Abstract
We consider the effects of herding from the point of view of well-mixing: we assume namely that populations that herd have less than well-mixed interactions. For a single population, this leads to a hyperbolic model which is intermediate between exponential and logistic growth. We show that using this form of population growth for the prey in a Lotka-Volterra model stabilises the coexistence equilibrium. For the effect of predators, we consider a trade-off for the prey population, whereby predation pressure is reduced but so is access to resources. For simplicity, the reduced access to resources is modelled as a reduction in carrying capacity with increasing predator pressure. The predator escape is modelled in terms of the degree of herding, which gets stronger with increasing predator pressure. We apply this to two types of predator: a specialist which goes extinct if the prey does, and a generalist which survives in the absence of prey. We find that in both cases, Hopf bifurcations are possible: stable limit cycles arise at large enough prey carrying capacities in the case of a specialist predator, but unstable limit cycles arise at low enough carrying capacities in the case of the generalist predator.
The authors “Ezio Venturino” and “Iulia Martina Bulai” are member of the research group GNCS of INdAM.
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Acknowledgements
This paper was written during a visit of EV at UCT. The invitation of HL and his support are kindly acknowledged. The research of EV has been partially supported by the Dipartimento di Matematica “Giuseppe Peano” research project “Metodi numerici e computazionali per le scienze applicate”. The research of IMB has been partially supported by the INdAM GNCS project “Finanziamento Giovani Ricercatori 2018/2019”. The authors warmly thank the referee for the comments provided, that allowed a substantial improvement of the paper.
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Laurie, H., Venturino, E., Bulai, I.M. (2020). Herding Induced by Encounter Rate, with Predator Pressure Influencing Prey Response. In: Aguiar, M., Braumann, C., Kooi, B., Pugliese, A., Stollenwerk, N., Venturino, E. (eds) Current Trends in Dynamical Systems in Biology and Natural Sciences. SEMA SIMAI Springer Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-41120-6_4
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