Fig 1.
Schematic of the eco-evolutionary games with general dynamic environments.
(Top) The group strategic behaviors are described by a nonlinear evolutionary public goods game. (Top, Bottom) The influence of environmental changes consists of two prominent aspects: the global environmental fluctuations that directly affect the synergy and discounting of the group payoffs, characterized by the nonlinear factor w, and the asymmetric environmental feedbacks that drive the local strategy-dependent feedback-evolving game, characterized by the multiplication factor of cooperators rc.
Fig 2.
Effects of varying modeling parameters on equilibrium states of local game-environment evolution in static global environments.
Panel (a) shows the combined influence of w and rd on the stability of the boundary fixed point (1, α). Panel (b) and (c) show the detailed influence of group size N on the equilibrium states revealed by and ϵ*, respectively. Panel (d) presents the trends of the interior fixed point
as θ changes (the dashed blue line indicates that the value of
is out of parameter range [α, β]). Panel (e) and (f) show how the threshold of the relative feedback speed ϵ*, which is the minimum value leading to the interior stability, is affected by different parameters. Parameters: N = 4 in (a, d-f), α = 1.5, β = 3.5 in (a, c, e) and rd = 0.8 in (f).
Fig 3.
Phase plane dynamics of local game-environment (x − rc) evolution with N = 4, α = 1.5, β = 3.5, θ = 2, rd = 1.
We set w = 0.8, 1, 1.2 to describe the scenarios of discounting, linear, and synergy PGG in each column, and the corresponding ϵ* are 0.29, 0.44, 1.06, respectively. We therefore choose ϵ = 0.09, 0.24, 0.86 for the first row such that ϵ < ϵ*, and ϵ = 0.79, 0.94, 1.56 for the third row such that ϵ > ϵ*. The blue, pink and gray areas represent the basin of attraction of different fixed points and (
), respectively. The stable coexistence of cooperators and defectors only occurs when ϵ > ϵ* (the third row), as predicted theoretically, while a synergistically enhanced global environment significantly promotes the emergence of full cooperation (the third column).
Fig 4.
Local game-environment evolution in a discretely varying global environment.
We use a periodic piecewise function, w1(t), to describe the global environmental fluctuations. We uniformly select initial points on the x − rc plane and plot the corresponding dynamical trajectories by numerically solving Eq 9. Trajectories that eventually evolve to (x* = 0), or circulate along an interior closed orbit are distinguished by blue, pink and orange, respectively. The emergence of cyclic evolution of group cooperation and local environment (orange areas), which cannot be observed in static global environments, indicates that the global environmental fluctuations could fundamentally alter the evolutionary outcomes in local feedback-evolving game. In all panels, N = 4, α = 1.5, β = 3.5, θ = 0.5, rd = 0.6.
Fig 5.
Emergence of cyclic evolution of group cooperation and local environment under periodically changing global environments given by w1(t).
Panel (a) shows local game-environment evolution similar to Fig 4. Panel (b) displays four typical dynamic trajectories in detail, particularly the interior closed yet irregular orbits. The last row presents time evolution of the frequency of cooperators x and the multiplication factor of cooperators rc under different initial conditions, corresponding to the colored trajectories in panel (b). Results show that the formation of cyclic evolution under discrete global environment is due to the fact that the two fixed points are in the mutual attraction domains (b-d). Parameters: N = 4, α = 1.5, β = 3.5, θ = 1, rd = 1.2, T = 10π, ϵ = 4.
Fig 6.
Game-environment evolution under different sizes of group N and distribution ratio of the expected total payoffs of the cooperators and defectors θ.
Trajectories on x − rc phase plane eventually evolve to x* = 0, or circulate along an interior closed orbit, which are distinguished by blue, pink and orange, respectively. Panels (a) and (b) correspond to N = 5 and 6 with θ = 0.5, rd = 0.6, T = 2π and ϵ = 9, while panels (c) and (d) correspond to θ = 1 and 1.5 with N = 4, rd = 1, T = 10π and ϵ = 6. In all panels, α = 1.5, β = 3.5.
Fig 7.
Local game-environment evolution under frequency-dependent global environment.
We assume the global environment is in a simple threshold-function manner denoted by w4(t). In specific, when the frequency of cooperators x(t) is smaller than the threshold K, the global environment becomes synergistically enhanced in order to facilitate cooperation, while on the contrary, when x(t) is larger than K, the global environment becomes discounted. The first row shows the overall evolutionary trends. Trajectories that eventually evolve to full defection, the stable interior fixed point and the boundary equilibrium point on the x-axis are distinguished by blue, grey and green, respectively. The second row shows the detailed evolutionary trajectories of several representative initial points in the phase plane. The plane is divided into two regions by the threshold K, with orange on the left representing w(t) = 1.3 and blue on the right representing w(t) = 0.7. The vertical dotted and solid lines represent and x = K, respectively, and the triangle represents the stable interior fixed point. Unlike the time-dependent cases, the internal periodic orbit disappears, instead, an interior stable equilibrium emerges under all thresholds. For comparison, the parameters are the same as in Fig 4: N = 4, α = 1.5, β = 3.5, θ = 0.5, rd = 0.6, ϵ = 6.