Population Models with Phenotypic Diversity and Fluctuating Environment Illustrate Parameter Regimes of Adaptation vs. Extinction

A species may survive long-term environmental fluctuations by maintaining a sufficient phenotypic variability to span the changes. Bet-hedging and adaptive mutatability (or mutator strains) are among possible mechanisms. We construct simple population dynamics models for these strategies in a unified framework, with a stochastic environment model and a specifiable environmental carrying capacity, identifying different parameter regimes: Quantitatively, how much hedging is needed or what mutation rate is required? How much selection burden re wild-type is tolerable for the strategy to work? Is the diverse population susceptible to invasion by a reverting wild type better adapted in the short run, but producing an extinction outcome in the long run? We give answers that are quantitative in the context of the specific idealized models, but that can also be qualitatively illuminating to more realistic cases.