Interacting Locally Regulated Diffusions
How Migration Helps to Survive Competition
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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 attrac...
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.
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