Invasion dynamics and attractor inheritance

Abstract

We study the dynamics of a population of residents that is being invaded by an initially rare mutant. We show that under relatively mild conditions the sum of the mutant and resident population sizes stays arbitrarily close to the initial attractor of the monomorphic resident population whenever the mutant has a strategy sufficiently similar to that of the resident. For stochastic systems we show that the probability density of the sum of the mutant and resident population sizes stays arbitrarily close to the stationary probability density of the monomorphic resident population. Attractor switching, evolutionary suicide as well as most cases of "the resident strikes back" in systems with multiple attractors are possible only near a bifurcation point in the strategy space where the resident attractor undergoes a discontinuous change. Away from such points, when the mutant takes over the population from the resident and hence becomes the new resident itself, the population stays on the same attractor. In other words, the new resident "inherits" the attractor from its predecessor, the former resident.

Joint evolution of predator body size and prey-size preference

“…The biological interpretation of this inequality is clear: the mutant's effective birth rate has to exceed the dilution rate. After successful invasion, the mutant generally replaces the resident (Geritz et al 2002); around evolutionary branching points they can coexist, leading to a dimorphic predator population (Metz et al 1992(Metz et al , 1996Geritz et al 1997Geritz et al , 1998.…”

Evolutionary Games in Space

“…This approach has led to the discovery of shifting evolutionary isoclines and the slowing down of the evolutionary path). Another approach is the use of pairwise resident-invader contests [23,22,21]. It has lead to many interesting insights regarding evolutionary branching, evolutionarily singular strategies, and the "inheritance" of a resident's attractor by a mutant invader [23,22,21].…”

“…Replacing v by the strategy u ij results in the fitness function for individuals of type ij. We note that one does not have to use the G−function approach to study strategy dynamics and a fitness-function based formalism can also be pursued [2,1,14,23,22,21]. In the following, we shall replace H ij (ũ,z)…”

Adaptive Phenotypic Diversification along a Temperature-Depth Gradient

“…B). In case of positive invasion fitness, the mutant generically replaces the resident (Geritz et al 2002) and the resident population's trait value shifts accordingly. This directional selection on the trait of a monomorphic resident population persists as long as the selection gradient…”