Bifurcations and Chaos in the Spatio-Temporal Evolution of Strategies Induced by a Nonlinear Public Goods Game

Classical evolutionary game models typically assume a fixed population size and neglect spatial migration among individuals. However, during the game, participants undergo birth and death processes through evolution, leading to continuous changes in total population size, while the spatial movement of individuals further affects their strategic interactions. Therefore, within the framework of nonlinear public goods games, we develop game dynamics models that simultaneously account for spatial diffusion and variable population size. We analyze the dynamical properties of the temporal system, including the existence, stability, and bistability of equilibria, as well as transcritical and Hopf bifurcations and their associated characteristics. In the context of spatial evolutionary games, we further investigate the spatial stability, Turing instability, Hopf bifurcation, Turing–Hopf bifurcation, and chaotic behavior of the diffusion system. Our findings demonstrate that the evolutionary outcome of the temporal system is determined by key sensitive parameters and initial conditions. Furthermore, diffusion induces new chaotic dynamics in the steady-state solutions of the time-evolving system, rendering the density variations of cooperators and defectors unpredictable. Several counterintuitive dynamical phenomena occur: a decrease in the birth rate or an increase in the death rate of defectors leads to a simultaneous increase in the density of both cooperators and defectors, whereas an increase in payoffs results in a reduction in their densities. Our findings provide a novel perspective on variable-population evolutionary game systems modeled by partial differential equations.