Short-term insurance versus long-term bet-hedging strategies as adaptations to variable environments

Abstract

Understanding how organisms adapt to environmental variation is a key challenge of biology. Central to this are bet-hedging strategies that maximize geometric mean fitness across generations, either by being conservative or diversifying phenotypes. Theoretical models have identified environmental variation across generations with multiplicative fitness effects as driving the evolution of bet-hedging. However, behavioral ecology has revealed adaptive responses to additive fitness effects of environmental variation within lifetimes, either through insurance or risk-sensitive strategies. Here, we explore whether the effects of adaptive insurance interact with the evolution of bet-hedging by varying the position and skew of both arithmetic and geometric mean fitness functions. We find that insurance causes the optimal phenotype to shift from the peak to down the less steeply decreasing side of the fitness function, and that conservative bet-hedging produces an additional shift on top of this, which decreases as adaptive phenotypic variation from diversifying bet-hedging increases. When diversifying bet-hedging is not an option, environmental canalization to reduce phenotypic variation is almost always favored, except where the tails of the fitness function are steeply convex and produce a novel risk-sensitive increase in phenotypic variance akin to diversifying bet-hedging. Importantly, using skewed fitness functions, we provide the first model that explicitly addresses how conservative and diversifying bet-hedging strategies might coexist.

Keywords: Cliff-edge effect; environmental canalization; environmental stochasticity; fluctuating selection; geometric mean fitness; variance-sensitivity.

Figure 1

Asymmetric fitness functions and uncertainty in fitness returns produces insurance. Both (A), environmental variation θ (arrows) moving the fitness function around (darker to lighter colors indicate increasing fluctuations in location parameter θ), and/or (B) phenotypic variation among individuals of a genotype (darker to lighter colored lines represent normal distributions of individuals with increasing standard deviations), can cause variation in the fitness returns. Insurance therefore takes the form of an adjustment of the mean phenotypic value μ_k away from the peak of the deterministic fitness function (0, indicated by black dotted line) toward the less steeply decreasing side, to avoid accidentally falling off the cliff‐edge (C). Colored lines in C depict arithmetic mean fitness of a genotype experiencing environmental or phenotypic variation with standard deviation corresponding to the colored curves in A and B (from darker to lighter, 0.5, 1, 1.5, and 2). Dotted lines indicate the peak of these curves, which moves farther away from 0 the more variation there is. See Methods text for more details.

Figure 2

Fitness surfaces for a symmetrical fitness function for genotypes with different values of the mean phenotype μ_k and variation in phenotype σ_k. Contour lines show long‐term arithmetic mean fitness (top row) and geometric mean fitness (bottom row). The position θ of the fitness function fluctuates stochastically between generations, θ∼N (0, σ_θ), the magnitude σ_θ of environmental fluctuations increases successively (from 0 to 3) in the different columns from left to right. Irrespective of the scale of these fluctuations and the phenotypic variation (σ_k), fitness is always maximized by a peak in the contours in the middle of the x‐axis, corresponding to a mean phenotypic value (μ_k) of zero, because the individual fitness function is a symmetrical normal distribution with a mean of 0 and width of ω = 1.

Figure 3

Fitness surfaces for an asymmetrical fitness function for genotypes with different values of the mean phenotype μ_k and phenotypic variation σ_k. Contour lines show long‐term arithmetic mean fitness (top row) and geometric mean fitness (bottom row). The position θ of the fitness function fluctuates stochastically between generations, θ∼N (0, σ_θ), the magnitude σ_θ of the fluctuations increasing from left to right. In contrast to Fig. 2, as soon as σ_θ > 0 fitness is always maximized here by a peak in the contours to the right of the middle of the x‐axis (where μ_k=0), because the individual fitness function is a skew normal distribution with a mode of 0, width of ω=1 and skew of α=5.

Figure 4

Conservative bet‐hedging (CBH), defined as the difference in the mean phenotype (μ_k) maximizing long‐term arithmetic fitness and the phenotype that maximizes geometric mean genotype fitness, for given phenotypic standard deviation (σ_k). Lines represent results for different values environmental variation σ_θ, as in Fig. 3. The individual fitness function w has a width of ω = 1 and skew of α = 5, also as in Fig. 3. Colored stars along the σ_k axis represent the optimal amount of DBH for the corresponding magnitude of environmental variation σ_θ and width of the fitness function ω.

Figure 5

Cross‐sections of fitness surfaces in the third column of Fig. 3 (σ_θ = 1). The red line is the fitness surface for arithmetic mean fitness, the blue line is the fitness surface for geometric mean fitness. Dotted vertical lines show the maxima for the respective functions. (A) A horizontal cross‐section taken in the trait (μ_k) dimension at σ_k = 0, hence the individual fitness function (black line) peaks at zero, but fluctuates over time (black arrows), which causes mean fitness to be maximized at positive values of mean phenotype for both arithmetic and geometric mean. (B) A vertical cross‐section taken in the phenotypic variance (σ_k) dimension at μ_k = –2, and so diversifying bet‐hedging is favored by geometric mean fitness, but arithmetic mean fitness also increases with phenotypic variance, because of Jensen's inequality (the individual fitness function is strongly convex at μ_k = –2) and hence what is known as a variance‐prone strategy – see text for details.