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15.2.5. Fitness and the mutation-selection equation
Selection acts on phenotypes. The fact that an individual survives and reproduces
(so perpetuating its genes) may depend on chance. We may however compute (and
sometime measure) the \propensity" of an individual to survive and produce viable
ospring. This quantity is termed \tness", and for our assumptions we may assume
that it is proportional to the average number of sons reaching the reproductive age
for a given phenotype u, and for a given time interval (generation).
We are now in the position of simulating the evolution of a population. We
may represent it as a set of N individuals x (j) , j = 1; N, characterized by their
genotype (that unambiguosly characterize the phenotype within our assumptions).
The spatial structure of the system may be that of a regular graph (or continuous
space with an interaction range), or a social network that may evolve with the
system (and in this case may be related to the genotype of the organisms), or a
\well stirred" environment where all interactions are equally probable.
In each generation, each individual interact with one or more other individuals,
accumulating a \score" (tness), that may determine its survival or, equivalently,
the probability of producing ospring.
In the simplest arrangement, the time evolution (discrete generations) of an
asexual population is given by the following phases:
1. Scoring phase: each individual is allowed to interact with others, according
with its spatial connectivity (and possibly to displacement like evasion, pursuit,
etc.). Their interactions depend on the relative phenotypes, and contribute to
the accumulation of a score (generally reset to a default value at the beginning
of the phase). The score may be negative or positive.
2. Survival phase: individuals can survive with a probability that is a monotonic
function of the score.
3. Reproduction phase: empty locations may be colonized by neighboring ones.
Colonization implies copying the genome with errors (mutations).
The previous model represents a whole ecosystem, but one is often interested
only on some subpart of an ecosystem, like one or few species. If the consistency
of a species does not aect others (for instance, one can assume that a species
of herbivores does not modify the abundance of grass), then one can neglect to
simulate the invariable species, by changing the default value of the score.
Let me be more explicit. Assume that a positive score increases survival, and a
negative one decreases it. The presence (interaction) of grass, increases the survival
of herbivores, while the encounter with a predator decreases it. If one is interested
in the simulation of the interplay among grass, herbivores and carnivores, then all
three species should be simulated. But if one assumes that grass is abundant and
not modied by herbivores, one can neglect to include the vegetable substrate, and
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