Abstract
Game theory ideas provide a useful framework for studying evolutionary dynamics in a well-mixed environment. This approach, however, typically enforces a strictly fixed overall population size, deemphasizing natural growth processes. We study a competitive Lotka-Volterra model, with number fluctuations, that accounts for natural population growth and encompasses interaction scenarios typical of evolutionary games. We show that, in an appropriate limit, the model describes standard evolutionary games with both genetic drift and overall population size fluctuations. However, there are also regimes where a varying population size can strongly influence the evolutionary dynamics. We focus on the strong mutualism scenario and demonstrate that standard evolutionary game theory fails to describe our simulation results. We then analytically and numerically determine fixation probabilities as well as mean fixation times using matched asymptotic expansions, taking into account the population size degree of freedom. These results elucidate the interplay between population dynamics and evolutionary dynamics in well-mixed systems.
Highlights
Recent advances in experimental evolution open new directions for quantitative studies of evolutionary dynamics [1, 2]
Microbial experiments demonstrate an intricate feedback between evolutionary and population dynamics [3,4,5], theoretical understanding is often limited to evolutionary dynamics in a fixed population size, mostly within the framework of evolutionary game theory and population genetics [6,7,8,9,10,11,12]
We focus on the strong mutualism scenario, where conventional replicator dynamics with genetic drift fails to predict the fixation probability, due to a strong coupling between evolutionary and population dynamics
Summary
Recent advances in experimental evolution open new directions for quantitative studies of evolutionary dynamics [1, 2]. For positive α’s (first quadrant of Fig. 1), a stable fixed point corresponding to a species coexistence appears at f ∗ = α1/(α1 + α2), lying between the unstable fixed points f = 0 and f = 1 This scenario is commonly referred to as the “snowdrift game” in game theory or mutualism in the context of evolution [7, 14, 15]. The emphasis is on parameter values such that an attractive line of approximately fixed population size dominates the long-time dynamics This limit enables us to identify the mapping between the model and the replicator dynamics.
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