A modified Leslie-Gower predator-prey model with ratio-dependent functional response and alternative food for the predator

In this work, a modified Leslie-Gower predator-prey model is analyzed, considering an alternative food for the predator and a ratio-dependent functional response to express the species interaction. The system is well defined in the entire first quadrant except at the origin (0,0). Given the importance of the origin (0,0) as it represents the extinction of both populations, it is convenient to provide a continuous extension of the system to the origin. By changing variables and a time rescaling, we obtain a polynomial differential equations system, which is topologically equivalent to the original one, obtaining that the non-hyperbolic equilibrium point (0,0) in the new system is a repellor for all parameter values. Therefore, our novel model presents a remarkable difference with other models using ratio-dependent functional response. We establish conditions on the parameter values for the existence of up to two positive equilibrium points; when this happen, one of them is always a hyperbolic saddle point, and the other can be either an attractor or a repellor surrounded by at least one limit cycle. We also show the existence of a separatrix curve dividing the behavior of the trajectories in the phase plane. Moreover, we establish parameter sets for which a homoclinic curve exits, and we show the existence of saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, and homoclinic bifurcation. An important feature in this model is that the prey population can go to extinction; meanwhile, population of predators can survive because of the consumption of alternative food in the absence of prey. In addition, the prey population can attain their carrying capacity level when predators go to extinction. We demonstrate that the solutions are non-negatives and bounded (dissipativity and permanence of population in many other works). Furthermore, some simulations to reinforce our mathematical results are shown, and we further discuss their ecological meanings. Copyright (c) 2017 John Wiley & Sons, Ltd.