Version of notes for discussion I led in Sebastain Schreiber's for PBG 271 Research Conference in Ecology on "Life history evolution in stochastic environments".

### Reading Childs et al and Stearn's piece on Bernoulli

Childs, Metcalf and Rees (2010) present a plant-focused review of bet-hedging, including a review of essential theory. As in many papers on bet-hedging, these authors distiguish between conservative and diversified strategies.

#### Geometic mean fitness concept

When fitness is temporally variable, the appropriate measure of evolutionary success is the expected geometric mean growth rate of a genotype. This is because fitness, like population growth, is an inherently multiplicative process that is very sensitive to occasional low values. -- Childs et al 2011

Let's remind ourselves why.

Following roughly the notation of Frank and Slatkin (1990), let \(x_i\) the frequency of genotype \(i\) and \(R_i(t)\) the mean fitness of genotype \(i\) in year \(t\), then genotype frequency \(i\) changes as

Setting \(y(t) = x_1(t)/x_2(t)\), we get

Hence, \(y(t) = y(0) \prod_{i=0}^{t-1} \lambda(i)\) where \(\lambda(t) = R_1(t)/R_2(t)\) is a (non-standard) relative fitness measure.

We can rearrange this and take logs to see \(\tfrac{1}{t} \log (y(t)/y(0)) = \tfrac{1}{t} \sum_{i=0}^{t-1} \log \lambda(i)\). Applying the law of large numbers, \(E[\log \lambda(t)]\) determines whether genotype 1 "wins" (i.e. expectation is >0) or "loses" (i.e. expectation is <0).But this (\((\prod_t \lambda(i))^{\tfrac{1}{t}}\)) is the geometric mean fitness!

#### So the stochastic growth rate *is* the geometric mean fitness

And these are the appropriate measure of success for long-term evolution in a varying environment. Stochastic growth rate \(\rho\) (in Childs' notation) with \(\lambda(i)\) fitness in generation \(i\) is

Which we just saw above is the geometic mean fitness.

#### So how does this lead to a mean-variance tradeoff?

As Childs et al note, Jensen's inequality here implies the geometic mean fitness is less than the arithmetic mean \(\rho \le \log E[\lambda(i)]\).

More specifically, as Sebastian mentioned last week, we can take a Taylor approximation,

With \(f = \log\) this gives

implying the magnitude of the decrease from arithmetic to geometic mean fitness is proportional to the squared coefficient of variation in fitness. In fact, if \(\lambda(i) \approx 1 + \epsilon_i\) with \(\epsilon\) a "small" random variable with mean zero this gives the common approximation to the stochastic growth rate \(\lambda_S \approx \bar \lambda - \tfrac{1}{2} \sigma_\lambda^2\).

This leads to Slatkin's (1974) coining of *evolutionary bet-hedging*
to denote the resulting "tradeoff" where strategies with lower
arithmetic mean fitness can invade against those with higher
arithmetic mean fitness (if they have lower variance).

### Discussion points

- is "tradeoff" a good term here? doesn't evolution just favor the maximizer of geometric mean fitness?
- evidence for bet hedging in nature. what is needed and what has been shown?
- BH in structured populations -- IPMs are suggested as means for analyzing, what are obstacles to getting there (e.g. data limitation)?
- conservative versus diversified BH: how to connect this terminology to our framework? Any concise model of "conservative" BH?
- whose fitness is it?