# A phenotypic view of evolution Evolution in Structured Populations

## On partitioning fitness

I realized last week that I should probably explain more clearly what I meant by temporal components of fitness. As a result, a discussion of variance will have to wait until next week (I actually am flying to California for a National Academies Keck Futures Initiative meeting next week, so the next update may be two weeks).

What I am using is Arnold and Wade’s (1984, Evolution 38:709) episodes of selection paper. In this paper they partition episodes of selection using a very simple idea. They start with the multivariate breeders equation, but in reality there is no need for that. Lets forgo the opportunity to be cool and use matrices, and instead use the simple univariate breeders equation:

Where (Yea, you know the drill): z̄’ is the mean phenotype between generations, VA is the additive genetic variance, VP is the phenotypic variance, and z̄* is the mean phenotype within generations due to selection.

Another point (yea, I know: boring) but one that bears repeating, is that z̄ etc. are real quantities with units attached to them. For example, the average weight of men in the United States is z̄ = 195.5 pounds. It is important to remember that, all equations aside, we really are talking about the real world!

From our perspective the important part of this equation is the selection differential S:

What Arnold and Wade pointed out was that you could easily divide this selection vector into multiple episodes of selection by simply adding and subtracting the mean phenotype at various moments in the history of the organism:

In other words, it is quite easy to divide an overall selection differential into multiple episodes of selection. So, that tells us about selection, but does not tell us about fitness. However, that is easy if we further expand our equation. In particular, the mean of the population after selection, but before reproduction, is found by taking the fitness-weighted sum of the traits:

What can be seen from this is that indeed each episode of selection is associated with an episode specific relative fitness.

It is a bit involved, but Arnold and Wade show that absolute fitness can be partitioned easily as long as the absolute fitnesses are multiplicative. That is:

where capital Wi etc. is the absolute fitness of the ith individual. Absolute fitness is the fitness of an individual on an arbitrary scale – for example, it might be the number of eggs a frog lays ranging from zero to the maximum clutch size possible – whereas the breeders equation uses relative fitness, which has a mean of one. For each episode of selection the relative fitness must be calculated as  .

What it means for fitnesses to be multiplicative is that they need to be conditioned on previous events. Examples of appropriate conditional fitnesses would be:

W1 = the probability of being born alive

W2 = the probability of surviving to adulthood given that the individual was born alive

W3 = the probability of successfully mating given that the individual survived to adulthood.

W4 = the number of eggs the individual lays given it successfully mated

Note that in each case the fitness is a function of previous events. Thus, survival to adulthood is only measured on those individuals that are born alive, etc.

The beauty of this is that as I pointed out last week it is probably impossible to measure true fitness. What this paper tells us is that even if we can’t actually measure total fitness, as long as we are honest about what temporal component we measure, and the temporal component of fitness is multiplicative (e.g., we measure survival to adulthood only on live born individuals) we are safe. We simply need to make sure that we are aware that other episodes of selection, and other temporal fitness components affect the evolution of the trait. For example, Arnold and Wade measured selection for mating success in male frogs. They found that mating selection favored larger males. One could immediately ask why frogs aren’t huge. A reasonable answer would be that there are selection episodes other than mating that keeps the males size in check.

The net result is that we can define “fitness” in conceptual terms that cannot be realistically measured, and then when we actually need a definition that can be used in actual research or modeling just use a valid temporal fitness component and call it good. Last week I defined fitness as the probability of founding a lineage that persists until some arbitrary point (such as a speciation event) in the distant future. First off, there is an obvious problem with this definition for sexual species, so that was definitely not the last word in fitness definitions. But for the purposes of this discussion it is also a useless definition from a practical perspective. However, using this idea of partitioning fitness it is easy to see we can have this useless conceptual definition, and at the same time use a temporal component of fitness that Can be measured. For example, to the list above we can add an additional component of fitness:

W5 = the probability an egg gives rise to a lineage that survives to the arbitrary point in the future times the number of eggs laid.

Now we have our unworkable conceptual definition of fitness, and we can still, say focus on sexual selection and W3, the probability of successfully mating given that the individual survived to adulthood. In short, we can have our cake and eat it too. We can (although apparently I can’t) develop a comprehensive definition of fitness that will satisfy the most demanding critic, and still develop a temporal component of fitness that is usable in experimental settings.