Two-Species Migration and Clustering in Two-Dimensional Domains

We extend two-species models of individual aggregation or clustering to two-dimensional spatial domains, allowing for more realistic movement of the populations compared with one spatial dimension. We assume that the domain is bounded and that there is no flux into or out of the domain. The motion of the species is along fitness gradients which allow the species to seek out a resource. In the case of competition, species which exploit the resource alone will disperse while avoiding one another. In the case where one of the species is a predator or generalist predator which exploits the other species, that species will tend to move toward the prey species, while the prey will tend to avoid the predator. We focus on three primary types of interspecies interactions: competition, generalist predator–prey, and predator–prey. We discuss the existence and stability of uniform steady states. While transient behaviors including clustering and colony formation occur, our stability results and numerical evidence lead us to believe that the long-time behavior of these models is dominated by spatially homogeneous steady states when the spatial domain is convex. Motivated by this, we investigate heterogeneous resources and hazards and demonstrate how the advective dispersal of species in these environments leads to asymptotic steady states that retain spatial aggregation or clustering in regions of resource abundance and away from hazards or regions or resource scarcity.