Evol Ecol Res 11: 637-650 (2009)     Full PDF if your library subscribes.

Evolutionary stabilities in multidimensional-traits and several-species models

J. Apaloo and S. Butler

Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada

Correspondence: J. Apaloo, Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia B2G 2W5, Canada.
e-mail: japaloo@stfx.ca


Questions: Co-evolutionary models with one to multidimensional strategies can result in stable coalitions of many strategies. Are coalition strategies that are evolutionarily stable (ESS) and neighbourhood invader (NIS) also convergence stable? What is the implication for co-evolutionary models?

Mathematical methods: Optimization of the fitness function and dynamical systems based on the selection gradient of the fitness function.

Key assumptions: The fitness of a trait depends on its strategy value and on the environment consisting of strategy values of other traits and their population sizes. Co-evolutionary strategies in a close neighbourhood of a singular point of the canonical equation maintain stable population dynamic equilibria.

Conclusions: In single-species evolutionary games with a multidimensional strategy set, a strategy that is both an ESS and NIS is also a strong convergence stable strategy and thus convergence stable. In co-evolutionary games, this implication is not guaranteed and there can never be a strategy that is strong NIS. Therefore, ‘fast evolution’, which can occur in single-species evolution when a singular point is both ESS and NIS, may not occur in co-evolutionary models.

Keywords: absolute convergence stability, convergence stability, evolutionary stability, evolutionarily stable strategy, neighbourhood invader strategy, strong convergence stability, strong neighbourhood invader stability.

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