A Calculating the coevolved dispersal distances

The calculations for the baseline model (monoculture evolution in the presence of moderate environmental variation) were presented in Snyder (2006). Here I show how to calculate the coevolved dispersal distances. To find the coevolved dispersal distances, we find the evolutionarily stable (ESS) annual dispersal distance as a function of the perennial dispersal distance, then find the ESS perennial dispersal distance as a function of the annual dispersal distance. At the intersection of these two curves, neither species will be driven to adjust its dispersal strategy in response to the other’s strategy, for each species is on its evolutionarily stable curve: this point is an evolutionarily stable coalition. Finally, we need to verify that evolutionary dynamics cause the system to converge to this point: i.e., that the intersection is not only evolutionarily stable but convergently stable, in the sense of Geritz et al. (1998). To find, e.g., the ESS annual dispersal distance as a function of the perennial dispersal distance, consider an annual invader in the presence of annual and perennial residents. The invader is a mutant with dispersal distance a little larger or smaller that of the annual resident. If the invader has a positive long-run growth rate, it will replace the former resident and become subject to invasion attempts by new mutants. The process stops when the best strategy the annual invader could adopt is that of the annual resident — i.e., invader growth is maximized when the invader dispersal distance equals the resident dispersal distance. Signifying the invader long run growth rate by ri(da, dp, di), where da is the resident annual distance, dp is the resident perennial distance, and di is the invader distance, the ESS annual dispersal distance da is given by the condition ∂ri(da, dp, di) ∂di ∣∣∣∣ di=da=d∗a = 0. (1)