A Leslie-Gower-type predator-prey model with sigmoid functional response

摘要

In this work, a continuous-time predator-prey model of Leslie-Gower type considering a sigmoid functional response is analysed. Using the MatLab package some simulations of the dynamics are shown. Conditions for the existence of equilibrium points, their nature and the existence of at least one limit cycle in phase plane are established. The existence of a separatrix curve dividing the behaviour of trajectories is proved. Thus, two closed trajectories can have different omega-limits being highly sensitive to initial conditions. Moreover, for a subset of parameter values, it can be possible to prove that the point (0,0) can be globally asymptotically stable. So, both populations can go to extinction, but simulations show that this situation is very difficult. According to our knowledge no previous work exists analysing the model presented here. A comparison of the model here studied with the May-Holling-Tanner model shows a difference on the quantity of limit cycles.