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?