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.
Highlights
Ecological interactions create selection pressures that may change those very interactions
The resulting trait-substitution sequences describe directional coevolution, characterized well by a set of ordinary differential equations called the canonical equations of adaptive dynamics theory (Dieckmann and Law 1996), which have a form similar to Lande’s equations of quantitative genetics theory (Lande 1979)
Mutation rates are sufficiently small relative to the timescale of the population dynamics, so that the evolutionary dynamics are reduced to trait-substitution sequences resulting from repeated mutant invasions
Summary
Ecological interactions create selection pressures that may change those very interactions. For higher-dimensional traits or more than one resident, obtaining formal conditions for the occurrence of evolutionary branching is difficult (but see Ito and Dieckmann (2014) for a special case) This is largely because in these more complex community dynamics it is not easy to analyze the outcomes of mutant. The remaining cases to be analyzed are (a) only some residents are similar to each other and (b) the population densities of some residents are very small so that they may go extinct as a result of the invasion Both cases are likely to occur in multispecies coevolution, including processes involving multiple evolutionary branching and taxon cycles, commonly observed in numerical simulations of trait-mediated community dynamics (e.g., Doebeli and Dieckmann 2000; Ito and Dieckmann 2007).
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