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An Inclusive Fitness Example
We’ve seen that the effect of a social behaviour can be broken down into the expression,
WI =
N
−1
X
i=0
∂wi
Roi .
∂z0
This is the inclusive fitness effect. It says that the effect of a behaviour can be decomposed into the effects
on the fitness of each member of the population, weighted by the relatedness between that recipient and the
actor. This is a very mysterious object and a clarifying example is in order.
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Example on a 4-cycle:
Consider a (haploid, asexually-reproducing) population that resides in the environment depicted in figure 1.
Each vertex is occupied by a single individual. Births and deaths follow a Moran process: at some instant
an individual dies at random. The neighbours then compete for the vacated site. Eacch can occupy the site
with probability proportional to their relative fitness. For example, if the individual at 1 dies, then 0 and 2
compete to place an offspring at that site. Individual 0 places an offspring with probability w0 /(w0 + w2 );
individual 2 does with probability w2 /(w0 + w2 ).
0
1
3
2
Figure 1: A four-cycle population.
Suppose individual 0 suddenly behaves altruistically, giving a benefit b to each of its neighbours at a cost
2c. We are interested in evaluating if this altruistic gene will spread. We do this with an inclusive fitness
analysis. There are two components to such an analysis:
1. How is the fitness of oeach member of the population affected by the altruistic act?
2. How is each individual related to the actor?
These two questions can be answered in any order. I will start with the second.
1.1
Relatedness on the 4-cycle
We need to calculate
R0i =
cov(x0 , xi )
.
cov(x0 , x0 )
As I mentioned before, this is a measure of the probability that 0 and i share a gene from a common ancestor.
But how?
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Denote the probability that individual i and j share a gene from a common ancestor by rij , and the
average of all rij by rave . It can be shown [?] that the Rij and rij relate via
Rij = lim
µ→0
rij − rave
.
1 − rave
Here, µ is the mutation rate from A to B and from B to A. We suppose that this is some small number, so
that µ ≈ 0. To find the Rij we establish a system of equations describing the evolutionary process in the
neutral population.
I’ll go on a tangent here for a bit, for the benefit of the uninitiated. Suppose that the gene B codes for
(B)usiness as usual and A codes for (A)ltruist. When I write neutral population I am assuming that the
altruist gene may be present in the population but that it is not yet “turned on”; an individual may have
the A gene, but does not yet act altruistically. Time goes on and individuals die and give birth at random.
There is no fitness advantage to either B or A. Then, suddenly, the gene activates in one individual, making
them behave altruistically. At this point they have, possibly, many relatives in the population. And so, this
altruistic act may be benefiting those that share the altruistic gene but have not yet acted altruistically. It
is in the neutral population, before A is active, that we calculate relatedness.
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