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Self-Replication and Localized Structures Interaction in a Nonlocal Model of Population Dynamics

A particular type of localized structure in a prototypical model for population dynamics interaction is studied. The model considers cooperative and competitive interaction among the individuals. Interaction at distance (or nonlocal interaction) and a simple random walk for the motion of the individuals are included. The system exhibits the formation of a periodic cellular pattern in some region of its parameter space. Inside this parameter region, it is possible to observe the localization of a single cell from the cellular pattern into an unpopulated background. The stability of this localized structure is discussed, as well as the destabilization process that gives rise to its own self-replication, inducing the propagation of the cellular pattern. The long distance interaction between these localized structures is also studied which results in a mutual repulsion.