Diffusion-driven instability of periodic solutions

Victor Ogesa Juma

University of British Columbia

"Diffusion-driven instability of periodic solutions"

Reaction-diffusion systems are fundamental in modeling the complex spatiotemporal dynamics in biological, chemical, and ecological phenomena. In this study, we investigate a bistable reaction-diffusion system motivated by the experimental observations on Rho-GEF-Myosin signaling network that controls cell contraction dynamics. Through a combination of numerical bifurcation analysis and simulations, we explore how diffusion alters the intrinsic dynamics of distinct temporal regimes exhibited by the underlying reaction kinetics. Our results demonstrate that: (i) diffusion can destabilize a uniform stable steady state, leading to classical Turing patterns; (ii) in oscillatory regimes, diffusion drives the system away from temporal periodicity into spatially heterogeneous oscillations, indicating far-from-equilibrium behavior; and (iii) in bistable regions, diffusion induces pattern formation, wave propagation, and oscillatory pulses. Floquet theory is used to quantify the diffusion-driven destabilization of a homogeneously stable limit cycle, identifying critical diffusion coefficients for diffusion-driven instability. These findings offer theoretical insights into diffusion-induced transitions and can contribute to the broader understanding of pattern formation and dynamic regulation in developmental and cellular biology.

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