© Canadian Mathematical Society 1994. The usual super-Brownian motion is a measure-valued process that arises as a high density limit of a system of branching Brownian partitles in which the branching methanism is critical. In this work we consider analogous processes that model the evolution of a system of two such populations in which there is inter-species competition or prédation. We first considera competition model in which inter-species collisions may result in casualties on both sides, Using a Girsanov approach, we oblain existence and uniqueness of the appropriate martingale problem in one dimension. In two and three dimensions we establish existence only. However, we do show that, in three dimensions, any solution will not he absolutely continuous with respect to the law of two independent super-Brownian motions. Although the supports of two independent super-Brownian motions collide in dimensions lour and live, we show that there is no solution lo the martingale problem in these cases. We next study a prédation model in which collisions only affect the "prey" species. Here we can show both existence and uniqueness in one, two and three dimensions. Again, there is no solution in four and live dimensions. As a tool for proving uniqueness, we obtain a representation of martingales for a super-proeessas stochastic integrals with respect to the related orthogonal martingale measure. W also obtain existence and uniqueness for a related single population model in one dimension in which panicles are killed at a rate proportional to Ihe local densily. This model appears as a limit oía rescaled contati process as the range (if interaction goes to infinity.

We consider a population living in a patchy environment that varies stochastically in space and time. The population is composed of two morphs (that is, individuals of the same species with different genotypes). In terms of survival and reproductive success, the associated phenotypes differ only in their habitat selection strategies. We compute invasion rates corresponding to the rates at which the abundance of an initially rare morph increases in the presence of the other morph established at equilibrium. If both morphs have positive invasion rates when rare, then there is an equilibrium distribution such that the two morphs coexist; that is, there is a protected polymorphism for habitat selection. Alternatively, if one morph has a negative invasion rate when rare, then it is asymptotically displaced by the other morph under all initial conditions where both morphs are present. We refine the characterization of an evolutionary stable strategy for habitat selection from Schreiber (Am Nat 180:17-34, 2012) in a mathematically rigorous manner. We provide a necessary and sufficient condition for the existence of an ESS that uses all patches and determine when using a single patch is an ESS. We also provide an explicit formula for the ESS when there are two habitat types. We show that adding environmental stochasticity results in an ESS that, when compared to the ESS for the corresponding model without stochasticity, spends less time in patches with larger carrying capacities and possibly makes use of sink patches, thereby practicing a spatial form of bet hedging. © 2014 Springer-Verlag Berlin Heidelberg.