Figure 1
From: A tale of two contribution mechanisms for nonlinear public goods
The adaptive dynamics of the cooperative investment with varying production and remaining functions g(s) and h(s).
The left column shows the trajectories obtained by numerical simulations (see Supplementary Information), the middle column the PIP. The singular strategies (dashed horizonal lines) are indicated where appropriate. The right column shows the production function g(s) (increasing) and the remaining function h(s) (decreasing) accrued in homogeneous populations. (a) A branching point 0.5. (b) A CESS 0.5. (c) A repeller 0.5. (d and e) Two singular strategies; in (d), a branching point 0.67 along with a repeller 0.29; in (e), a branching point 0.31 together with a repeller 0.69. We adopt the fixation probability ρ(y, x) − ρ(x, x) as the ‘invasion fitness’ and perform a single simulation in (a) and two distinct simulations in (b)–(e). The abscissa ‘time’ represents the number of updating steps divided by 5 × 103. Parameters: n = 10, N = 100, σ = 0.00005, u = 0.01, (a) g(s) = −11s2 + 22s and h(s) = 0.8s2 − 1.8s + 1, (b) g(s) = s2 + 10s and h(s) = −s2 + 1, (c) g(s) = s2 + 10s and h(s) = s2 − 2s + 1, (d) g(s) = −s3 − 2.9s3 + 14.6s and h(s) = −0.1s3 + 0.54s2 − 1.44s + 1, (e) g(s) = s3 − 5.9s2 + 16.2s and h(s) = 0.1s3 + 0.25s2 − 1.35s + 1.
