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The Diffusion of Passive Tracers in Laminar Shear Flow

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Dynamics of reaction-diffusion systems in a subdiffusive regime.

D Hernández¹, C Varea, R A Barrio

¹Instituto de Fisica, Universidad Nacional Autónoma de México (UNAM), Apartado Postal 20-364 01000 México, D.F., Mexico.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|April 28, 2009

Summary

Fractional time derivatives in reaction-diffusion systems lead to anomalous subdiffusion. This research relaxes Turing instability conditions, enabling pattern formation in non-activator-inhibitor systems.

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Area of Science:

Complex Systems
Nonlinear Dynamics
Mathematical Biology

Background:

Reaction-diffusion systems are fundamental to pattern formation.
Anomalous diffusion, particularly subdiffusion, deviates from classical Fickian diffusion.
Turing instabilities typically require specific activator-inhibitor kinetics.

Purpose of the Study:

To investigate reaction-diffusion dynamics with fractional time derivatives.
To analyze the implications of anomalous subdiffusion on pattern formation.
To explore the relaxed conditions for Turing instabilities.

Main Methods:

Mathematical modeling of reaction-diffusion systems.
Analysis of fractional time derivatives and their impact on diffusion.
Numerical simulations in two dimensions using a generic Turing model.

Main Results:

Fractional time derivatives induce anomalous subdiffusion (mean-square displacement ~ t^gamma, gamma<1).
The conditions for diffusion-driven instabilities, including Turing instabilities, are significantly relaxed.
Pattern formation (Turing instability) is possible in systems without activator-inhibitor kinetics.

Conclusions:

Fractional dynamics offer a broader framework for understanding pattern formation.
Subdiffusion can facilitate instabilities previously thought impossible.
This work expands the scope of systems exhibiting Turing patterns.