Population structure and life cycle
The population is made of a finite number n of demes which are homogeneously distributed on a lattice. For simplicity, we consider a closed one-dimensional lattice on a circle (see Fig. 1), with symmetrical migration from each deme in both directions. The extension to two-dimensional lattices is straightforward (see Gandon & Rousset, 1999, for examples). We consider models where, as shown on Fig. 1:
1 individuals affect the fecundity of other individuals, i.e. the relative number of gametes they produce;
2 gamete dispersal occurs;
3 differential gamete death according to parental genotype may occur at this stage; and
4 further competition occurs between all surviving gametes that have dispersed to one deme, independent of genotype, so that only N of them develop into haploid adults.
The genotype of a haploid individual is a random variable with value a or A, and so is the associated phenotype, with value za or zA. The generic notation Z will be used for the latter random variables. A variable Z(i) may be defined for each individual i (i=1,…,Nn). Fitness is the expected number of adult offspring of an individual. The fitness W(i) of an individual i is defined as this expectation, evaluated conditionally on its phenotype and on the phenotypes of other individuals (this corresponds to neighbour-modulated fitness, Hamilton, 1970). We consider here models where interactions are spatially homogeneous, so that the fitness of each individual may be expressed as a function of its own phenotype, of the phenotype of other individuals in the same deme, of the phenotype of individuals one step further on the lattice, of the phenotype of individuals two steps further on the lattice, and so on. Thus, it is convenient to express fitness as a function of phenotypes of individuals at positions relative to that of the focal individual on the lattice, rather than considering absolute positions (i.e. the i index). All of the individuals will be considered in turn as the focal individual, and given individual i is taken as the focal individual, we write Z• for its phenotype Z(i), and z• for the realized value of Z(i). Likewise, the phenotypes of its neighbours at different distances form a vector of random variables, Z ≡ Z(i), whose elements will be indexed according to their position relative to the focal individual. Thus Z0 ≡ Z0(i) will be the average phenotype of neighbours in the same deme as individual i taken as the focal individual, Z1 ≡ Z1(i) will be the average phenotype of neighbours in a deme adjacent to the focal individual and Zn−1 ≡ Zn−1(i) will be the average phenotype of neighbours in the other adjacent deme, etc.
We consider models in which fitness takes the following form. Let gr be the relative contribution of an individual to gametes competing after phase (iii) in a deme r lattice steps apart. Migration rates between demes depend only on their relative position, so that the relative contribution of an individual in deme j to gametes competing in deme l may be written as gl−j (where gl−j ≡ gj−l). Let Zj be a vector obtained by circular permutation of the elements of Z and with first element Zj (for example, Z1=(Z1, Z2,…,Z0)). Then the expected number of adult offspring of focal individual i is
hence we consider the fitness function w (z•, z) written as
where z• is the variable describing the phenotype of a focal individual (thus z• is equivalent to y in Taylor & Frank, 1996), z ≡ (z0,z1,…,zn−1) where zj is the variable describing the average phenotype of individuals at j steps from this individual on the lattice (here including the focal individual itself for j=0), and zj is a vector obtained by circular permutation of the elements of z and with first element zj (for example, z1=(z1, z2,…,z0)). Examples are given below in eqns 19, 20 and above in eqn 7. This form of the fitness function follows from the assumptions of spatial homogeneity of dispersal and other interactions. The number of gametes produced is a function of (zj, zj) by the assumption of spatial homogeneity of interactions determining fecundity, and the l − j index in gl−j follows from the assumption of spatial homogeneity of backward dispersal rates: the fraction of gametes produced in deme j which is sent to deme l is a function of the distance l − j but not of l and j separately.
As fitness is the number of adult offspring, and the total number of adult offspring is also the number of adult parents, Nn, the average fitness over all individuals in the population (W¯ below) is necessarily 1, whatever may be the genetic composition of the parental population. Note that this does not appear to be always so for the ‘mean fitness’ (W¯) in, e.g. Frank (1997), and that the results below apply only when fitness is defined as here. Another consequence of this definition is that
where ∑n−1j = • is a sum over all variables of the fitness function. A simple formal argument is given by computing dw(z,…,z)/dz in two ways. First, the expected number of adults offspring of all individuals is 1 when all individuals are identical: w(z,…,z)=1 for any z. Thus the derivative is zero. Second we compute the same quantity as a total derivative dw(f•(z), f0(z),…, fn−1(z))/dz where we set set all functions fj(z)=z, i.e. as
Thus, the latter expression is 0. Sums of higher order partial derivatives are likewise shown to be zero. It will be very useful at several steps in our analysis. In particular, it implies that we can express any partial derivative as the sum of the others, and therefore we can eliminate one derivative from some expressions when convenient.