Interacting Locally Regulated Diffusions

In naturally reproducing populations one usually encounters anaverage number of more than one offspring per individual. However,given non-extinction, classical super-critical branching processesgrow beyond all bounds. This is unrealistic because of boundedresources. We consider a model for a population withsubpopulations living in separated regions and with migrationbetween the regions. In addition to the births and deathsin a super-critical branching mechanism, there are deaths resultingfrom competition between any two individuals in one subpopulation.We prove that there is exactly one attractive equilibrium distributionand that the system starting in any nontrivial initial measureconverges to this equilibrium distribution. The interpretation of thisresult is that the population stabilizes in the equilibrium state aftersome time whenever external events such as natural catastrophesdecimate the population. Furthermore, we establish a criterion onthe parameters for local extinction of the population.This book is written for mathematicians who are interested inpopulation models with competition ore more generally inpopulation dynamics.

Autorentext

Dr. phil. nat. Martin Hutzenthaler, 2004 diploma in mathematics at the University of Erlangen- Nürnberg, 2007 Ph. D. at the University of Frankfurt/Main.

Klappentext

In naturally reproducing populations one usually encounters an average number of more than one offspring per individual. However, given non-extinction, classical super-critical branching processes grow beyond all bounds. This is unrealistic because of bounded resources. We consider a model for a population with subpopulations living in separated regions and with migration between the regions. In addition to the births and deaths in a super-critical branching mechanism, there are deaths resulting from competition between any two individuals in one subpopulation. We prove that there is exactly one attractive equilibrium distribution and that the system starting in any nontrivial initial measure converges to this equilibrium distribution. The interpretation of this result is that the population stabilizes in the equilibrium state after some time whenever external events such as natural catastrophes decimate the population. Furthermore, we establish a criterion on the parameters for local extinction of the population. This book is written for mathematicians who are interested in population models with competition ore more generally in population dynamics.