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Chaos and unpredictability in evolutionary dynamics in discrete time.

Daniele Vilone¹, Alberto Robledo, Angel Sánchez

¹Grupo Interdisciplinar de Sistemas Complejos, Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Spain.

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Summary

The discrete-time replicator equation reveals complex dynamics like chaos and periodicity in two-strategy games, diverging from continuous-time models. These findings highlight unusual implications for population dynamics and initial condition sensitivity.

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Area of Science:

Evolutionary Game Theory
Dynamical Systems Theory

Background:

The replicator equation models strategy evolution in games.
Continuous-time models typically yield stable population equilibria.

Purpose of the Study:

To analyze the discrete-time replicator equation for two-strategy games.
To investigate deviations from continuous-time stationary properties.

Main Methods:

Studied a discrete-time version of the replicator equation.
Analyzed parameter-dependent stationary properties.
Observed attractor cascades and variations due to bimodality.

Main Results:

Discrete-time dynamics exhibit periodic and chaotic behavior for large parameters.
Period-doubling and chaotic-band-splitting cascades were observed.
Bimodality leads to more complex variations than unimodal maps.

Conclusions:

Discrete-time models present different stationary properties than continuous-time models.
Unusual physical implications arise from unphysical stationary solutions.
Sensitivity to initial conditions and fractal structures impact population uncertainty.