pubmed.ncbi.nlm.nih.gov

Spatial dynamics of ecological public goods - PubMed

  • ️Thu Jan 01 2009

Spatial dynamics of ecological public goods

Joe Yuichiro Wakano et al. Proc Natl Acad Sci U S A. 2009.

Abstract

The production, consumption, and exploitation of common resources ranging from extracellular products in microorganisms to global issues of climate change refer to public goods interactions. Individuals can cooperate and sustain common resources at some cost or defect and exploit the resources without contributing. This generates a conflict of interest, which characterizes social dilemmas: Individual selection favors defectors, but for the community, it is best if everybody cooperates. Traditional models of public goods do not take into account that benefits of the common resource enable cooperators to maintain higher population densities. This leads to a natural feedback between population dynamics and interaction group sizes as captured by "ecological public goods." Here, we show that the spatial evolutionary dynamics of ecological public goods in "selection-diffusion" systems promotes cooperation based on different types of pattern formation processes. In spatial settings, individuals can migrate (diffuse) to populate new territories. Slow diffusion of cooperators fosters aggregation in highly productive patches (activation), whereas fast diffusion enables defectors to readily locate and exploit these patches (inhibition). These antagonistic forces promote coexistence of cooperators and defectors in static or dynamic patterns, including spatial chaos of ever-changing configurations. The local environment of cooperators and defectors is shaped by the production or consumption of common resources. Hence, diffusion-induced self-organization into spatial patterns not only enhances cooperation but also provides simple mechanisms for the spontaneous generation of habitat diversity, which denotes a crucial determinant of the viability of ecological systems.

PubMed Disclaimer

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.

Chaotic pattern formation in spatial ecological public goods. A sequence of snapshots A–H demonstrates the spatial density distribution of cooperators (green) and defectors (red) over time (see

Movie S1

). The symmetry of the initial configuration A should be preserved in a deterministic system, but after some time it breaks down and disappears because of limitations of the numerical integration of Eq. 2. The exponential amplification of arbitrarily small disturbances characterizes chaotic systems. The initial configuration is a vacant L × L square (L = 400) with no flux boundaries and a homogeneous disk with radius L/10 in the center, where cooperators and defectors coexist at equal density (udisk = vdisk = 0.1 and u = v = 0 elsewhere). The parameters of the ecological public goods are N = 8, d = 1.2, b = 1, r = 2.34, c = 1. The multiplication factor r lies slightly below the Hopf bifurcation rHopf = 2.3658, such that the fixed point Q is unstable, and in the absence of space, the population disappears. Diffusion of defectors is twice that of cooperators (DC = 1, DD = 2). The color brightness indicates the density of cooperators (green) and defectors (red). The snapshots are taken at times t = 0 (A), 1,200 (B), 1,800 (C), 2,000 (D), 2,200 (E), 2,600 (F), 2,800 (G), 4,000 (H). The numerical integration uses a spatial grid with dx = 0.8 and step size dt = 0.01.

Fig. 2.
Fig. 2.

Typical stationary patterns for diffusion-induced instability (Turing patterns) (A) and diffusion induced coexistence (B) on an L × L square (no flux boundaries). The density variation of cooperators (green) and defectors (red) is shown along a cross-section (solid blue line). (A) In the absence of space, cooperators and defectors coexist for suitable initial configurations (r = 2.4 > rHopf = 2.3658). Diffusion destabilizes the spatially homogeneous state and induces stable and static heterogeneous strategy distributions, where individuals spontaneously aggregate in spots or striped patterns. (B) In the absence of space, the population goes extinct (r = 2.24 < rHopf). Diffusion stabilizes persistence of the population and coexistence of cooperators and defectors by inducing heterogeneous strategy distributions. In both scenarios, the parameters of the ecological public goods are n = 8, c = 1, d = 1.2, b = 1, DC = 1, DD = 10, with an initial configuration where densities are randomly drawn in [0, 0.1]. Numerical integration is performed on a spatial grid with L = 283, dx = 1.4, and a step size of dt = 0.1. The brightness of the colors indicates the strategy densities (Upper) and the dashed horizontal lines mark the densities at Q (Lower).

Fig. 3.
Fig. 3.

Diversity of spatial distributions in terms of the ratio of the diffusion of defectors to cooperators DD/DC and the multiplication factor r in spatial ecological public-goods games. The brightness of the colors indicates the density of cooperators (green) and defectors (red). In the absence of space, the population survives for r > rHopf. If cooperator diffusion exceeds defector diffusion, DD/DC ≤ 1, the dynamics is barely affected by space, except for a small chaotic region (blue frame) near rHopf. The dynamics becomes much richer for DD/DC > 1. For r < rHopf, the chaotic regime increases (blue frame) and is replaced by diffusion induced coexistence patterns for high DD/DC (red frame). For r > rHopf, the homogeneous spatial distributions (orange frame) are replaced by diffusion-induced instability (Turing patterns; green frame) for high DD/DC. For very large r, all patterns disappear. The parameters are N = 8, c = 1, d = 1.2, b = 1, DC = 1, L = 283, dx = 1.4, dt = 0.1 (rHopf = 2.3658) and an initial configuration with random cooperator and defector densities in [0, 0.1]. For a detailed phase plane diagram and other initial configurations see

SI Text

and

Figs. S1 and S5

.

Fig. 4.
Fig. 4.

Average global population density (A) and ratio of cooperators to defectors (B) as a function of the multiplication factor r in ecological public-goods interactions. The nonspatial stable equilibrium (solid lines; requires r > rHopf) is shown together with numerical results for the stationary spatial distributions (dots). The chaotic regime depicts the average and standard deviation of the time series from t = 5,000 to 10,000 with dt = 0.1, dx = 1.4 and L = 283. For small r, the population goes extinct, but for increasing r, the population persists and exhibits chaotic dynamics that change into quasistatic and static patterns emerging through diffusion-induced coexistence (r < rHopf). For r > rHopf, static patterns are triggered by diffusion-induced instability (Turing patterns) and relax into spatially homogeneous coexistence for high r. Diffusion supports cooperation by significantly increasing the persistence region of the population and in the chaotic regime, cooperator densities even exceed those of defectors. Snapshots illustrate typical patterns emerging in the different dynamical regimes. The brightness of the colors indicates the density of cooperators (green) and defectors (red). The parameters for the ecological public goods game are N = 8, c = 1, d = 1.2, b = 1, DC = 1, DD = 10 such that rHopf = 2.3658.

Similar articles

Cited by

References

    1. Lee KJ, McCormick WD, Ouyang Q, Swinney HL. Patterns formation by interacting chemical fronts. Science. 1993;261:192–194. - PubMed
    1. Gell-Mann M. The Quark and the Jaguar: Adventures in the Simple and the Complex. New York: Freeman; 1994.
    1. Pearson JE. Complex patterns in a simple system. Science. 1993;261:189–192. - PubMed
    1. Rietkerk M, Dekker SC, de Ruiter PC, van de Koppel J. Self-organized patchiness and catastrophic shifts in ecosystems. Science. 2004;305:1926–1929. - PubMed
    1. Wolfram S, editor. Theory and Applications of Cellular Automata. Singapore: World Scientific; 1986.

Publication types

MeSH terms

LinkOut - more resources