Temporal and spatio-temporal dynamics in predator-prey models

PhD Final Seminar.

Abstract:

In this final seminar, I will analyse temporal and spatio-temporal predator-prey models. I extend the Holling-Tanner predator-prey model by additionally considering an alternative food for predators and/or Allee effect(s) on the prey. The standard Holling-Tanner model is characterised by the use of a functional response proposed by Holling and a logistic form for the growth of the predator and prey population. The analysis of the models developed in this project complement the results of previous articles about the standard Holling-Tanner model with different functional responses.

I will show that there are significant differences with the well-known Holling-Tanner model using a combination of both mathematical and numerical analysis. The new models developed here exhibit rich dynamics for different parameter values. For the spatially independent case I prove the existence of a separatrix curve and an homoclinic curve in the phase plane dividing families of trajectories of different behaviours. When the homoclinic curve breaks it generates a non-infinitesimal limit cycle and different kinds of bifurcations, such as a saddle-node bifurcation, a Hopf bifurcation, a Bogadonov-Takens bifurcation and an homoclinic bifurcation.

The results of the analysis of the modified Holling-Tanner model with alternative food provides the basis for the spatial reaction-diffusion models. I identify different classes of spatially dependent problems where the underlying modified Holling-Tanner models drive the onset and persistence of spatial patterns in the predator and prey populations.