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Pattern formation in individual-based systems with time-varying parameters.

Peter Ashcroft1, Tobias Galla1

  • 1Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom.

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|February 4, 2014
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Summary
This summary is machine-generated.

Finite-time sweeps across symmetry-breaking bifurcations generate patterns. Slow parameter changes yield large patterns, while fast changes create many small domains, similar to defect formation.

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Area of Science:

  • Complex systems
  • Non-equilibrium physics
  • Theoretical biology

Background:

  • Symmetry-breaking bifurcations are critical transitions in many systems.
  • The Kibble-Zurek mechanism explains defect formation during phase transitions.
  • Intrinsic noise from discrete dynamics can trigger symmetry breaking.

Purpose of the Study:

  • To investigate pattern formation during finite-time sweeps across symmetry-breaking bifurcations in individual-based models.
  • To analyze the impact of sweep speed on pattern scale.
  • To theoretically predict the characteristic length scale of emergent patterns.

Main Methods:

  • Utilizing a linear-noise approximation to derive theoretical estimates for pattern length scales.
  • Applying the framework to models of opinion dynamics, evolutionary game theory, and cell differentiation.
  • Conducting numerical simulations to validate theoretical predictions.

Main Results:

  • Pattern scale depends on the speed of parameter sweeps across bifurcations.
  • Slow sweeps generate large-scale patterns, while fast sweeps result in numerous small domains.
  • Theoretical predictions for characteristic length scales are confirmed by simulations.

Conclusions:

  • The study provides a theoretical framework for understanding pattern formation in systems undergoing symmetry breaking.
  • Sweep speed is a crucial parameter influencing the spatial organization of patterns.
  • The findings are applicable to diverse fields, including social dynamics, evolution, and developmental biology.