in life history / evolution / stochasticity on Mon 12 January 2015

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

$$x_i(t+1) = \frac{R_i(t)x_i(t)}{R_1(t)x_1(t) + R_2(t)x_2(t)}$$

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

$$y(t+1)=[R_1(t)/R_2(t)]y(t)$$

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

$$\rho = E [ \log (\lambda(i)) ]$$

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,

$$E[f(x)] \approx f( E[x]) + \tfrac{1}{2} f''(E[x]) {\rm Var}(x).$$

With $$f = \log$$ this gives

$$E[\log (\lambda)] \approx \log (E[\lambda]) - \tfrac{1}{2 E[\lambda]^2} {\rm Var}(\lambda),$$

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?