Spatial Interactions and Oscillatory Tragedies of the Commons

A tragedy of the commons (TOC) occurs when individuals acting in their own self-interest deplete commonly held resources, leading to a worse outcome than had they cooperated. Over time, the depletion of resources can change incentives for subsequent actions. Here, we investigate long-term feedback between game and environment across a continuum of incentives in an individual-based framework. We identify payoff-dependent transition rules that lead to oscillatory TOCs in stochastic simulations and the mean field limit. Further extending the stochastic model, we find that spatially explicit interactions can lead to emergent, localized dynamics, including the propagation of cooperative wave fronts and cluster formation of both social context and resources. These dynamics suggest new mechanisms underlying how TOCs arise and how they might be averted.

Figure 1

Coevolutionary dynamics of strategies and resources in replicator and IBM dynamics. (left) The dynamics with “IBM2,” in which offspring of the focal player replace the opponent. (right) The dynamics with “IBM3,” in which offspring of the focal player replaces a random individual. In both panels, parameter space is divided according to the sign of $R_0-T_0$, $S_0-P_0$. In each section in the parameter space, a phase diagram with different $A_0$ is shown, where the abscissa represents $x$ and the ordinate denotes $n$. Light gray trajectories are mean field solutions and black trajectories denote IBM dynamics where arrows denote the flow of time. Visualized IBM trajectories are the average of 100 replicates with the same parameter set, except for oscillatory dynamics, given phase differences that can arise due to demographic noise. Common parameters for all replicates: $\theta =2$, $\epsilon =0.5$, $\mathrm{\Delta }x=1$, and $\mathrm{\Delta }t=0.05$, $A_1=[3,0;5,1]$; $A_0$ varies by region. Full parameter list for $A_0$ in Fig. S1 in the Supplemental Material [7].

Figure 2

Strategy-resource dynamics given spatial interactions. Colors in each heat map denote the fraction of averted dynamics out of 20 replicates with different $A_0$’s. The horizontal axis of the heat maps are $S_0-P_0$ and the vertical ones are $R_0-T_0$. Each grid on the heat maps has increment 0.1. The diffusivity $D_n$ is shown in the title of each panel. Other parameters for all replicates are $L=100$, $\theta =2$, $\epsilon =0.5$, $\mathrm{\Delta }x=1$, and $\mathrm{\Delta }t=0.05$, $A_1=[3,0;5,1]$. The white lines mark out the boundary of different dynamics predicted by the mean field model. Full parameter list for $A_0$ in Fig. S2 in the Supplemental Material [7].

Figure 3

Spatiotemporal dynamics of resources and cooperators. The background color represents the environment, while a red square means a cooperator occupies the lattice site. The empty sites are occupied by defectors. (Top row) $D_n=0$, a circular wave of cooperative population propagates outward. (Middle row) $D_n=1$, a few small patches of cooperators move around and divide. (Bottom row) $D_n=\infty$, a large cooperator cluster expands and shrinks over time with increasing amplitude until extinction.