Evolutionary bet‐hedging reconsidered: What is the mean–variance trade‐off of fitness?

Bet-hedging in parasitoids: when optimization is not the best strategy to cope with climatic extremes

Bet-hedging occurs when unreliable environments select for genotypes exhibiting a lower variance in fitness at the cost of a lower mean fitness for each batch of progeny. This means that at the level of the genotype, the production of mostly non-optimal phenotypes may be favored when at least some phenotypes are successful. As extreme unreliable climatic events are increasing because of climate change, it is pertinent to investigate the potential of bet-hedging strategies that allow insects to cope with climate change. Evidence for bet-hedging is scarce in most insects, including parasitoids, but the unique lifestyle and biology of parasitoids leads to the expectation that bet-hedging may occur frequently. Here, we evaluate a range of parasitoid traits for which a bet-hedging strategy could be envisioned even if bet-hedging has not been identified as such yet. Under-identification of bet-hedging in nature could have resulted from a major focus of studies on parasitoid life history evolution and foraging behavior on optimality models, predicting how mean fitness can be maximized. Most environmental factors, however, vary unpredictably. Life history and behavioral adaptations are thus expected to be affected by environmental stochasticity. In this paper, we review different aspects of parasitoid behavior, physiology, and life histories and ask the question whether parasitoid traits could have evolved under selection by environmental stochasticity.

Beyond the (geometric) mean: stochastic models undermine deterministic predictions of bet hedger evolution

Bet hedging is a ubiquitous strategy for risk reduction in the face of unpredictable environmental change where a lineage lowers its variance in fitness across environments at the expense of also lowering its arithmetic mean fitness. Classically, the benefit of bet hedging has been quantified using geometric mean fitness (GMF); bet hedging is expected to evolve if and only if it has a higher GMF than the wild-type. We build upon previous research on the effect of incorporating stochasticity in phenotypic distribution, environment, and reproduction to investigate the extent to which these sources of stochasticity will impact the evolution of real-world bet hedging traits. We utilize both individual-based simulations and Markov chain numerics to demonstrate that modeling stochasticity can alter the sign of selection for the bet hedger compared to deterministic predictions. We find that bet hedging can be deleterious at small population sizes and beneficial at larger population sizes. This non-monotonic dependence of the sign of selection on population size, known as sign inversion, exists across parameter space for both conservative and diversified bet hedgers. We apply our model to published data of bet hedging strategies to show that sign inversion exists for biologically relevant parameters in two study systems: Papaver dubium, an annual poppy with variable germination phenology, and Salmonella typhimurium, a pathogenic bacteria that exhibits antibiotic persistence. Taken together, our results suggest that GMF is not enough to predict when bet hedging is adaptive.

A long-term alternative formula for a stochastic stock price model

This study presents a long-term alternative formula for stock price variation described by a geometric Brownian motion on the basis of median instead of mean or expected values. The proposed method is motivated by the observation made in remote fields, where optimalizty of bet-hedging or diversification strategies is explained based on a measure different from expected value, like geometric mean. When the probability distribution of possible outcomes is significantly skewed, it is generally known that expected value leads to an erroneous picture owing to its sensitivity to outliers, extreme values of rare occurrence. Since geometric mean, or its counterpart median for the log-normal distribution, does not suffer from this drawback, it provides us with a more appropriate measure especially for evaluating long-term outcomes dominated by outliers. Thus, the present formula makes a more realistic prediction for long-term outcomes of a large volatility, for which the probability distribution becomes conspicuously heavy-tailed.

A new long-term measure of sustainable growth under uncertainty

The trade-off between short-term success and long-term sustainability is a common subject of great importance both in the biological evolution of organisms and in the economic activities of human beings. In evolutionary biology, bet-hedging theories have described it as the trade-off between the (within-generation) arithmetic mean fitness and the (between-generation) geometric mean fitness of a genotype. Accordingly, bet-hedging strategies observed in various organisms are regarded as optimizing the geometric mean fitness. To increase the geometric mean fitness signifies to suppress the between-generation variance in the mean fitness. Thus, this view is consistent with mean-variance portfolio analysis in which the standard deviation of a portfolio is regarded as a measure of risk. In the present study, we provide yet another measure of long-term sustainability, which is based on minimization of the probability of extinction/bankruptcy that randomly varying population/asset size after a long time becomes less than a certain small value. We present results for representative examples to show that the present criterion gives a qualitatively similar but quantitatively different prediction from the traditional ones. In particular, we emphasize that maximizing survival probability (i.e. minimizing extinction probability) is equivalent neither to maximizing geometric mean fitness nor to minimizing variance in mean fitness, while these three are consistently related to each other.