Bifurcation analysis of a prey-predator coevolution model

We show in this paper how numerical bifurcation analysis can be used to study the evolution of genetically transmitted phenotypic traits. For this, we consider the standard Rosenzweig-MacArthur prey-predator model [The American Naturalist, 97 (1963), pp. 209-223] and, following the so-called adaptive dynamics approach, we derive from it a second-order evolutionary model composed of two ODEs, one for the prey trait and one for the predator trait. Then, we perform a detailed bifurcation analysis of the evolutionary model with respect to various environmental and demographic parameters. Surprisingly, the evolutionary dynamics turn out to be much richer than the population dynamics. Up to three evolutionary attractors can be present, and the bifurcation diagrams contain numerous global bifurcations and codimension-2 bifurcation points. Interesting biological properties can be extracted from these bifurcation diagrams. In particular, one can conclude that evolution of the traits can be cyclic and easily promote prey species diversity.