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Free keywords:
Demographic Stochasticity; Diffusion Theory; Evolutionary
Games; Fixation Probability; Weak Selection; Quantitative Biology; Populations and Evolution; MSC 60J60; MSCV91A22; MSC 92D25
Abstract:
We study the fixation probability of a mutant type when introduced into
a resident population. As opposed to the usual assumption of constant population
size, we allow for stochastically varying population sizes. This is
implemented by a stochastic competitive Lotka-Volterra model. The competition
coefficients are interpreted in terms of inverse payoffs emerging from an
evolutionary game. Since our study focuses on the impact of the competition
values, we assume the same birth and death rates for both types. In this general
framework, we derive an approximate formula for the fixation probability
ϕ of the mutant type under weak selection. The qualitative behavior of ϕ
when compared to the neutral scenario is governed by the invasion dynamics
of an initially rare type. Higher payoffs when competing with the resident
type yield higher values of ϕ. Additionally, we investigate the influence of
the remaining parameters and find an explicit dependence of ϕ on the mixed
equilibrium value of the corresponding deterministic system (given that the
parameter values allow for its existence).
