Lotka–Volterra approximations for evolutionary trait-substitution processes

Journal of Mathematical Biology, vol. 80, pp. 2141–2226

Abstract

A set of axioms is formulated characterizing ecologically plausible community dynamics. Using these axioms, it is proved that the transients following an invasion into a sufficiently stable equilibrium community by a mutant phenotype similar to one of the community's finitely many resident phenotypes can always be approximated by means of an appropriately chosen Lotka-Volterra model. To this end, the assumption is made that similar phenotypes in the community form clusters that are well-separated from each other, as is expected to be generally the case when evolution proceeds through small mutational steps. Each phenotypic cluster is represented by a single phenotype, which we call an approximate phenotype and assign the cluster's total population density. We present our results in three steps. First, for a set of approximate phenotypes with arbitrary equilibrium population densities before the invasion, the Lotka-Volterra approximation is proved to apply if the changes of the population densities of these phenotypes are sufficiently small during the transient following the invasion. Second, quantitative conditions for such small changes of population densities are derived as a relationship between within-cluster differences and the leading eigenvalue of the community's Jacobian matrix evaluated at the equilibrium population densities before the invasion. Third, to demonstrate the utility of our results, the 'invasion implies substitution' result for monomorphic populations is extended to arbitrarily polymorphic populations consisting of well-recognizable and -separated clusters.

Authors 3

1. Hiroshi C. Ito corresponding

   The Graduate University for Advanced Studies, SOKENDAI · International Institute for Applied Systems Analysis

   Affiliation as printed

   Department of Evolutionary Studies of Biosystems, The Graduate University for Advanced Studies (Sokendai), Hayama, 240-0193, Kanagawa, Japan. hiroshibeetle@gmail.com

   Evolution and Ecology Program, International Institute for Applied Systems Analysis, Schlossplatz 1, 2361, Laxenburg, Austria. hiroshibeetle@gmail.com

   Evolution and Ecology Program, International Institute for Applied Systems Analysis, Schlossplatz 1, 2361, Laxenburg, Austria

2. Ulf Dieckmann

   The Graduate University for Advanced Studies, SOKENDAI · International Institute for Applied Systems Analysis

   Affiliation as printed

   Department of Evolutionary Studies of Biosystems, The Graduate University for Advanced Studies (Sokendai), Hayama, 240-0193, Kanagawa, Japan

   Evolution and Ecology Program, International Institute for Applied Systems Analysis, Schlossplatz 1, 2361, Laxenburg, Austria

3. Johan A. J. Metz Aachen Mathematical Institute Institute of Biology, Leiden University Naturalis Biodiversity Center

   Leiden University · International Institute for Applied Systems Analysis · Naturalis Biodiversity Center

   Affiliation as printed

   Evolution and Ecology Program, International Institute for Applied Systems Analysis, Schlossplatz 1, 2361, Laxenburg, Austria

   Mathematical Institute and Institute of Biology, Leiden University, P.O. Box 9512, 2300 RA, Leiden, The Netherlands

   Naturalis Biodiversity Center, P.O. Box 9517, 2300 RA, Leiden, The Netherlands