Nonlinear eco-evolutionary games with global environmental fluctuations and local environmental feedbacks
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.