Dynamics in a Filippov-Gause Predator-Prey Model with Hunting Cooperation or Competition Among Predators for High or Low Prey Densities

A Filippov-Gause predator-prey model is proposed, which suggests that cooperation and competition among predators depend on prey abundance. In this case, when the prey abundance is high, predators cooperate in hunting, but when the prey abundance decreases and falls below a critical level, predators compete for scarce food. The proposed model is analyzed in terms of the existence and stability of its equilibria and its pseudo-equilibrium, which is located on the differentiable curve separating the two vector fields. Given the conditions found which determine the existence of cycles and limit cycles, contrasted with a bifurcation diagram, we find that the proposed model can have an asymptotically stable limit cycle around the pseudo-equilibrium and two positive inner equilibria, together with an unstable limit cycle around the pseudo-equilibrium. Similarly, if the model has a pseudo-equilibrium and no positive inner equilibria, then we could have at least two limit cycles, one stable and one unstable, surrounding the pseudo-equilibrium.