Bulletin of Mathematical Biology

, Volume 76, Issue 12, pp 3070–3087 | Cite as

Evolution of Density-Dependent Cooperation

  • Anne SeppänenEmail author
  • Kalle Parvinen
Original Article


Cooperation is surprisingly common in life despite of its vulnerability to selfish cheating, i.e. defecting. Defectors do not contribute to common resources but take the advantage of cooperators’ investments. Therefore, the emergence and maintenance of cooperation have been considered irrational phenomena. In this study, we focus on plastic, quantitative cooperation behaviour, especially on its evolution. We assume that individuals are capable to sense the population density in their neighbourhood and adjust their real-valued investments on public goods based on that information. The ecological setting is described with stochastic demographic events, e.g. birth and death, occurring at individual level. Individuals form small populations, which further constitute a structured metapopulation. For evolutionary investigations, we apply the adaptive dynamics framework. The cost of cooperative investment is incorporated into the model in two ways, by decreasing the birth rate or by increasing the death rate. In the first case, density-dependent cooperation evolves to be a decreasing function of population size as expected. In the latter case, however, the density-dependent cooperative investment can have a qualitatively different form as it may evolve to be highest in intermediate-sized populations. Indeed, we emphasize that some details in modelling may have a significant impact on the results obtained.


Adaptive dynamics Structured metapopulation Public-goods cooperation 



This study was funded by the Academy of Finland, project number 128323 to K.P., and by the Kone Foundation, a grant to A.S. The authors wish to thank Samuel Alizon and anonymous reviewers for comments.


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Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  2. 2.Evolution and Ecology ProgramInternational Institute for Applied Systems AnalysisLaxenburgAustria

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