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Modeling herd behavior in population systems. (English) Zbl 1225.49037

Summary: We show that under suitable simple assumptions the classical two populations system may exhibit unexpected behavior. Considering a more elaborated social model, in which the individuals of one population gather together in herds, while the other one shows a more individualistic behavior, we model the fact that interactions among the two occur mainly through the perimeter of the herd. We account for all types of populations’ interactions, symbiosis, competition and the predator-prey interactions. There is a situation in which competitive exclusion does not hold: the socialized herd behavior prevents the competing individualistic population from becoming extinct. For the predator-prey case, sustained limit cycles are possible, the existence of Hopf bifurcations representing a distinctive feature of this model compared with other classical predator-prey models. The system’s behavior is fully captured by just one suitably introduced new threshold parameter, defined in terms of the original model parameters.

MSC:

49N75 Pursuit and evasion games
92D25 Population dynamics (general)
93C15 Control/observation systems governed by ordinary differential equations
34H05 Control problems involving ordinary differential equations
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