The goal is to understand Equation 4.2 of Rodrigues and Kokko (2016).
To understand the technique, I read through Taylor et al. (2007), and Taylor and Frank (1996), specifically examples 4, 4a and 4b. Very briefly, we begin with a matrix whose elements represent the genetic contribution of class to class . Then for some mutant genic value , the fitness derivative is
where is the left eigenvector of and is the reproductive values, and is the right eigenvector of and is the class frequencies. In the Appendix Taylor and Frank show us that the reproductive values and class frequencies at the steady state may be used, so one need only concern one’s self with taking derivatives of the elements of the contributions matrix itself.
Given Equation 3 of Taylor and Frank (1996), the derivatives can be expanded as
where is the phenotype of the actor, is the phenotype of the neighbour, and is the relatedness of the neighbour to the actor, which scales the effect of the correlated change in the actor’s phenotype to its effect on .
After working through the examples in Taylor and Frank (1996), I was able to rederive the results in
Rodrigues and Kokko (2016). I will use the survival-survival scenario as an example.
From the appendix, the contributions matrix is
where is fecundity in class , is survival in class , is the probability that a dispersing offspring joins a group, and is the probability of being joined by a disperser. I have introduced a `hat’ to indicate which survival refers to the survival of the partner, as this makes the maths easier to follow later.
First, the helping behaviour only manifests when the individual has a partner, so we need only take derivatives of the and .
Second, the cost and benefit mentioned in the paper are to be interpreted as the derivatives of survival with respect to the donor’s and recipient’s helping phenotype respectively
Setting gives the condition for invasion of the helping strategy given in Equation 4.2 in the text.