Evol Ecol Res 11: 489-515 (2009)     Full PDF if your library subscribes.

Evolutionary game theory: ESS, convergence stability, and NIS

Joseph Apaloo1, Joel S. Brown2 and Thomas L. Vincent3

1Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada,  2Department of Biological Sciences, University of Illinois, Chicago, Illinois, USA and  3Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona, USA

Correspondence: J. Apaloo, Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, PO Box 5000, Antigonish, Nova Scotia B2G 2W5, Canada.
e-mail: japaloo@stfx.ca


Question: How are the three main stability concepts from evolutionary game theory – evolutionarily stable strategy (ESS), convergence stability, and neighbourhood invader strategy (NIS) – related to each other? Do they form a basis for the many other definitions proposed in the literature?

Mathematical methods: Ecological and evolutionary dynamics of population sizes and heritable strategies respectively, and adaptive and NIS landscapes.

Results: Only six of the eight combinations of ESS, convergence stability, and NIS are possible. An ESS that is NIS must also be convergence stable; and a non-ESS, non-NIS cannot be convergence stable. A simple example shows how a single model can easily generate solutions with all six combinations of stability properties and explains in part the proliferation of jargon, terminology, and apparent complexity that has appeared in the literature. A tabulation of most of the evolutionary stability acronyms, definitions, and terminologies is provided for comparison.

Key conclusions: The tabulated list of definitions related to evolutionary stability are variants or combinations of the three main stability concepts.

Keywords: adaptive landscape, convergence stability, Darwinian dynamics, evolutionary game stabilities, evolutionarily stable strategy, neighbourhood invader strategy, strategy dynamics.

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