A tale of two contribution mechanisms for nonlinear public goods

Abstract

Amounts of empirical evidence, ranging from microbial cooperation to collective hunting, suggests public goods produced often nonlinearly depend on the total amount of contribution. The implication of such nonlinear public goods for the evolution of cooperation is not well understood. There is also little attention paid to the divisibility nature of individual contribution amount, divisible vs. non-divisible ones. The corresponding strategy space in the former is described by a continuous investment while in the latter by a continuous probability to contribute all or nothing. Here, we use adaptive dynamics in finite populations to quantify and compare the roles nonlinearity of public-goods production plays in cooperation between these two contribution mechanisms. Although under both contribution mechanisms the population can converge into a coexistence equilibrium with an intermediate cooperation level, the branching phenomenon only occurs in the divisible contribution mechanism. The results shed insight into understanding observed individual difference in cooperative behavior.

Figure 1. 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.

Figure 2. The adaptive dynamics of the cooperative probability with varying production functions g(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) accrued in homogeneous populations. (a) A repeller 0.67. (b) A CESS 0.35. (c and d) Two singular strategies; in (c), a CESS 0.73 along with a repeller 0.27; in (d), a CESS 0.16 together with a repeller 0.84. We adopt the fixation probability ρ(y, x) − ρ(x, x) as the ‘invasion fitness’ and perform two distinct simulation in (a and b) and three distinct simulations in (c) and (d). The abscissa ‘time’ represents the number of updating steps divided by 10⁵. Parameters: N = 100, n = 10, σ = 0.00001, u = 0.01, (a) g(s) = 0 for 0 ≤ s < 1, g(s) = 3.5 for s = 1, (b) g(s) = 0 for s = 0, g(s) = 3.5 for 0 < s ≤ 1, (c) g(s) = 0 for s = 0, g(s) = 1.75 for 0 < s < 0.5, g(s) = 3.5 for s = 1, (d) g(s) = 0 for 0 ≤ s < 0.5, g(s) = 1.75 for s = 0.5, g(s) = 3.5 for 0.5 < s ≤ 1.