The ESS under spatial variation with applications to sex allocation.

I derive a new approximation which uses the backward Kolmogorov equation to describe evolution when individuals have variable numbers of offspring. This approximation is based on an explicit fixed population size assumption and therefore differs from previous models. I show that for individuals to accept an increase in the variance of offspring number, they must be compensated by an increase in mean offspring number. Based on this model and any given set of feasible alleles, an evolutionary stable strategy (ESS) can be found. Four types of ESS are possible and can be discriminated by graphical methods. These ESS values depend on population size, but population size can be reinterpreted as deme size in a structured population. I adapt this theory to the problem of sex allocation under variable returns to male and female function and derive the ESS sex allocation strategy. I show that allocation to the more variable sexual function should be reduced, but that this effect decreases as population size increases and as variability decreases. These results are compared with results from exact matrix models and computer simulations, all of which show strong congruence.