Bet-hedging as an evolutionary game: the trade-off between egg size and number

Abstract

Bet-hedging theory addresses how individuals should optimize fitness in varying and unpredictable environments by sacrificing mean fitness to decrease variation in fitness. So far, three main bet-hedging strategies have been described: conservative bet-hedging (play it safe), diversified bet-hedging (don't put all eggs in one basket) and adaptive coin flipping (choose a strategy at random from a fixed distribution). Within this context, we analyse the trade-off between many small eggs (or seeds) and few large, given an unpredictable environment. Our model is an extension of previous models and allows for any combination of the bet-hedging strategies mentioned above. In our individual-based model (accounting for both ecological and evolutionary forces), the optimal bet-hedging strategy is a combination of conservative and diversified bet-hedging and adaptive coin flipping, which means a variation in egg size both within clutches and between years. Hence, we show how phenotypic variation within a population, often assumed to be due to non-adaptive variation, instead can be the result of females having this mixed strategy. Our results provide a new perspective on bet-hedging and stress the importance of extreme events in life history evolution.

Figure 1.

An illustration of the parameters (‘genes’), σ_b, , and σ_w, in the females’ ‘genome’. σ_b is the standard deviation in a uniform distribution with mean and corresponds to between year variation in the mean propagule size produced by each female. Every year, each female will ‘choose’ a value of her mean propagule size (), and the distribution within the clutch around that mean has a standard deviation of σ_w, independently of whether she produces a clutch with normally or uniformly distributed propagule sizes. The dashed line corresponds to an arbitrary m_min, which varies from one time step to another. All propagule sizes smaller than m_min (grey area) will not survive.

Figure 2.

The rate of evolutionary change of the two variation parameters σ_w and σ_b, i.e. within- and between-clutch variation, respectively. The arrows indicate the direction of the mean rate of change from 10 000 simulations of 10 time steps each. A long (short) arrow corresponds to quick (slow) evolution. The solid and dashed lines are the nullclines of σ_w and σ_b evolutionary rates, respectively. The block arrows represent the principal directions of evolutionary change in each region of parameter space. The nullclines intersect at a convergent stable evolutionary equilibrium (solid circle). Only the case of intermediate adult survival is fully shown, but the convergent stable strategies of low survival (s = 0.01, square) and high survival (s = 0.4, triangle) are also depicted. (a) Normal within-clutch distribution. (b) Uniform within-clutch distribution. See text for further model and simulation details.

Figure 3.

The distributions of propagule sizes corresponding to the convergent stable strategies marked in figure 2a. Each depicted distribution is a combination (convolution) of between- and within-clutch variation of propagule size, for the cases of low survival (s = 0.01, solid line), intermediate survival (s = 0.1, long-dashed line) and high survival (s = 0.4, short dashed line). For comparison, the distribution of the minimal viable propagule size, m_min, is also shown (dotted line).