This thesis starts from the Matsuda and Abrams paper 'Timid Consumers: Self-Extinction Due to Adaptive Change in Foraging and Anti-predator Effort.' Matsuda and Abrams show an example of evolutionary suicide due to the evolution of prey timidity in a predator-prey model with a Holling type II functional response. The key assumption they use to obtain evolutionary suicide is that the predator population size is kept constant. In this thesis, we relax this assumption by introducing a second type of prey to the model and investigate whether evolutionary suicide may still occur according to the evolution of timidity in the first prey species. To study this in the long-term, we use the theory of adaptive dynamics.
Firstly, we analyse the limit case where the predator dynamics depend only upon the second prey species. Predators still hunt the evolving prey either as a snack or for entertainment without gaining any energy. Under this hypothesis, our model reproduces Matsuda and Abrams' results both qualitatively and quantitatively. Moreover, the introduction of the second type of prey allows for the appearance of limit cycles as dynamical attractors. We detect a fold bifurcation in the stability of the limit cycles when the first type of prey timidity increases. Thus, we are able to construct an example of evolutionary suicide on a fold bifurcation of limit cycles. Furthermore, we perform critical function analysis on the birth rate of the evolving prey as a function of prey timidity. We derive general conditions for the birth rate function that assure the occurrence
of evolutionary suicide.
Secondly, we analyse the full model without making any simplifying assumptions. Because of the analytical complexity of the system we use numerical bifurcation analysis to study bifurcations of the internal equilibria. More specifically, we utilize the package MatCont to carry out equilibria continuation. In this way, we are able to estimate the range of parameters where the results of Matsuda and Abrams' model hold. Starting from the parameter set that reproduce Matsuda and Abrams' results quantitatively we track the fold bifurcation and show that evolutionary suicide occurs for a considerably wide range of parameters. Moreover, we find that in the full model evolutionary suicide may also occur through a subcritical Hopf bifurcation.
