34

4

Evolution

Problem. Outline how Hamilton’s rule suggests conditions under which coöperative

behaviour can evolve.

4.1.2

Models of Evolution

Typical approaches assume a constant population of upper MM individuals, each of whose

inheritable characteristics are encoded in a string (the genome bold ss, synonymous with

genotype) of upper NN symbols, s Subscript i Baseline comma i equals 1 comma ellipsis comma upper Nsi, i = 1, . . . , N. upper NN is fixed, and environmental condi-

tions are supposedly fixed too. All of the individuals at generation tt are replaced by

their offspring at generation t plus 1t + 1. The state of the population can be described by

specifying the genomes of all individuals. Typically, values of upper MM and upper NN are chosen

such that the occupancy numbers of most possible genomes are negligibly small;

for example, if upper N tilde 10 Superscript 6N ∼10⁶ and upper M tilde 10 Superscript 9M ∼10⁹, upper M much less than 2 Superscript upper NM ≪2^N, the number of possible genomes

assuming binary symbols. In classical genetics, attention is focused on a few char-

acteristic traits governed by a few alleles, each of which will be carried by a large

number of individuals and each of which acts independently of the others (hence,

“bean bag genetics”); modelling is able to take much better account of the epistatic

interactions between different portions of the genome (which surely corresponds

better to reality).

The model proceeds in three stages (cf. evolutionary computing, Sect. 4.3):

Reproduction: Each individual produces a certain number of offspring; the individ-

ual alphaα at generation tt is the offspring of an individual (the parent)

that was living at generation t minus 1t −1 and which is chosen at random

among the upper MM individuals of the population.

Mutation: Each symbol is modified (flipped in the case of a binary code) at a

rate muμ; the rate is constant throughout each genome and is the same

from generation to generation.

Selection: The genome is evaluated to determine its fitness upper W left parenthesis bold s right parenthesis equals normal e Superscript upper F bold s divided by upper CW(s) = e^Fs/C, ¹⁶

which, in turn, determines the number of offspring.upper CC is the selective

temperature.

The topography of a fitness landscape is obtained by associating a height upper F left parenthesis bold s right parenthesisF(s) with

each point bold ss in genotype space. Various fitness landscapes have been studied in

the literature; limiting cases are those lacking epistatic interations (i.e., interactions

between genes) and those with very strong epistatic interations (one genotype has

the highest fitness; the others are all the same). In the latter case the population

may form a quasispecies (the term is due to EigenSuperscript 45⁴⁵), consisting of close but not

identical genomes. Distances between genomes ss and s primes^, are conveniently given by

the Hamming distance:

16 The fitness of a phenotypic trait is defined as a quantity proportional to the average number of

offspring produced by an individual with that trait, in an existing population. In the model, the fitness

of a genotypebold ss is proportional to the average number of offspring of an individual possessing that

genotype.