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Question: What is the general quantitative relationship between adaptive phenotypic diversity, or bet-hedging, and the environmental uncertainty that selects for it?

Mathematical methods: Building on the fitness set approach introduced by Levins, we develop a graphical heuristic for determining the optimal amount of diversity in a fluctuating environment. We use as our optimality criterion the expected long-term growth rate of a lineage.

Key insights: Each of the phenotypes in a polyphenic population may be seen as investing a certain proportion of its reproductive effort in each of the possible environments. A bet-hedging lineage that produces the phenotypes in just the right proportions-so that the overall reproductive investment in each environment matches the environmental frequencies-grows faster on average than other lineages. How much faster it grows than the resident population, and thus the strength of selection towards the optimal bet-hedging strategy, depends on how far the residents are from the optimal investment profile.

Predictions: A rigorous empirical demonstration that bet-hedging is adaptive requires a comparison of the degree of phenotypic diversification in similar populations subject to varying levels of environmental uncertainty. We confirm that bet-hedging should be observed only within a certain range of environmental variation; when the environment is more predictable than this, a phenotypic generalist would do better. We furthermore provide a simple method to calculate this range, based on the shape of the fitness trade-offs. Within this range, we predict a linear relationship between the frequency of phenotypes and the frequency of environments, independent of the shape of the trade-offs.