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    Persistent cooperation, a strategy where individuals pay extra to secure their contributions, significantly hinders defection in structured populations. This finding contrasts with well-mixed populations, suggesting spatial structure aids cooperation.

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    Area of Science:

    • Evolutionary Biology
    • Social Science
    • Game Theory
    • Computational Simulation

    Background:

    • Cooperative behavior is a key area of study in evolutionary biology and social science.
    • The Public Goods Game (PGG) models cooperation but often results in the dominance of defection.
    • Previous research explored cooperation in well-mixed populations; this study examines structured populations.

    Purpose of the Study:

    • To investigate the evolutionary dynamics of persistent cooperation in lattice-structured populations.
    • To analyze the impact of persistent cooperators on the prevalence of pure cooperators and defectors.
    • To compare cooperation dynamics in structured versus well-mixed populations.

    Main Methods:

    • Theoretical analysis of evolutionary dynamics on one-dimensional lattices.
    • Numerical simulations to illustrate phase diagrams, strategy frequencies, and spatial patterns.
    • Modeling three competing strategies: pure cooperators, defectors, and persistent cooperators.

    Main Results:

    • Approximate expressions for strategy fixation probabilities were derived for one-dimensional lattices.
    • Simulations on one- and two-dimensional lattices revealed phase diagrams and spatial patterns.
    • Persistent cooperators were found to suppress defector spread more effectively in structured populations than in well-mixed ones.

    Conclusions:

    • Structured populations generally facilitate the evolution of cooperation.
    • Persistent cooperation is a robust strategy for promoting cooperation, especially in spatial networks.
    • The findings highlight the importance of population structure in understanding the evolution of cooperation.