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

Abstract

In the present study, we have modified the traditional May-Holling-Tanner predator-prey model used to represent the interaction between least weasel and field-vole population by adding an Allee effect (strong and weak) on the field-vole population and alternative food source for the weasel population. It is shown that the dynamic is different from the original May-Holling-Tanner predator-prey interaction since new equilibrium points have appeared in the first quadrant. Moreover, the modified model allows the extinction of both species when the Allee effect (strong and weak) on the prey is included, while the inclusion of the alternative food source for the predator shows that the system can support the coexistence of the populations, extinction of the prey and coexistence and oscillation of the populations at the same time. Furthermore, we use numerical simulations to illustrate the impact that changing the predation rate and the predator intrinsic growth rate have on the basin of attraction of the stable equilibrium point or stable limit cycle in the first quadrant. These simulations show the stabilisation of predator and prey populations and/or the oscillation of these two species over time.

Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect

“…Obviously, the model proposed by Martinez-Jeraldo and Aguirre [35] is singular at x = 0. To avoid the singularity, Arancibia-Ibarra et al [4] and [6] introduced the conception of alternative food source for the predator and proposed the following modified May-Holling-Tanner models with respectively the single and multiple Allee effects, namely…”

“…The singularity at x = 0 has been removed in these two models. Based on numerical bifurcation package, Arancibia-Ibarra et al [4] and [6] detected a wide range of bifurcations including saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations, and gave the basin of attraction of the stable positive equilibrium in the first quadrant.…”