A note on the replicator equation with explicit space and global regulation

Alexander S. Bratus; Vladimir P. Posvyanskii; Artem S. Novozhilov; Alexander S. Bratus; Vladimir P. Posvyanskii; Artem S. Novozhilov

doi:10.3934/mbe.2011.8.659

Mathematical Biosciences and Engineering

2011, Volume 8, Issue 3: 659-676. doi: 10.3934/mbe.2011.8.659

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A note on the replicator equation with explicit space and global regulation

1.
Applied Mathematics–1, Moscow State University of Railway Engineering, Obraztsova 9, Moscow, 127994

Received: 01 July 2010 Accepted: 29 June 2018 Published: 01 June 2011
MSC : Primary: 35K57, 35B35, 91A22; Secondary: 92D25.

A replicator equation with explicit space and global regulation is considered. This model provides a natural framework to follow frequencies of species that are distributed in the space. For this model, analogues to classical notions of the Nash equilibrium and evolutionary stable state are provided. A sufficient condition for a uniform stationary state to be a spatially distributed evolutionary stable state is presented and illustrated with examples.
- evolutionary stable state,
- reaction-diffusion systems.,
- Nash equilibrium,
- Replicator equation
Citation: Alexander S. Bratus, Vladimir P. Posvyanskii, Artem S. Novozhilov. A note on the replicator equation with explicit space and global regulation[J]. Mathematical Biosciences and Engineering, 2011, 8(3): 659-676. doi: 10.3934/mbe.2011.8.659

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Abstract

A replicator equation with explicit space and global regulation is considered. This model provides a natural framework to follow frequencies of species that are distributed in the space. For this model, analogues to classical notions of the Nash equilibrium and evolutionary stable state are provided. A sufficient condition for a uniform stationary state to be a spatially distributed evolutionary stable state is presented and illustrated with examples.
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