Sharp thresholds limit the benefit of defector avoidance in cooperation on networks

Consider a cooperation game on a spatial network of habitat patches, where players can relocate between patches if they judge the local conditions to be unfavorable. In time, the relocation events may lead to a homogeneous state where all patches harbor the same relative densities of cooperators and defectors, or they may lead to self-organized patterns, where some patches become safe havens that maintain an elevated cooperator density. Here we analyze the transition between these states mathematically. We show that safe havens form once a certain threshold in connectivity is crossed. This threshold can be analytically linked to the structure of the patch network and specifically to certain network motifs. Surprisingly, a forgiving defector avoidance strategy may be most favorable for cooperators. Our results demonstrate that the analysis of cooperation games in ecological metacommunity models is mathematically tractable and has the potential to link topics such as macroecological patterns, behavioral evolution, and network topology.

Cooperation, behavior that leads to benefits for others at a cost to oneself, is widespread across biological systems, ranging from cells cooperating to form organisms, to cooperation among individuals in populations and among microbiotic and macrobiotic taxa in ecosystems. In many cases the costs of cooperation are high. Hence, how cooperative behavior persists in a population represents a fundamental question in biology (1–8). In general, cooperation is most likely to evolve and persist if there are mechanisms that directly or indirectly benefit cooperators’ reproductive success. Examples include kin selection, punishment of defectors who forgo the cooperative investment, or a direct self-benefit such as in cases of investment into a common good (4).Among the most general mechanisms that can favor cooperation is the notion of network or spatial reciprocity (1, 9–11). In classical examples of reciprocity, cooperation creates favorable conditions for other proximal cooperators (4). A result is the emergence of cooperative havens, where the rewards generated by mutual cooperation have enriched some physical or topological neighborhoods. The formation of cooperative neighborhoods in structured populations, where individuals interact with only a limited subset of the population, has traditionally been studied on networks, where each node represents an individual agent and an edge means that the two connected individuals play against each other (1, 10, 12–19). By assuming weak selection and treating space implicitly, the resulting systems can often be analyzed mathematically. Although this framework has become a powerful tool for conceptual understanding, it represents a strong abstraction from real-world ecology where interactions, and hence cooperative behaviors, occur often randomly within a location that is itself embedded in a larger spatial context (20–23). By focusing on spatially explicit models of cooperation, we gain the opportunity to understand feedbacks between the rules of the game, movement strategies, and long-term persistence of cooperation at larger scales (10, 20, 23–27).Here we study a model of cooperation in spatially structured populations inspired by ecological metacommunities (21–23, 28), where network nodes—instead of individuals—represent habitat patches containing many interacting individuals, and edges mean that two patches are connected by dispersal of those individuals (Fig. 1A). Each patch is a location where games are played, harboring cooperator and defector subpopulations which grow and shrink in time due to internal interactions and movement among locations. Metacommunity models allow one to represent the effects of physical spatial structure directly and explicitly. Moreover, they can be analyzed using master stability functions, which can be used to untangle the impacts of local dynamics and network structure (28–30). We use this ability to explore how different movement strategies impact the outcomes of a cooperation game as a function of network structure.Open in a separate windowFig. 1.Emergence of a heterogeneous stationary state on a two-patch network. (A) Schematic of the spatial game, showing local payoff (Π) relationships among cooperators and defectors occupying the same patch (gray circles) and the dispersal route between them. (B) Difference in equilibrium densities of both types in patches 1 and 2 as link strength is varied. Arrows refer to the example time series shown in C and D. Initial conditions were uniformly drawn from 10−4,10−3], and the patch with the largest initial cooperator density is patch 1. (C) The homogeneous steady state, with the same equilibrium densities of C and D across locations. (Inset) Network showing the proportions of each type in each patch. (D) The same game but with faster diffusion (larger δ), showing emergence of a heterogeneous steady state with higher cooperator densities in patch 1. Parameters are R = 3, S=2, T = 5, P=0.2, μ = 1, and α=3.