Figure 3.

Influences of ecological constraints on the evolutionarily stable amount of help offered by helpers (h*, squares) and the tolerance threshold of breeders, below which they will evict helpers (p*, triangles) when density dependence acts through: (a) the probability of inheriting a territory; or (b) the probability of surviving dispersal (eviction). Solid lines and symbols represent low helping costs, g=0.1. Dashed lines and open symbols represent higher helping costs, g=0.2. When the probability of finding a territory after eviction is low, ecological constraints can be considered to be high. Because values for (b) could not be found analytically, values presented are the ±s.d. for 15 simulations (see Appendix A). Ecological constraints do not influence the amount of help offered, and only slightly influence the tolerance threshold in (a) (and not at all in (b)). Fixed parameters differ from those used in figures 2 and 4 to illustrate that stable groups will form at high c, and they differ between (a) and (b) because differences in the mechanism of density dependence lead to different parameter values under which grouping is stable. In (a), c=0.9, s=0.01, δ=0.01, S_b=0.95, S_n=0.7, F₀=2, S_f=0.1, K=0.4. In (b), c=0.9, s=0.001, δ=0.01, S_b=0.99, S_n=0.9, F₀=1, S_f=0.1, β=0.7.