Heterogeneity induces spatiotemporal oscillations in reaction-diffusion systems.

Area of Science:

• Mathematical Biology
• Chemical Kinetics
• Pattern Formation

Background:

• Reaction-diffusion (RD) systems are fundamental to understanding pattern formation in biological and chemical systems.
• Turing instabilities typically generate stationary patterns, but their dynamics in heterogeneous environments are less understood.
• Previous models often assume spatial homogeneity, limiting their applicability to complex natural systems.

Purpose of the Study:

• To investigate a novel instability in activator-inhibitor RD systems with spatial heterogeneity.
• To characterize the periodic creation, translation, and destruction of spike solutions.
• To explore the underlying mechanisms and theoretical limitations in explaining these spatiotemporal oscillations.

Main Methods:

• Analysis of the Gierer-Meinhardt system in the shadow limit.
• Numerical exploration of the system's behavior.
• Investigation of spike movement using asymptotic theory.
• Examination of bifurcations and stability of stationary states.
• Testing robustness with the Schnakenberg RD model and localized heterogeneity.

Main Results:

• A new instability arises in spatially heterogeneous RD systems, leading to oscillatory spike dynamics.
• This oscillatory behavior occurs within the Turing space, not associated with Hopf bifurcations.
• Asymptotic theory predicts spike speed but fails to explain the oscillations.
• Oscillations are driven by destabilization of steady spikes due to endogenous activator production.
• The oscillation period approaches infinity at the instability edge, distinct from typical limit cycle bifurcations.
• Nearby stationary states are Turing unstable or exhibit saddle-node bifurcations.

Conclusions:

• Spatially heterogeneous RD systems exhibit robust, ubiquitous spatiotemporal oscillations.
• Current theoretical tools, including spike stability analysis and shadow-limit asymptotics, are insufficient to explain these phenomena.
• This highlights challenges in understanding robust pattern emergence from Turing mechanisms in heterogeneous environments.
• Further mathematical analysis is needed to elucidate these complex dynamics.