Evolution of cooperation has been a major issue in evolutionary biology. Cooperation is observed not only in dyadic interactions, but also in social interactions involving more than two individuals. It has been argued that direct reciprocity cannot explain the emergence of cooperation in large groups because the basin of attraction for the 'cooperative' equilibrium state shrinks rapidly as the group size increases. However, this argument is based on the analysis of models that consider the deterministic process. More recently, stochastic models of two-player games have been developed and the conditions for natural selection to favour the emergence of cooperation in finite populations have been specified. These conditions have been given as a mathematically simple expression, which is called the one-third law. In this paper, we investigate a stochastic model of n-player games and show that natural selection can favour a reciprocator replacing a population of defectors in the n-player repeated Prisoner's Dilemma game. We also derive a generalized version of the one-third law (the {2/[n(nC1)]} 1/(nK1) law). Additionally, contrary to previous studies, the model suggests that the evolution of cooperation in public goods game can be facilitated by larger group size under certain conditions.
The histories of individuals impact the dynamics of their populations. Matrix projection models (MPMs) are used to analyse population dynamics, but are not structured to incorporate these influences. Historical MPMs (hMPM) were developed to incorporate these impacts, but their complexity has left them little used. We developed r package lefko3 to provide simple, quick methods to estimate and analyse hMPMs, as well as ahistorical MPMs.
Package lefko3 handles the entire workflow from dataset organization to the construction and analysis of hMPMs. Dataset management functions reorganize most demographic data formats, and matrix creation functions estimate both raw and function‐based matrices. Vital rates may be modelled as mixed or generalized linear models, with model selection protocols involving current best practices.
The core kernels are binaries allowing even matrices with over 10,000 rows and columns to be estimated quickly without parallelization. We also include functions to conduct basic deterministic projection analyses.
Package lefko3, available on CRAN, dramatically reduces the difficulties in testing the impacts of individual history on population dynamics. We provide three vignettes to showcase how hMPMs can be developed and analysed.
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