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

Abstract

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 ω-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.

Keywords:

• predator–prey model
• functional response
• bifurcation
• separatrix curve
• heteroclinic orbit
• stability

2000 AMS Subject Classifications:

• 92D25
• 34C23
• 58F14
• 58F21

Acknowledgements

The authors thank the members of the Grupo de Ecología Matemática from the Instituto de Matemáticas at the Pontificia Universidad Católica de Valparaíso, for their valuable comments and suggestions. This work was partially financed by the Fondecyt 1120218 and DIEA-PUCV 124.730/2012 projects.