Diffusive instability in hyperbolic reaction-diffusion equation with different inertia

Area of Science:

• Mathematical Biology
• Chemical Kinetics
• Nonlinear Dynamics

Background:

• Reaction-diffusion systems are fundamental models in biological and chemical pattern formation.
• Hyperbolic systems incorporate inertia, offering distinct dynamics from traditional parabolic models.
• Understanding instabilities is crucial for predicting pattern emergence.

Purpose of the Study:

• To analyze wave, Turing, and Hopf instabilities in a 2D hyperbolic reaction-diffusion system with differing inertia.
• To theoretically and numerically determine criteria for these instabilities.
• To compare the behavior of hyperbolic systems with their parabolic counterparts.

Main Methods:

• Theoretical analysis of stability criteria.
• Numerical simulations of the reaction-diffusion system.
• Application of the Schnakenberg system as a local interaction model.

Main Results:

• Wave instability can occur in hyperbolic systems with different inertias and equal diffusivities, a scenario impossible in parabolic systems.
• Wave instability is possible in two-species hyperbolic systems with identical inertia if diffusion coefficients differ.
• Turing instability is independent of inertia, but local system stability is inertia-dependent.

Conclusions:

• Inertia plays a significant role in wave instability in hyperbolic reaction-diffusion systems.
• The distinct dynamics of hyperbolic systems allow for instabilities not observed in parabolic systems.
• The findings provide insights into pattern formation mechanisms influenced by inertial effects.