Lotka–Volterra approximations for evolutionary trait-substitution processes

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.

The adaptation front equation explains diversification hotspots and living-fossilization

“…For simplicity, we consider a trait space scaled beforehand so that the mutational covariance matrix is given by the identity matrix multiplied by 2 , i.e., the mutation is isotropic with its average size (standard deviation) given by . We also assume that is sufficiently small so that in many cases evolutionary dynamics proceeds as directional evolution, evolutionary branching, and extinction of well-separated phenotypic clusters (27,36).…”

Desempeño Del Modelo De Lotka-Volterra Y Holling Aplicado a Sistemas Presa-Depredador

“…Además, si el tamaño de la población de depredadores es cero y el tamaño de la población de presas es positivo, el tamaño de la población de depredadores sigue siendo cero y la población de presas crece sin límite. Por último, si el tamaño de la población de depredadores es positivo y el tamaño de la población de presas es cero, el tamaño de la población de presas sigue siendo cero, mientras que la población de depredadores se reduce hacia el tamaño cero (Ito et al, 2020). Este es el escenario actual sobre el modelo de Lotka-Volterra.…”

Metacommunities, fitness and gradual evolution

“…The "invasion implies substitution"-principle is a formal characterization of the spread of mutations affecting quantitative traits in the context of density and frequency-dependent selection (Rousset, 2004;Meszéna et al, 2005;Dercole and Rinaldi, 2008;Lehmann and Rousset, 2014;Dercole, 2016;Cantrell et al, 2017;Ito et al, 2020), and it consists of two propositions (Priklopil and Lehmann, 2020). The first says that if a mutation in the focal species results in a phenotype with a closely similar expression to its ancestral (resident) form, a successful invasion of the mutant allele will generically lead to its fixation.…”

“…In this paper we provide sufficient conditions for selective sweeps when evolution occurs in an asexual focal species that resides in a wider metacommunity (Wilson, 1992;Leibold et al, 2004), and where socially interacting individuals are structured into discrete classes that determine their physiological state, such as age and size, as well as their local biotic and abiotic environment. We provide these conditions in terms of an "invasion implies substitution"-principle, which is a formal characterization of the invariance of selection with respect to a population-wide frequency of a mutant allele that affects quantitative traits in the context of density-and frequency-dependent selection (Hamilton, 1964;Rousset, 2004;Meszéna et al, 2005;Dercole and Rinaldi, 2008;Lehmann and Rousset, 2014;Dercole, 2016;Cantrell et al, 2017;Ito et al, 2020;Cai and Geritz, 2020;Priklopil and Lehmann, 2020). The principle consists of two propositions (Priklopil and Lehmann, 2020).…”