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The use of predator-prey models in theoretical ecology has a long history,and the model equations have largely evolved since the original Lotka-Volterrasystem towards more realistic descriptions of the processes of predation,reproduction and mortality. One important aspect is the recognition of the factthat the growth of a population can be subject to an Allee effect, where theper capita growth rate increases with the population density. Including anAllee effect has been shown to fundamentally change predator-prey dynamics andstrongly impact species persistence, but previous studies mostly focused onscenarios of an Allee effect in the prey population. Here we explore apredator-prey model with an ecologically important case of the Allee effect inthe predator population where it occurs in the numerical response of predatorwithout affecting its functional response. Biologically, this can result fromvarious scenarios such as a lack of mating partners, sperm limitation andcooperative breeding mechanisms, among others. Unlike previous studies, weconsider here a generic mathematical formulation of the Allee effect withoutspecifying a concrete parameterisation of the functional form, and analyse thepossible local bifurcations in the system. Further, we explore the globalbifurcation structure of the model and its possible dynamical regimes for threedifferent concrete parameterisations of the Allee effect. The model possesses acomplex bifurcation structure: there can be multiple coexistence statesincluding two stable limit cycles. Inclusion of the Allee effect in thepredator generally has a destabilising effect on the coexistence equilibrium.We also show that regardless of the parametrisation of the Allee effect,enrichment of the environment will eventually result in extinction of thepredator population.

Bifurcation analysis of the predator-prey model with the Allee effect in the predator