TEORÍA DE APRENDIZAJE CONSTRUCTIVISTA

pared to the strength of fluctuations. Consider a population of sizeN and the differences in fitness is given by theselection strengths. The strength of demographic fluctuations decreases with increasing N and hence fluctuations play only a minor role compared to selection by fitness differences for large N. As a rule of thumb, random drift is of minor importance for sN 1 but dominates the dynamics for sN 1. See also the following Section 4.2 where this relation is discussed in more detail.

To summarize, random drift alone is certainly not sufficient to substantially drive evolutionary progress in nature. However, it can be an important part and its role has to be determined from case to case. Here, we consider the role of fluctuations specifically for scenarios where fitness is frequency-dependent.

4.2 A stochastic description of evolutionary dynamics

To take demographic fluctuations in mathematical approaches into account, one has to go beyond the replicator model introduced in Sec. 2, and has to start with a stochastic model, based on birth and death events. We here introduce a common model, the Moran process, see [90, 91, 92, 93]. A similar but time-discrete description is given by the Fisher-Wright process [38, 94, 37, 81], a more detailed introduction can also be found in the Diploma thesis of the author, [95].

Let us again consider only two different traitsAandB. The model assumes a fixed population sizeN in a well mixed population. The number of individuals belonging to typeAandB are given by NA and NB =N −NA respectively. As illustrated in Fig. 4.1, the rates are given

by, ΓB→A= φA ¯ φ NA N NB N ,and ΓA→B= φB ¯ φ NB N NA N . (4.1)

For example, an individual of typeAreplaces one individual of typeB according to its abun- dance NA and its fitness φA. Within a fixed time, the generation time, N such replacing

events occur such that, on average, every individual in the population is replaced once during that time. The full stochastic dynamics is described by a master equation, giving the tem- poral change of the probabilityP(NA, NB;t) for the population to consist ofNA individuals

belonging to traitA. The dynamics can easily be expanded to cases involving more than two types, see for example [95].

In many circumstances, a diffusion-approximation, where the number of individuals belonging to TypeA andB can be described by continuos variables and the dynamics follows a Fokker- Planck equation, works very well. Within this approach, the probability densityP(x;t) for a fraction x = NA

N of type A to be present in the system is described by the Fokker-Planck

equation, ∂tP(x;t) =−∂xα(x)P(x;t) | {z } selection +1 2∂2xβ(x)P(x;t) | {z } f luctuatons , (4.2)

Figure 4.1:The Moran process. A fixed number N of individuals change the traits they belong to by replacing events. This can be illustrated by an urn-model. With respect to the fitness and abundance of different types, one individual is chosen for reproduction. The offspring individual then randomly replaces another individual. Here two different types are denoted as red and blue.

or a corresponding Langevin equation. With the rates, Eqs. (4.1),α and β are given by, α= φAx−φ_¯B(1−x) φ ,and β= 1 N φAx+φB(1 +x) ¯ φ . (4.3)

The first term describes directed drift. In the deterministic limit N → ∞ this is the only remaining term and the dynamics is then given by the replicator equation (2.5), ˙x = α(x). The second term describes the impact of demographic fluctuations, here given by a diffusion term. It induces deviations from the deterministic solutions.

In this continuous description the role of demographic fluctuations is obvious: The strength of fluctuations scale with 1/√N. Thus, for very largeN, fluctuations (via the corresponding diffusion term) affect the dynamics only slightly, leading to centered distributions around the deterministic trajectories. Or, in the other marginal case, where the population size is very small, fluctuations completely dominate the dynamics and fitness differences (and the correspondingαterm) are negligible. In the first extreme, one hasDarwinian selection, while for the second case there isneutral evolution. In between, there is a crossover between both scenarios. It is given when both terms in Eq. (4.2) have about equal weights and balance each other. For the case of frequency independent fitness with a fitness differenceφA−φB =sand

the average fitness ¯f = 1 this is given if,

sN _∼1. (4.4)

ForsN 1, evolution is effectively neutral, while for sN 1, evolution is Darwinian. This condition has been stated by Kimura already [79]. In population genetics literature it is often stated as 2sNe= 1 [79, 36]. However, note that it only gives the rough position of the

crossover. Further, if fitness-terms are more involved, more complex relations might hold. See also the first manuscript at the end of this chapter.

The diffusion approach based on the generalized Moran process, Eqs. (4.2) and (4.3), or the Kimura equation [79, 37],

∂tP(x, t) =−s∂xx(1−x) +µ∂x(1−2x)P(x, t) + ₂1

Ne

4.3 Frequency-dependent scenarios 23