From gambling to growing in uncertain environments
In unpredictably varying environments, it is advantageous for individuals in a population to accept a reduction of their short-term reproductive success in exchange for longer-term risk reduction. This phenomenon called bet-hedging, protects the population from potential damages associated with environmental variations. It is universally present in biology for instance in bacteria resistance to antibiotics, in plants delaying germination or in virus evolution.
The idea of bet-hedging is perhaps best illustrated using Kelly's model, originally introduced in the context of gambling models such as horse races. The gambler strives to optimize his/her capital growth by placing appropriate bets similarly to the biological population which invests in appropriate phenotypes to grow and survive. Following this idea, we first analyze the trade-off between the average growth rate of the capital of the gambler and the risk the gambler takes in Kelly's model . Secondly, we discuss how to extend that model to describe adaptive strategies of gambling .
Then, we turn to the modeling of a biological population embedded in fluctuating environments. Assuming no sensing mechanism, we focus on the simplest non-trivial case, i.e. two randomly switching phenotypes subjected to two stochastically switching environments. Since the optimal asymptotic (long term) growth rate was studied elsewhere; we focus on finite time growth rate fluctuations. An exact asymptotic expression for the variance, alongside with approximations valid in different regimes, are tested numerically. Our simulations of the dynamics suggest a close connection between this variance and the extinction probability, understood as risk for the population. Motivated by our previous analysis of Kelly's gambling model, we study the trade-off between the average growth rate and the variance of the biological population . 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.
 L. Dinis et al., Phase transitions in optimal betting strategies, EPL 131, 60005 (2020).
 A. Despons et al., Adaptive strategy in Kelly's horse race model, arXiv:2201.03387 (2022)
 L. Dinis et al., Pareto-optimal trade-off for phenotypic switching in a stochastic environment, https://www.biorxiv.org/content/10.1101/2022.01.18.476793v1, (2022)
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