Spatial patterns of competing random walkers

• The dynamics of a class of individual-based models with birth and death rates which implement competitive interactions is reviewed. Individuals perform Gaussian and Lévy flights.
• Competition produces clustering and pattern instabilities.
• Random walks delay such instabilities, and also lead to competitive advantage of the least diffusing species.

We review recent results obtained from simple individual-based models of biological competition in which birth and death rates of an organism depend on the presence of other competing organisms close to it. In addition the individuals perform random walks of different types (Gaussian diffusion and Lévy flights). We focus on how competition and random motions affect each other, from which spatial instabilities and extinctions arise. Under suitable conditions, competitive interactions lead to clustering of individuals and periodic pattern formation. Random motion has a homogenizing effect and then delays this clustering instability. When individuals from species differing in their random walk characteristics are allowed to compete together, the ones with a tendency to form narrower clusters get a competitive advantage over the others. Mean-field deterministic equations are analyzed and compared with the outcome of the individual-based simulations.