119 research outputs found
Interacting diffusions and trees of excursions: convergence and comparison
We consider systems of interacting diffusions with local population
regulation. Our main result shows that the total mass process of such a system
is bounded above by the total mass process of a tree of excursions with
appropriate drift and diffusion coefficients. As a corollary, this entails a
sufficient, explicit condition for extinction of the total mass as time tends
to infinity. On the way to our comparison result, we establish that systems of
interacting diffusions with uniform migration between finitely many islands
converge to a tree of excursions as the number of islands tends to infinity. In
the special case of logistic branching, this leads to a duality between the
tree of excursions and the solution of a McKean-Vlasov equation.Comment: Published in at http://dx.doi.org/10.1214/EJP.v17-2278 the Electronic
Journal of Probability (http://ejp.ejpecp.org
Ergodic behavior of locally regulated branching populations
For a class of processes modeling the evolution of a spatially structured
population with migration and a logistic local regulation of the reproduction
dynamics, we show convergence to an upper invariant measure from a suitable
class of initial distributions. It follows from recent work of Alison Etheridge
that this upper invariant measure is nontrivial for sufficiently large
super-criticality in the reproduction. For sufficiently small
super-criticality, we prove local extinction by comparison with a mean field
model. This latter result extends also to more general local reproduction
regulations.Comment: Published at http://dx.doi.org/10.1214/105051606000000745 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Costly defense traits in structured populations
We propose a model for the dynamics of frequencies of a costly defense trait.
More precisely, we consider Lotka-Volterra-type models involving a prey (or
host) population consisting of two types and a predator (or parasite)
population, where one type of prey individuals - modeling carriers of a defense
trait - is more effective in defending against the predators but has a weak
reproductive disadvantage. Under certain assumptions we prove that the relative
frequency of these defenders in the total prey population converges to
spatially structured Wright-Fisher diffusions with frequency-dependent
migration rates. For the many-demes limit (mean-field approximation) hereof, we
show that the defense trait goes to fixation/extinction if and only if the
selective disadvantage is smaller/larger than an explicit function of the
ecological model parameters.Comment: 48 page
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