Abstract
Branching measure-valued diffusion models are investigated that can be regarded as pairs of historical Brownian motions modified by a competitive interaction mechanism under which individuals from each population have their longevity or fertility adversely affected by collisions with individuals from the other population. For 3 or fewer spatial dimensions, such processes are constructed using a new fixed-point technique as the unique solution of a strong equation driven by another pair of more explicitly constructible measure-valued diffusions. This existence and uniqueness is used to establish well-posedness of the related martingale problem and hence the strong Markov property for solutions. Previous work of the authors has shown that in 4 or more dimensions models with the analogous definition do not exist.
Highlights
1.1 BackgroundConsider populations of two different species which migrate, reproduce and compete for the same resources
Continuous space point processes have recently been used by mathematical biologists to model such competing species
The biologists are dealing with organisms of a given size and finite interaction range, from a mathematical perspective it is natural to consider a scaling limit in which the interaction becomes purely local and the total population becomes large
Summary
Consider populations of two different species which migrate, reproduce and compete for the same resources. Super–Brownian models in which the mortality or fertility of individuals is subject to local effects are relatively easy to construct and analyse in one dimension. The interacting model solves a martingale problem that looks like the one for a pair of independent super–Brownian motions except for the addition of tame “drift” terms The law of such a process can be constructed by using Dawson’s Girsanov theorem (see [5]) to produce an absolutely continuous change in the law of a pair of independent super–Brownian motions (see Section 2 of [20]). Models for two populations in which the branching rate is subject to local interactions are studied in [10]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
