatial nature and unique to this problem. We also make a few modifications to the model to simulate environmental biases.;Another problem of great interest is: under what conditions does a discrete map with a constant perturbation exhibit this 'break of chaos' behaviour? We study this problem by focusing on unimodal functions that relate to population dynamics. For all of the models we study, we observe 2-cycle dynamics for large enough choice of growth parameter (and 3-cycle dynamics for a special case). We outline the consequences of this study to the field of population dynamics, and we mention applications to synchronization and cellular automata.