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The dynamical theory of coevolution: a derivation from stochastic ecological processes

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Abstract

In this paper we develop a dynamical theory of coevolution in ecological communities. The derivation explicitly accounts for the stochastic components of evolutionary change and is based on ecological processes at the level of the individual. We show that the coevolutionary dynamic can be envisaged as a directed random walk in the community's trait space. A quantitative description of this stochastic process in terms of a master equation is derived. By determining the first jump moment of this process we abstract the dynamic of the mean evolutionary path. To first order the resulting equation coincides with a dynamic that has frequently been assumed in evolutionary game theory. Apart from recovering this canonical equation we systematically establish the underlying assumptions. We provide higher order corrections and show that these can give rise to new, unexpected evolutionary effects including shifting evolutionary isoclines and evolutionary slowing down of mean paths as they approach evolutionary equilibria. Extensions of the derivation to more general ecological settings are discussed. In particular we allow for multi-trait coevolution and analyze coevolution under nonequilibrium population dynamics.

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Author notes

  1. Ulf Dieckmann

    Present address: Wissenschaftskolleg zu Berlin, Wallotstrasse 19, D-14193, Berlin, Germany

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  1. Theoretical Biology Section, Institute of Evolutionary and Ecological Sciences, University of Leiden, Kaiserstraat 63, 2311 GP, Leiden, The Netherlands

    Ulf Dieckmann

  2. Department of Biology, University of York, YO1 5DD, York, UK

    Richard Law

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  1. Ulf Dieckmann
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  2. Richard Law
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Dieckmann, U., Law, R. The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biology 34, 579–612 (1996). https://doi.org/10.1007/BF02409751

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