Author(s) :
Jochen Blath
,
Alison Etheridge
,
Mark Meredith
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 16-2008
MSC 2000
- 60K35 Interacting random processes; statistical mechanics type models; percolation theory
-
60J80 Branching processes
-
60J85 Applications of branching processes
-
60J70 Applications of diffusion theory
-
92D25 Population dynamics
Abstract :
Note: This paper is the full version of Blath, Etheridge & Meredith
(2007). It has also successfully undergone the peer-reviewing process of Annals of Applied Probability, but proved too long to be published in its entirety. It contains full technical details and additional remarks.
We propose two models of the evolution of a pair of competing
populations. Both are lattice based. The first is a compromise
between fully spatial models, which do not appear amenable to analytic
results, and interacting particle system models, which don't, at
present, incorporate all the competitive strategies that a population
might adopt. The second is a simplification of the first in which
competition is only supposed to act within lattice sites and the total
population size within each lattice point is a constant. In a special
case, this second model is dual to a branching-annihilating random
walk.
For each model, using a comparison with oriented percolation, we show
that for certain parameter values both populations will coexist for
all time with positive probability. As a corollary we deduce survival
for all time of branching annihilating random walk for sufficiently
large branching rates.
We also present a number of conjectures relating to the role of
space in the survival probabilities for the two populations.
Keywords :
competing species, coexistence, branching annihilating random walk, interacting diffusions, regulated population, heteromyopia, stepping stone model, survival, Feller diffusion, Wright-Fisher diffusion