The viscosity of a population is the tendency of offspring to remain near their place of birth. Taylor 1992 ("Altruism in viscous populations") provides a model to study how viscosity affects the evolution of altruism in a patch-structured population. In Taylor's model, the marginal inclusive fitness is given by \begin{equation} W = bR -c - s^{2}(b-c)R \end{equation} where $R$ is the relatedness of a mother to random offspring born on her breeding patch and $s$ is the probability the offspring will remain on the natal patch. The altruistic act creates $b$ offspring with average relatedness $R$ and $-c$ offspring with relatedness 1.
My question is about the term $s^{2}(b-c)R$. Taylor explains this term as follows:
these b - c extra offspring remain on the natal patch with probability s and will therefore displace s(b - c) random individuals competing for the next generation breeding spots on that patch, and these will be native to that patch with probability s and in this case will have average relatedness R to the actor. Thus these displaced individuals will have average relatedness sR (p. 353)
My question is: Why is the average relatedness $sR$ (as opposed to $R$)? In other words, why is the term that accounts for competition among relatives $s^2(b-c)R$---as opposed to $s(b-c)R$?
Answer
The $sR$ that your looking at is the average relatedness of the next generation. This assumes that the new immigrants into the the population are completely unrelated. So if the population is completely viscous ($s=1$) the average relatedness of the next generation equals that of the current generation. On the other hand, if the population is not viscous ($s=0$) the next generation would be entirely unrelated because it would only be made up of unrelated immigrants.
The $s(b-c)$ is the number if individuals that are displaced by new offspring. Again if all of the new generation sticks around ($s=1$) then the number of displaced individuals equals the number of new individuals. And if all new progeny leave ($s=0$), there are no displaced individuals.
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