An Introduction to Genetic Algorithms - Boente

Chapter 3: Genetic Algorithms in Scientific Models
Like all mathematical models in population genetics, Kirkpatrick's model makes a number of assumptions that
allow it to be solved analytically: each organism has only one gene of interest; the population is assumed to be
infinite; each female chooses her mate by examining all the males in the population; there are no evolutionary
forces apart from natural selection and sexual selection on one locus (the model does not include mutation,
genetic drift, selection on other loci, or spatial restrictions on mating). In addition, the solution gives only the
equilibrium dynamics of the system, not any intermediate dynamics, whereas real systems are rarely if ever at
an equilibrium state. Relaxing these assumptions would make the system more realistic and perhaps more
predictive of real systems, but would make analytic solution intractable.
Collins and Jefferson proposed using computer simulation as a way to study the behavior of a more realistic
version of the model. Rather than the standard approach of using a computer to iterate a system of differential
equations, they used a genetic algorithm in which each organism and each interaction between organisms was
simulated explicitly.
The simulation was performed on a massively parallel computer (a Connection Machine 2). In Collins and
Jefferson's GA the organisms were the same as in Kirkpatrick's model (that is, each individual consisted of
two chromosomes, one with a p gene and one with a t gene). Females expressed only the p gene, males only
the t gene. Each gene could be either 0 or 1. The population was not infinite, of course, but it was large:
131,072 individuals (equal to twice the number of processors on the Connection Machine 2). The initial
population contained equal numbers of males and females, and there was a particular initial distribution of 0
and 1 alleles for t and p. At each generation a certain number of the t = 1 males were killed off before
reproduction began; each female then chose a surviving male to mate with. In the first simulation, the choice
was made by sampling a small number of surviving males throughout the population and deciding which one
to mate with probabilistically as a function of the value of the female's p gene and the t genes in the males
sampled. Mating consisted of recombination: the p gene from the female was paired with the t gene from the
male and vice versa to produce two offspring. The two offspring were then mutated with the very small
probability of 0.00001 per gene.
This simulation relaxes some of the simplifying assumptions of Kirkpatrick's analytic model: the population is
large but finite; mutation is used; and each female samples only a small number of males in the population
before deciding whom to mate with. Each run consisted of 500 generations. Figure 3.10 plots the frequency of
t = 1 genes versus p = 1 genes in the final population for each of 51 runs—starting with various initial t = 1, p
= 1 frequencies—on top of Kirkpatrick's analytic solution. As can be seen, even when the assumptions are
relaxed the match between the simulation results and the analytic solution is almost perfect.
The simulation described above studied the equilibrium behavior given
Figure 3.10: Plot of the t = 1 (t1) frequency versus the p = 1 (p1) frequency in the final population (generation
500) for 51 runs (diamonds) of Collins and Jefferson's experiment. The solid line is the equilibrium predicted
by Kirkpatrick's analytic model. (Reprinted by permission of publisher from Collins and Jefferson 1992. ©
1992 MIT Press.)
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