Author(s) :
Jochen Blath
,
Alison Etheridge
,
Mark Meredith
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 26-2006
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 :
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 r\^ole 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