Taming the diffusion approximation through a controlling-factor WKB method

Abstract

The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied. This failure becomes severe in situations where the direction of selection switches sign. Here we employ the WKB (large-deviations) method, which requires only the logarithm of a given quantity to be smooth over its state space. Combining the WKB scheme with asymptotic matching techniques, we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations, applicable for much wider regimes of parameters. We also introduce a scalable (independent of population size) WKB-based numerical technique. The method is applied to a central problem in population genetics and evolution, finding the chance of ultimate fixation in a zero-sum, two-types competition.

References in corpus (17)

• Genomic evidence of rapid and stable adaptive oscillations over seasonal time scales in Drosophila
• Statistical Mechanics of Ecological Systems: Neutral Theory and Beyond
• WKB theory of large deviations in stochastic populations
• Extinction Rates for Fluctuation-Induced Metastabilities : A Real-Space WKB Approach
• Population extinction in a fluctuating environment
• Fixation in finite populations evolving in fluctuating environments
• Evolution of a Fluctuating Population in a Randomly Switching Environment
• Resist or perish: fate of a microbial population subjected to a periodic presence of antimicrobial
• The dynamics of simple gene-network motifs subject to extrinsic fluctuations
• Species coexistence in a neutral dynamics with environmental noise
• An iterative action minimizing method for computing optimal paths in stochastic dynamical systems
• Eco-Evolutionary Dynamics of a Population with Randomly Switching Carrying Capacity
• Noise enhanced persistence in a biochemical regulatory network with feedback control
• Theory of time-averaged neutral dynamics with environmental stochasticity
• A comprehensive phase diagram for logistic populations in fluctuating environment
• Impact of Colored Environmental Noise on the Extinction of a Long-Lived Stochastic Population: Role of the Allee Effect
• Phase diagram for a logistic system under bounded stochasticity