Appendix D. Simulations of Model 1 (for Oreaster reticulatus) and Model 2 (for Strongylocentrotus droebachiensis) with different gradients in food distribution.
Model 1: Oreaster reticulatus
To test whether the step function used in our simulations was a necessary condition for the formation of a grazing front, we ran simulations of the sea star model with gradual increases in food concentration between the area denuded of algal film (algal biomass = 0) and the area with algal film at carrying capacity (algal biomass = 1) by changing the initial values of algal biomass in the cells. These gradients occurred over distances of 6.5 (steep) and 13 m (shallow) and represent 25 and 50 cells of the simulation array, respectively. We ran these simulations with 1000 sea stars for 1000 time steps (22 h).
Fronts formed with both gradients tested, and not just the step function, although the density of sea stars at the front decreased with the gradient (Fig. D1A). In all cases, the density of sea stars at the front increased with time and densities tended to converge after 1000 time steps.
Model 2: Strongylocentrotus droebachiensis
We also ran similar simulations with the sea urchin model, with gradual increases in food between the barrens (0 g kelp/m2) and the mature kelp bed (4000 g kelp/m2). We used gradients with increases in kelp biomass of 160 (steep) and 40 (shallow) g kelp/m2 over a meter of bottom. These gradients occurred over distances between the barrens and mature kelp beds of 25 and 100 m, respectively. We used an initial density of 71 urchins/m2 in these simulations and ran the simulations for 400 days.
As for the sea star model, fronts formed with both food gradients tested and the densities of urchins at the front, although initially lower for the gradients, converged with the step function at the end of the simulations (Fig. D1B).
FIG. D1. (A) Maximum density of sea stars (Oreaster reticulatus) at the front during the 1000 time steps of the simulations, and (B) maximum density of urchins (Strongylocentrotus droebachiensis) at the front during the 400 days of a simulation with different food gradients.