Oscillatory evolutionarily stable state and limit cycle in replicator dynamics
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866909981618470912 |
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| author | Chakraborty, Suman Dubey, Vikash Kumar Madhok, Vaibhav Chakraborty, Sagar |
| author_facet | Chakraborty, Suman Dubey, Vikash Kumar Madhok, Vaibhav Chakraborty, Sagar |
| contents | The idea of evolutionarily stable state (ESS) of a population is a cornerstone of evolutionary game theory; moreover, it coincides with the game-theoretic concept of Nash equilibrium. Such a state corresponds to a strategy adopted by the population such that a rare mutant strategy cannot invade the population. In parallel, the dynamical formulation of evolutionary game theory -- particularly through replicator dynamics embodying the tenet of survival of the fittest -- provides a framework for modelling frequency-dependent selection over time. While it is well known that an ESS corresponds to stable fixed point in replicator dynamics, the evolutionary game-theoretic characterization of limit cycles is unknown. Here we fill this lacuna by defining oscillatory ESS (OESS) which we prove to be a stable limit cycle. We also show when an OESS is unique and if there are multiple OESSes, then what their locations are in the phase space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01996 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Oscillatory evolutionarily stable state and limit cycle in replicator dynamics Chakraborty, Suman Dubey, Vikash Kumar Madhok, Vaibhav Chakraborty, Sagar Adaptation and Self-Organizing Systems Theoretical Economics Populations and Evolution The idea of evolutionarily stable state (ESS) of a population is a cornerstone of evolutionary game theory; moreover, it coincides with the game-theoretic concept of Nash equilibrium. Such a state corresponds to a strategy adopted by the population such that a rare mutant strategy cannot invade the population. In parallel, the dynamical formulation of evolutionary game theory -- particularly through replicator dynamics embodying the tenet of survival of the fittest -- provides a framework for modelling frequency-dependent selection over time. While it is well known that an ESS corresponds to stable fixed point in replicator dynamics, the evolutionary game-theoretic characterization of limit cycles is unknown. Here we fill this lacuna by defining oscillatory ESS (OESS) which we prove to be a stable limit cycle. We also show when an OESS is unique and if there are multiple OESSes, then what their locations are in the phase space. |
| title | Oscillatory evolutionarily stable state and limit cycle in replicator dynamics |
| topic | Adaptation and Self-Organizing Systems Theoretical Economics Populations and Evolution |
| url | https://arxiv.org/abs/2601.01996 |