Pareto-optimal trade-off for phenotypic switching of populations in a stochastic environment

Finding optimal survival strategies of living systems embedded in fluctuating environments generally involves a balance between phenotypic diversification and sensing. If we neglect sensing mechanisms, it is known that slow, resp. fast, environmental transitions favor a regime of heterogeneous, resp. homogeneous, phenotypic response. We focus here on the simplest non-trivial case, i.e. two randomly switching phenotypes subjected to two stochastically switching environments. The optimal asymptotic (long term) growth rate of this model was studied elsewhere; we further expand these results by discussing finite time growth rate fluctuations. An exact asymptotic expression for the variance, alongside with approximations valid in different regimes, are tested numerically in details. Our simulations of the dynamics suggest a close connection between this variance and the extinction probability, understood as risk for the population. Motivated by an earlier trade-off analysis between average capital growth rate and risk in Kelly’s gambling model, we study the trade-off between the average growth rate and the variance in the present model. Despite considerable differences between the two models, we find similar optimal trade-off curves (Pareto fronts), suggesting that our conclusions are robust, and broadly applicable in various fields ranging from biology/ecology to economics.