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A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator

Claudio Arancibia-Ibarra, ,School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, GP Campus, Brisbane, Queensland 4001 Australia, Facultad de Educación, Universidad de Las Américas, Av. Manuel Montt 948, Santiago, Chile, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster, ,Department of Computer Science, The University of South Dakota, Vermillion, SD 57069, South Dakota, USA, ,School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, GP Campus, Brisbane, Queensland 4001 Australia
2017 Discrete and continuous dynamical systems. Series B  

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We study a predator-prey model with Holling type I functional response, an alternative food source for the predator, and multiple Allee effects on the prey. We show that the model has at most two equilibrium points in the first quadrant, one is always a saddle point while the other can be a repeller or an attractor. Moreover, there is always a stable equilibrium point that corresponds to the persistence of the predator population and the extinction of the prey population. Additionally, we show
more » ... hat when the parameters are varied the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations. We use numerical simulations to illustrate the impact changing the predation rate, or the non-fertile prey population, and the proportion of alternative food source have on the basins of attraction of the stable equilibrium point in the first quadrant (when it exists). In particular, we also show that the basin of attraction of the stable positive equilibrium point in the first quadrant is bigger when we reduce the depensation in the model. 2010 Mathematics Subject Classification. Primary: 65L07, 92B05; Secondary: 37C75.
doi:10.3934/dcdsb.2020148 fatcat:zr342lplmbfxha2obnfc6lievi
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Claudio Arancibia-Ibarra, ,School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, GP Campus, Brisbane, Queensland 4001 Australia, Facultad de Educación, Universidad de Las Américas, Av. Manuel Montt 948, Santiago, Chile, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster, ,Department of Computer Science, The University of South Dakota, Vermillion, SD 57069, South Dakota, USA, ,School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, GP Campus, Brisbane, Queensland 4001 Australia. "A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator." Discrete and continuous dynamical systems. Series B 22.11 (2017) 0-0