Advanced points about the Fisher model.

Transcription

1 Advanced points about the Fisher model. 1. Strictly speaking, the solution for a* is not an ESS, because the second derivative is equal to zero. That is a consequence of the assumption of infinite population size, and as such, there are linear gains associated with allocation to male function. We are about to study that. 2. The assumption of infinite population size also means that no single individual that deviates from a*=0.5 affects the sex ratio. So, if the population is at equilibrium (a*=0.5), the sex ratio can drift. But if sex ratio drifts away from 0.5, then there is an advantage to individuals that are closer to 0.5 than the population mean. Thus the population mean will tend to be moved by selection back towards the equilibrium (converge on the equilibrium). Hence the equilibrium solution is Convergence Stable. In other words, the equilibrium is a CSS. We will graph that. 1

2 3. This tendency to move toward the attractor of a*=0.5 when the population mean is away from 0.5 is called convergence stability (or continuous stability). Mathematically, the attractor is convergence stable strategy (a CSS) if: 2 W i a res < 0 The Fisherian sex ratio is a CSS. It is not an ESS, but it is a CSS. 4. Finally, the sex ratio model is very good at showing an important point: as the sex ratio gets closer to the attractor of a*=0.5, the selection differential becomes less step, and drift becomes more likely. We will study this in more detail today. End of advanced points. Now: play the barnacle game 2

3 The barnacle game: to get an intuitive feeling for Local Mate Competition, we will play the barnacle game. Local Mate Competition: The Barnacle Game Rules: you have 100 units of resources 1. One egg costs 10 units 2. One sperm costs 1 unit. So you could make 10 eggs and no sperm, Or you could make 9 eggs and 10 sperm, Or 5.5 eggs and 45 sperm Or 0 eggs and 100 sperm. Get into groups of two and cross-fertilize. Then count the number of zygotes that you contributed to. Goal, maximize the number of offspring. Now groups of 3 (or 4?) 3

4 Hamilton s tweak: W.D. Hamilton knew that sex ratios were often female biased, even though the sexes were apparently equally costly. He wrote a very famous paper in 1967 called Extraordinary Sex Ratios (Hamilton 1967). In this paper, Hamilton relaxed the assumptions of infinite population size and random mating. Instead organism could be mating in small groups, like wasps or barnacles. This means that an individual s allocation to sons, or to male function in hermaphrodites, could greatly affect the mean of the group. He called this local mate competition. Here we will study Eric Charnov s (1980) extension to the model, but the ideas are the same As an historical note, Hamilton also introduced this idea as an unbeatable strategy. It was, in fact, the same idea as for an ESS in games against the field. But Hamilton s unbeatable strategy pre-dated the concept of the ESS by six years. ESS caught on, but unbeatable strategy did not. Literature Cited Hamilton, W. D Extraordinary sex ratios. Science 156: (paper posted on course web site). Charnov, E. L Sex allocation and local mate competition in barnacles. Marine Biology Letters 1:

5 Local mate competition in barnacles (relaxing Fisher s assumption of large population size and random mating). 1. We assume that Bateman s principle holds: (i) fitness gains through females (or female function in hermaphrodites) is limited only by resources. All eggs are fertilized. (ii) fitness through males (or male function in hermaphrodites) is determined by the number of females per male in the local population (or the number of eggs fertilized through male function in hermaphrodites). 2. We assume that there is no self-fertilization and no kin mating. This would be true in barnacles, as they are widely dispersed. define variables a i = allocation to male function by i th (our target) individual 1- a i = allocation to female function by i th (our target) individual a res = allocation to male function all others in the mating group a* = allocation to male function at equilibrium K = number of mates in the local mating group C eggs = cost per egg C sperm = cost per sperm 5

6 Now write W male, W female, and W i for our target individual W male = (number of eggs available)*(proportion of the target male s sperm in mating pop) W female = (number of eggs produced) W male = KR(1 a ) (Ra res i / C ) sperm C eggs ((K 1)Ra res / Csperm) + (Ra i / Csperm) W female = R(1 a i ) C eggs W i = W female + W male 6

7 W i = R C eggs (1 a res )a i KR C eggs (a i a res (K 1)) 2 + (1 a res )KR C eggs (a i a res (K 1)) To get the candidates for equilibrium, set a i = a res = a* W i = (a* + K 1 2a * K)R a * C eggs K K 1 Now solve for a* when the first derivative is = 0. We get: a* =! 2K 1 a * = K 1 2K 1 The second derivative is <0, so the solution is evolutionarily stable. The condition for convergent stability is also met. So the solution is an ESS and a CSS 7

8 Advanced point. W i = W m + W f So, at equilibrium W m thus + W f = 0 W m = W f Meaning that (at equilibrium) the marginal gain through male fitness is equal to the marginal loss through female function, or vice versa. If the population is not at a* then, if a small increase in a i increases fitness through male function more than it decreases fitness through female function, then selection will favor an increase in allocation to male function. And vice versa. 8

9 Allocation in barnacles to male and female function in small (N=2) and large (N>3) group sizes. From Raimondi & Martin. (1991) American Naturalist (see web site for pdf) 9

10 The frequency of males produced by a foundress wasp increases with the number of foundresses. the results are very close to the predicted values. (Show L567 lecture 10 LMC supplement.pdf See also L567 lecture 10 LMC clalcs.xls and local mate competition S318 exercise.docx.) 10