Fig 4. Effects of directed migration on population densities and frequency of cooperation as compared to unstructured populations.

In the absence of directed migration (undirected migration only) homogeneous coexistence results with densities Q = (u_eq, v_eq). The two panels depict the ratio of the average spatial population density to the population density in unstructured populations, $r_p=(\overline{u}+\overline{v}):(u_{\text{eq}}+v_{\text{eq}})$ (black solid line), and the ratio of the average cooperator frequency to their frequency in unstructured populations, $r_c=\overline{u}/(\overline{u}+\overline{v}):u_{\text{eq}}/(u_{\text{eq}}+v_{\text{eq}})$ (green solid line) for (A) cooperator aggregation, A_C, and (B) spreading defectors, R_D. (A) increasing A_C gives rise to four dynamical regimes: (i) homogeneity maintained, (ii) formation of smooth patterns, (iii) discontinuities emerge, and (iv) population expansion prevented. (B) increasing R_D only results in two regimes: (i) homogeneity (ii) formation of smooth patterns. In order to facilitate comparisons of regions (i) and (ii) across panels the large effects of discontinuities in (iii) and (iv) refer to the scale on the right of panel (A). In either case, heterogeneous distributions increase (average) population densities and cooperator frequencies. In (A), once discontinuities develop, populations consist exclusively of cooperators. The decrease in r_p for large A_C relates to the fact that populations are unable to expand and hence the ratio depends on the initial configuration. In contrast, r_c remains unaffected because defectors are absent and cooperators are at the maximum frequency. Parameters as in Fig 3.