### Abstract

- Top of page
- Abstract
- Introduction
- Large populations
- Finite populations
- Conclusions
- Acknowledgments
- References

In mixed or ‘bet-hedging’ strategies, offspring phenotypes are taken randomly from a distribution determined by the genotype and shaped by evolution. Offspring of a single parent represent a finite sample from this distribution, and therefore are subject to variability because of sampling. Contrary to a recent article by A.M. Simons (2007; *J. Evol. Biol.***20**: 813–817), I show that selection does not favour the production of many offspring just to reduce sampling variability when such mixed strategies are used in large populations.

### Introduction

- Top of page
- Abstract
- Introduction
- Large populations
- Finite populations
- Conclusions
- Acknowledgments
- References

In a recent paper in this journal, Simons (2007) claimed that natural selection acts directly to increase the number of offspring when the parent uses a mixed strategy, i.e. when offspring of the same parent exhibit random phenotypes chosen from a certain distribution. Such mixed strategies, often referred to as ‘bet-hedging’ strategies, are favoured in unpredictable stochastic environments (Levins, 1962; Slatkin, 1974; Bull, 1987; Philippi & Seger, 1989; Haccou & Iwasa, 1995; Sasaki & Ellner, 1995), and they are considered in this context by Simons (2007). Mixed strategies can, however, also evolve under asymmetric competition (Geritz, 1995) and elsewhere (Leimar, 2005; Matessi & Gimelfarb, 2006), and Simons’ argument actually does not depend on the reason why a mixed strategy evolved in the first place.

Simons (2007) argues that with more offspring, the phenotypic variability of the offspring of one parent will be less affected by sampling and will therefore be closer to the mixing distribution coded by the parent's genotype. Consequently, there will be less year-to-year variation between the realized distribution of offspring (each year being closer to the genetically determined distribution). Simons uses a simulation model of a specific system to demonstrate that reducing the year-to-year sampling variation increases the geometric mean fitness. Therefore, everything else held constant, the long-term population growth rate increases *more than proportionally* to the number of offspring. In models of optimal allocation between offspring size and offspring number (e.g. Smith & Fretwell, 1974), the growth rate is assumed to be proportional to offspring number. Direct selection on offspring number, i.e. deviation from proportionality, would amount to a hitherto unrecognized mechanism and would alter the optimal balance between offspring size and offspring number whenever offspring phenotype follows a mixed strategy.

In this note, I set up a simple analytical model to show that Simons’ (2007) claim is incorrect and that in large populations there is no direct selection on offspring number due to sampling stochasticity. Unlike simulation studies, which are necessarily bound to make rather specific assumptions, the results of the present analysis are valid under very broad conditions. In finite populations, there may be a weak effect similar to the one proposed by Simons, but this cannot be studied using his method based on the geometric mean fitness.

### Large populations

- Top of page
- Abstract
- Introduction
- Large populations
- Finite populations
- Conclusions
- Acknowledgments
- References

Assume that a parent produces *n* offspring. The phenotypes of the offspring are random variables taken from a genetically determined distribution. Hence the fitness accrued by the *i*th offspring of the *k*th parent, *ξ*_{i,k}(*t*), is also a random variable, which may also depend on year *t* as in stochastic environments. *ξ*_{i,k}(*t*) is taken from the distribution *f*_{t}(*ξ*), which can be determined for any specific model of interest, but the detailed properties of which are not important here. I denote the expectation and the variance of *f*_{t}(*ξ*) by *μ*_{t} and , respectively. *f*_{t}(*ξ*) realistically depends on population size *N*(*t*) in order to avoid exponential population growth.

Simons (2007) describes his calculation of the geometric mean fitness only verbally. His description appears to translate into calculating the average fitness of the offspring of an arbitrary parent *k* in year *t*,

- (3)

and taking the geometric mean of *w*_{k}(*t*) over time. The geometric mean of *w*_{k}(*t*) times *n* would be the geometric mean growth rate that characterizes long-term fitness of a parental genotype. Simons argues that more offspring reduces the variance of *w*_{k}(*t*) over time and hence increases the geometric mean of *w*_{k}(*t*). This implies that increasing *n* increases the geometric mean growth rate more than proportionally. [Technically, Simons did not multiply with *n* but showed simply that the geometric mean of *w*_{k}(*t*) increases with *n*, in order to isolate the presumed advantage of increased diversification among the offspring of a single parent from the obvious proportional advantage of having more offspring; see p. 815 of Simons (2007).]

There are two problems with Simons’ argument. Firstly, the geometric mean of *w*_{k}(*t*) characterizes the long-term growth of a highly unrealistic population, where each parent within a year has the same average offspring fitness as the arbitrarily chosen parent *k* such that *w*_{k}(*t*) = *w*(*t*) for all *k*. Because offspring of many parents amount to many independent samples from distribution *f*_{t}(*ξ*), this is as if repeated sampling would turn up exactly the same sample average in a large number of samples. In reality, offspring of different parents are not carbon copies of each other. Hence one should not repeat the same *n* offspring fitness values *N*(*t*) times to construct the offspring population, but take *N*(*t*)*n* independent values from the distribution *f*_{t}(*ξ*) as in eqns (1)–(2). When *N*(*t*) is large, the year-to-year variation caused by randomly sampling *N*(*t*)*n* offspring is negligibly small.

Secondly, under biologically realistic conditions the geometric mean of *w*_{k}(*t*) is zero for all strategies. To see this, note that *f*_{t}(0) > 0 because there is a nonzero probability that a given offspring dies before reproduction and thus accrues zero fitness. Therefore, all the *n* offspring of a parent in year *t* die with probability [*f*_{t}(0)]^{n}, which is a positive probability as long as *n* is finite. In the year when this happens, the value of *w*_{k}(*t*) is zero. Because the geometric mean is defined over an infinitely long-time period, there will be a year when *w*_{k}(*t*) is zero with probability 1, which makes the geometric mean of *w*_{k}(*t*) zero. As this is true for any conceivable strategy, the geometric mean of *w*_{k}(*t*) is useless as a measure of fitness.

Because in Simons’ simulations there seem to be only *n* independent offspring, there must have been years when all offspring died. Instead of an infinitely long time, however, Simons calculated the geometric mean fitness over only 30 years; hence, a number of simulations could avoid extinction. Occasional extinctions are not visible in Fig. 2 of Simons (2007) because it shows the arithmetic average of 100 estimations of the geometric mean fitness.

Notice that environmental stochasticity did not play any important role in the above argument. Indeed, if *f*(*ξ*) is independent of time *t* (and hence its mean *μ* and variance *σ*^{2} are constant numbers), then eqns (1)–(2) simplify to *N*(*t*+1) = *N*(*t*)*n**μ* and the long-term growth rate is *n**μ*, proportional to *n* as expected. In Simons derivation, *w*_{k}(*t*) still varies from year to year because of sampling stochasticity; it takes value zero with probability [*f*(0)]^{n}, which makes its long-term geometric mean zero with probability 1. Environmental stochasticity is one possible reason why a mixed strategy *f*(*ξ*) with positive variance *σ*^{2} is advantageous over pure strategies with zero variance, but there exist other evolutionary routes to mixed strategies as well (see e.g. Geritz, 1995; Leimar, 2005; Matessi & Gimelfarb, 2006). The present results hold independently of what selects for a mixed strategy or whether a mixed strategy is advantageous at all.

### Finite populations

- Top of page
- Abstract
- Introduction
- Large populations
- Finite populations
- Conclusions
- Acknowledgments
- References

The above argument does not apply to small populations, where the total number of offspring per generation, *N*(*t*)*n*, cannot be considered infinite. For permanently small populations, the geometric mean growth rate is inappropriate as a measure of long-term fitness: Because there is a small but positive probability that every individual of a finite population dies before reproduction, the geometric mean growth rate of any strategy is zero and the population will eventually go extinct with probability 1.

Extinction may nevertheless take a long time unless the population is very small, and therefore it makes sense to model natural selection conditionally on nonextinction. Gillespie (1974) showed that in finite populations living in stable environments, selection acts to decrease the variance in the number of offspring per parent. The strength of this effect is proportional to the inverse of population size and is therefore weak unless the population is very small and prone to extinction. If parents with a mixed strategy have more offspring, the realized phenotypic distribution of the offspring of each parent comes closer to the distribution determined by the genotype, and therefore the variance of the number of surviving offspring decreases. Selection for decreased variance in offspring number may thus lead to a weak effect similar to the one proposed by Simons (2007), but only in finite populations. Because this effect disappears as population size becomes large, this cannot be studied using the geometric mean growth rate. Methods suitable to investigate evolution in finite populations are discussed e.g. by Proulx (2000); Proulx & Day (2001) and Rousset (2003).

### Conclusions

- Top of page
- Abstract
- Introduction
- Large populations
- Finite populations
- Conclusions
- Acknowledgments
- References

Using a general analytical approach, I showed that selection in large populations does not favour more offspring only in order to reduce the variability because of sampling when the offspring phenotypes are taken randomly from a genetically determined distribution as in case of mixed or ‘bet-hedging’ strategies. This result is due to the fact that one has to average not only over the offspring of one parent but also over all parents of a given genotype which reproduce in the same year. Consequently, the size of the sample is not the size of a family but the size of the entire offspring population. In large populations, therefore, the sampling variability is vanishingly small.

There may be direct selection on offspring number in small populations (assuming the population escapes extinction) in order to decrease the variance in fitness accrued by phenotypically variable offspring. This effect can, however, not be studied using the geometric mean growth rate, which is zero for all strategies in finite populations.

This is not to argue that environmental stochasticity could not influence the optimal allocation between offspring size and number. If, for example, large offspring are less sensitive to harsh environmental conditions, then increasing offspring size is an alternative to bet hedging via dispersal or dormancy (Venable & Brown, 1988). I do not claim either that the annual growth rate *n**μ* must always be proportional to offspring number *n*. For example, many offspring may saturate predators which must spend time handling each individual prey, and therefore *μ* may increase with *n*. This note shows that only reduced sampling variability within a family is not sufficient to generate direct selection for increased offspring number in large populations.