Nonlinear Public Goods Game in Dynamical Environments

TL;DR

This work integrates dynamic environment feedback into a stochastic nonlinear public goods game in a well-mixed population, enabling coevolution of environmental state $p$ and cooperation level $x$. By combining mixed nonlinear PGG payoffs (synergy vs. discounting) with environment-driven payoff switching and a linear feedback from cooperation to the environment, the authors derive analytic phase diagrams and identify up to seven equilibria, including interior fixed points and limit cycles. Key insights show that environmental sensitivity $\theta$ and feedback speed $\epsilon$ govern the emergence and stability of cooperation, coexistence, and oscillatory regimes, with asymmetric nonlinearities $(\delta_s,\delta_d)$ further enriching the dynamics. The results generalize evolutionary game theory for real-world settings where environment and cooperation coevolve, offering actionable implications for ecological management, resource sharing, and policy design under nonlinear, stochastic, and dynamic conditions.

Abstract

The evolutionary mechanisms of cooperative behavior represent a fundamental topic in complex systems and evolutionary dynamics. Although recent advances have introduced real-world stochasticity in nonlinear public goods game (PGG), such stochasticity remains static, neglecting its origin in the external environment as well as the coevolution of system stochasticity and cooperative behavior driven by environmental dynamics. In this work, we introduce a dynamic environment feedback mechanism into the stochastic nonlinear PGG framework, establishing a coevolutionary model that couples environmental states and individual cooperative strategies. Our results demonstrate that the interplay among environment feedback, nonlinear effects, and stochasticity can drive the system toward a wide variety of steady-state structures, including full defection, full cooperation, stable coexistence, and periodic limit cycles. Further analysis reveals that asymmetric nonlinear parameters and environment feedback rates exert significant regulatory effects on cooperation levels and system dynamics. This study not only enriches the theoretical framework of evolutionary game theory, but also provides a foundation for the management of ecological systems and the design of cooperative mechanisms in society.

Figures (7)

• Figure 1: Model schematic. In a well-mixed population, groups of size $N$ are formed randomly each round to play a nonlinear PGG, with strategy evolution governed by replicator dynamics. The environmental state $p$ coevolves with population composition, modulating game outcomes probabilistically: each interaction becomes a discounting PGG (dPGG, probability $p$) with diminishing marginal returns to cooperation, or a synergistic PGG (sPGG, probability $1-p$) with increasing marginal returns. Crucially, cooperation elevates $p$ while defection reduces it, establishing a closed feedback loop where strategies alter their future payoff environment.
• Figure 2: The phase diagrams of the $r-\delta$ parameter plane with fixed $p$. When $r<N$, only two outcomes occur across $\delta$: full defection (D) or bistability between full cooperation and full defection (C/D). When $r>N$, the steady-state type depends on $p$: for small p in panel (a), most of the space converges to full cooperation (C), whereas for large p in panel (b), additional regimes appear, including interior coexistence (C+D) and mixed bistability between C and C+D. Color coding: red denotes D; yellow denotes C/D bistability; blue denotes C; green denotes C+D; light blue denotes C/C+D bistability. Parameters: $N=5$; (a) $p=0.2$; (b) $p=0.8$.
• Figure 3: Eco-evolutionary dynamics when no interior equilibrium exists. (Top row) Phase portraits showing vector fields (arrows) and streamlines in the $x-p$ plane, with color scale indicating flow speed magnitude. (Bottom row) Corresponding temporal dynamics from initial condition $(x_0,p_0)=(0.7,0.1)$, with cooperator frequency $x(t)$ (solid blue) and environmental state $p(t)$ (dashed red). Common parameters: $N=5$, $\theta=2$, $\epsilon=0.1$. Subplot-specific parameters: (a,e) $r=1$, $\delta=0.4$; (b,f) $r=2$, $\delta=0.4$; (c,g) $r=7$, $\delta=0.2$; (d,h) $r=9$, $\delta=0.1$.
• Figure 4: Eco-evolutionary dynamics with interior equilibria. Similar to Fig. \ref{['boundary']}, top row shows phase portraits in the x–p plane, and bottom row shows corresponding temporal dynamics from initial condition $(x_0,p_0)=(0.7,0.3)$. The panels illustrate the three outcome types: (a,d) Corner attraction: convergence to $M_1$ (full defection) with an unstable interior equilibrium; (b,e) persistent oscillations: a stable interior limit cycle; (c,f) interior stability: stabilization at $M_7$ with intermediate levels of cooperation and environment. Common parameters: $N=5$, $\theta=2$, $\epsilon=0.1$. Subplot-specific parameters: (a,d) $r=4$, $\delta=0.4$; (b,e) $r=6$, $\delta=0.8$; (c,f) $r=7$, $\delta=0.5$.
• Figure 5: The phase diagrams of the $r-\delta$ parameter plane with dynamic environments. The parameter space is partitioned by analytic boundaries (white curves), with the horizontal line at $r=N$ providing a fundamental demarcation. For $r<N$ (regions e, f and g), the system evolves toward $M_1$, representing full defection in a depleted environment. For $r>N$, four distinct regimes emerge: (a) full cooperation $M_4$ in an enriched environment; (b) boundary coexistence $M_6$; (c) interior fixed point $M_7$ with intermediate cooperation and environment ($x_7=1/(1+\theta)$); and (d) interior limit cycle. Parameters: $N=5$, $\theta=2$.