Description

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.