Spatio-temporal evolution of cooperation: multistability, pattern formation, and chaos in resource-driven...

In biological systems, cooperative behavior forms the foundation for the survival and prosperity of many organisms. However, the finite nature of resources often drives selfish individuals to exploit resources through deceptive tactics, thereby instigating conflicts between collective and individual interests. These strategic interactions not only alter the availability of environmental resources but also feedback on the strategic choices of populations, leading to the co-evolution of environmental resources and behavioral strategies. By integrating population dynamics with replicator dynamics, we develop models for both well-mixed and spatially heterogeneous distributions that incorporate resource feedback mechanisms to analyze the intricate interplay between cooperative behavior and resource dynamics across temporal and spatial scales. Our findings reveal complex evolutionary dynamics, including rich multistability, transcritical and Hopf bifurcations in the temporal system, alongside spatial stability, Turing instability, Turing-Hopf bifurcation, and chaotic behavior in the spatial diffusion system. In homogeneous distributions, payoffs result in stable periodic solutions, while heterogeneous distributions disrupt stable periodicity and lead to chaotic dynamics. Notably, increasing the initial density of cooperators, the rate of resource growth, and reducing the initial resource stock are favorable for sustaining cooperation. Interestingly, high payoffs for cooperators and low payoffs for defectors do not necessarily promote cooperative behavior, as evolutionary outcomes also depend on resource abundance. We provide the conditions that sustain cooperation, revealing the critical role of resource dynamics and spatial diffusion in shaping the evolution of cooperative strategies. Our findings have important implications for studying ecosystem management, conservation biology, and animal social behavior.