Shape effects on herd behavior in ecological interacting population models

•Several dynamical systems modeling two-populations interactions gathering in herds.•Interaction occurs on the perimeter in 2D.•Interaction occurs on the total surface area for populations living in 3D.•Here we even accommodate the model for herds that assume fractal shapes.•Populations symbiosis, competition and predator-prey are considered.•Stable solution is independent of the shape of the herd for competition.

In this paper, we introduce several dynamical systems modeling two-populations interactions. The main idea is to assume that the individuals of one of the populations gather together in herds, thus possess a social behavior, while individuals of the second population show a more individualistic attitude. We model the fact that the interaction between the two populations occurs mainly through the perimeter of the herd in a 2D space or through the total surface area for populations that live in a 3D space. This idea has already been explored earlier, but here we even accommodate the model for herds that assume fractal shapes. We account for all types of the populations intermingling: symbiosis, competition and predator-prey interactions. In the cases of obligated mutualism for the individualistic population and of competition, the stable solution attained by the populations is independent of the shape of the herd.