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Spatial instabilities in reaction random walks with direction-independent kinetics
1Department of Chemistry, Southern Methodist University, Dallas, Texas 75275-0314, USA. whorsthe@mail.smu.edu
Summary
We explore spatial instabilities in reaction-walk systems, finding new oscillatory patterns not seen in traditional reaction-diffusion models. These instabilities, driven by particle movement, create unique spatial structures.
Area of Science:
- Complex Systems
- Nonlinear Dynamics
- Mathematical Biology
Background:
- Reaction-diffusion systems are standard models for pattern formation.
- Brownian motion is the typical diffusion model, assuming infinite propagation speed.
- Persistent random walks offer an alternative diffusion model with finite propagation speed.
Purpose of the Study:
- To investigate spatial instabilities in reaction systems using persistent random walks.
- To compare instabilities in reaction-walk systems with those in reaction-diffusion systems.
- To derive and analyze new types of pattern formation.
Main Methods:
- Developing evolution equations for reaction random walks.
- Analyzing bifurcations from the homogeneous steady state.
- Investigating stability properties of one-variable systems.
Main Results:
- Identified two types of transport-driven instabilities.
- One instability yields stationary patterns, analogous to Turing instability.
- A novel instability in the ballistic regime produces oscillatory spatial patterns.
Conclusions:
- Reaction-walk systems exhibit unique spatial instabilities absent in reaction-diffusion systems.
- Finite propagation speed in random walks leads to new pattern-forming mechanisms.
- Oscillatory spatial patterns emerge in the ballistic regime of reaction-walk systems.