Mathematics – Probability
Scientific paper
2007-03-05
Mathematics
Probability
117 pages; Ph.D. thesis (2005) at the University of British Columbia, Canada
Scientific paper
This work deals with two problems arising in mathematical ecology. The first problem is concerned with diploid branching particle models and its behavior when rapid stirring is added to the interaction. The particle models involve two types of particles, male and female, and branching can only occur when both types of particles are present. We show that if the branching rate is sufficiently large, this particle model has a nontrivial stationary distribution, i.e. one that does not concentrate all weight on the all-0 state, using a comparison argument due to R. Durrett. We also show extinction for small branching rates, thereby establishing the existence of a phase transition. We then add two different rapid stirring mechanisms to the interactions and show that for the particle models with rapid stirring, there also exist nontrivial stationary distribution(s); for this, we analyze the limiting PDE and establish a condition on the PDE that guarantees existence of nontrivial stationary distributions for sufficient fast stirring. The second problem deals with a model of sympatric speciation, i.e. speciation in the absence of geographical separation, originally proposed by U. Dieckmann and M. Doebeli in 1999. We modify their original model to obtain several constant-population particle models. We concentrate on a continuous-time model that converges to a deterministic dynamical system as the number of particles becomes large. We establish various results regarding whether speciation occurs by studying the existence of bimodal stationary distributions for the limiting dynamical system.
No associations
LandOfFree
Equilibrium States of Two Stochastic Models in Mathematical Ecology does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Equilibrium States of Two Stochastic Models in Mathematical Ecology, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Equilibrium States of Two Stochastic Models in Mathematical Ecology will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-113995