Adaptive dynamics of saturated polymorphisms

We study the joint adaptive dynamics of n scalar-valued strategies in ecosystems where n is the maximum number of coexisting strategies permitted by the (generalized) competitive exclusion principle. The adaptive dynamics of such saturated systems exhibits special characteristics, which we first demonstrate in a simple example of a host-pathogen-predator model. The main part of the paper characterizes the adaptive dynamics of saturated polymorphisms in general. In order to investigate convergence stability, we give a new sufficient condition for absolute stability of an arbitrary (not necessarily saturated) polymorphic singularity and show that saturated evolutionarily stable polymorphisms satisfy it. For the case , we also introduce a method to construct different pairwise invasibility plots of the monomorphic population without changing the selection gradients of the saturated dimorphism.Peer reviewe

Coevolution of cannibalistic predators and timid prey: evolutionary cycling and branching

We investigate the coevolution of cannibalistic predators and timid prey, which seek refuge upon detecting a predator. To understand how the species affect each other’s evolution, we derived the ecological model from individual-level processes using ordinary differential equations. The ecological dynamics exhibit bistability between equilibrium and periodic attractors, which may disappear through catastrophic bifurcations. Using the critical function analysis of adaptive dynamics, we classify general trade-offs between cannibalism and prey capture that produce different evolutionary outcomes. The evolutionary analysis reveals several ways in which cannibalism emerges as a response to timidity of the prey. The long-term coevolution either attains a singularity, or becomes cyclic through two mechanisms: genetical cycles through Hopf bifurcation of the singularity, or ecogenetical cycles involving abrupt switching between ecological attractors. Further diversification of cannibalism occurs through evolutionary branching, which is predicted to be delayed when simultaneous prey evolution is necessary for the singularity’s attainability. We conclude that predator-prey coevolution produces a variety of outcomes, in which evolutionary cycles are commonplace.Peer reviewe