Cooperation in Harsh Environments and the Emergence of Spatial Patterns

This paper concerns the confluence of two important areas of research in mathematical biology: spatial pattern formation and cooperative dilemmas. Mechanisms through which social organisms form spatial patterns are not fully understood. Prior work connecting cooperation and pattern formation has often included unrealistic assumptions that shed doubt on the applicability of those models toward understanding real biological patterns. I investigated a more biologically realistic model of cooperation among social actors. The environment is harsh, so that interactions with cooperators are strictly needed to survive. Harshness is implemented via a constant energy deduction. I show that this model can generate spatial patterns similar to those seen in many naturally-occuring systems. Moreover, for each payoff matrix there is an associated critical value of the energy deduction that separates two distinct dynamical processes. In low-harshness environments, the growth of cooperator clusters is impeded by defectors, but these clusters gradually expand to form dense dendritic patterns. In very harsh environments, cooperators expand rapidly but defectors can subsequently make inroads to form reticulated patterns. The resulting web-like patterns are reminiscent of transportation networks observed in slime mold colonies and other biological systems.