On Conditions for Evolutionary Stability for a Continuously Varying Character

The two conditions for stability of an evolutionary equilibrium, the m-stability and the δ-stability conditions, are discussed. The m-stability condition is a condition for the convergence of the population toward the equilibrium, and the δ-stability condition coresponds to a local version of the classic evolutionarily stable strategy (ESS) condition. Together the two conditions provide the condition for a continuous stable strategy. The convergence stability condition corresponds to the requirement for convergence due to initial increase of rare alleles in a monomorphic population, and the local ESS stability condition corresponds to the stability of a monomorphic population at the evolutionary equilibrium against the increase of rare alleles. In this way, an evolutionary equilibrium that is convergence stable, but not local ESS stable, will tend to become polymorphic. The local ESS stability condition therefore contributes more to a description of the dynamics of variation at an evolutionary equilibrium than to the description of the stability of the evolutionary equilibrium. However, the characterization of a polymorphic evolutionary equilibrium cannot be reached by studying the initial increase of rare variant alleles, just as this method cannot describe all aspects of the convergence stability either. Combining these analyses provides a powerful tool in the initial exploration of evolutionary equilibriums of complicated systems, and convergence stable and local ESS unstable equilibriums point toward very interesting polymorphic evolutionary stable states. The analysis is illustrated on a model for intraspecific exploitative competition that may show monomorphic and polymorphic evolutionarily stable equilibriums.