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| In this talk we will explore the mathematical properties of a predator-prey model, where the prey population is structured according to a certain life history trait. The trait distribution within the prey population in the model is the result of interplay between genetic inheritance and mutation, as well as selectivity in the consumption of the predator. The evolutionary processes are considered to take place on the same time scale as ecological dynamics, i.e. we consider the evolution to be rapid. We investigate the existence of a coexistence stationary state in the model and carry out stability analysis of this state. We establish a number of biologically significant results. Amongst them we prove that the coexistence stationary state is stable when the saturation in the predation term is low. For a class of kernels describing genetic inheritance and mutation we show that stability of the predator-prey interaction would require a selectivity of predation according to the life trait. Finally, we derive expressions for the Hopf-bifurcation curve which can be used for constructing bifurcation diagrams in the parameter space without the need for a direct numerical simulation of the underlying integro-differential equations.
This talk is based on joint work with A. Yu Morozov (University of Leicester). |
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