Herding Induced by Encounter Rate, with Predator Pressure Influencing Prey Response

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.