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In the following we will study the evolution of the distribution Φ(θ). For this purpose, we introduce a technique from evolutionary quantitative genetics,15 _{first analyzed by}

Lande (1976). This approach provides a tractable method to study an evolutionary process within a continuously polymorphic population, i.e. to address the evolution of a type distribution.

Consider a large population which is heterogeneous along one trait α. The trait value is normally distributed with mean ¯α and variance σ2_{, α ∼F(α,α, σ¯} 2_{). To sim-}

plify notation, we will denote the distribution function byF(α) and the corresponding density function by f(α). Let the fitness of an α-type for a given distribution with mean ¯α be givenw(α,α¯). The mean fitness of the population is then

¯

w=

Z

w(α,α¯)dF(α). (9)

Here we allow for frequency dependent fitness. Fitness is called frequency dependent, if the fitness of a α-type does also depend on the composition of the population.16 _In

economic terms, frequency dependence is given if one group of subjects – respectively the strategy played by theseα-types – creates an externality on other subjects’ fitness. Consider for example a predator that hunts on a prey structured in different size classes and all predators are specialized to hunt on only one size class. If hunting has a significant effect on the prey populations, then the evolutionary success of a predator is dependent on how many other predators are specialized on the same class.17

Within one generation, the change in the trait mean value in response to selection is defined as

∆¯α= ¯αs−α¯, (10)

15_{Compare Falconer and Mackay (1995) and Roff (1997) for an introduction to quantitative genetics.}

A critical review is provided by Pigliucci and Schlichting (1997).

16_{Of course, frequency dependent fitness depends also on the variance. In order to ease notation,}

we have suppressed this variable inw(.).

17_{A trait with frequency independent fitness would be the case of different breathing technics,}

which lead to different capabilities in oxygen uptake. Whatever distribution of effective and ineffec- tive breathers we assume, their cumulative impact on atmospheric oxygen is insignificant to change the amount of oxygen potentially available to other individuals. Hence, the fitness of each type is independent of the distribution of the different types.

where ¯αs, the mean trait value after selection, is defined as ¯ αs = Z αw(α,α¯) ¯ w dF(α). (11)

The logic expressed in (11) is similar to standard replicator dynamics. While the initial frequency of a type wasf(α), the post-selection frequency of this α-type, w(α,_w¯α¯)f(α), will be higher for types with above-average fitness. Hence, in the computation of ¯αs, more successful types will get a higher weight than less successful types. If, for example, the studied trait is body size and e.g. large individuals (with high α-values) are more fit than short individuals, the average body size in the population would increase due to natural selection: ¯αs>α¯ and ∆¯α >0.

The analysis so far describes selection within one generation. In order to address the (inter-generational) evolution of the traitα, we follow Lande (1976) and introduce the following structure of reproduction: First, only selected individuals produce the next generation of offspring. Second, sexual reproduction is assumed with random partner choice. That is, two random subjects who survived selection mate, produce offsprings and die thereafter. For a reasonable selective pressure18 _{this mechanism transforms}

the initial distribution back to a normal distribution with constant variance σ2 _but

a different mean. Starting from a normal distribution with mean α, selection will first lead to a distribution, which deviates from the normal distribution. The mean of this (non-normal) distribution after selection is given by ¯αs from (11). After random mating and reproduction, however, the distribution ofα-values in the new generation is again normal withF(α,α¯s, σ2). While the variance is unaffected by this process, the mean of the distribution changes from ¯α to ¯αs. The direction of evolution is therefore determined by selection, characterized in (10) and (11). This now allows us to analyze the evolutionary process in more detail.

From (9) we can derive the change in mean fitness from a marginal change in ¯α,

∂w¯ ∂α¯ = Z w(α,α¯)∂f(α) ∂α¯ dα+ Z ∂w(α,α¯) ∂α¯ dF(α). (12)

While the first term characterizes the direct change in the mean fitness due to a change in the composition of the population, the second term depicts the indirect, frequency dependent fitness impact. From the density of the normal distribution we can easily compute ∂f(α)/∂α¯. Substituting in (12) and rearranging yields19

18_{For a detailed discussion see Lande (1976).}

∆¯α= σ2 ¯ w Z w(α,α¯)∂f(α) ∂α¯ dα (13)

which can be also expressed as

∆¯α= 1 ¯

w Z

[w(α,α¯) (α−α¯)]dF(α). (14)

The right hand side in equation (13) respectively (14) characterizes pace and direction of the evolutionary process. As ¯w > 0, the direction of the evolutionary change in the mean trait value, ¯α, is determined by the sign of the integral in (14). Note, that the integral term represents only the direct change in mean fitness (the first term in equation (12)). From (14) therefore follows that the evolution of ¯α is independent of the frequency dependent fitness change associated with a variation in ¯α. If the direct fitness change is positive, the distribution will evolve towards a higher mean ¯α. An evolutionary equilibrium is reached if ∆¯α= 0. Such an equilibrium is characterized by

Z

[w(α,α¯e_{) (α−α¯}e_{)]dF(α) = 0, (15)}

where ¯αe _{denotes the equilibrium mean trait value.}

4.4

The Indirect Evolution of Conditional