Delayed bifurcation phenomena in reaction-diffusion equations: persistence of canards and slow passage through Hopf bifurcations

TL;DR

This paper demonstrates that delayed bifurcation phenomena like canards and slow passage through Hopf bifurcations, known in ODEs, also occur in reaction-diffusion PDEs, with implications for spatio-temporal dynamics in biological systems.

Contribution

It extends the understanding of delayed bifurcations from ODEs to PDEs, showing their persistence and providing a quantitative framework for predicting delayed stability loss.

Findings

Delayed bifurcation phenomena persist in reaction-diffusion PDEs.

Spatio-temporal canard dynamics are observed in simulations.

A formula for the buffer curve predicting delayed oscillation onset is derived.

Abstract

In the context of a spatially extended model for the electrical activity in a pituitary lactotroph cell line, we establish that two delayed bifurcation phenomena from ODEs ---folded node canards and slow passage through Hopf bifurcations--- persist in the presence of diffusion. For canards, the single cell (ODE) model exhibits canard-induced bursting. Numerical simulations of the PDE reveal rich spatio-temporal canard dynamics, and the transitions between different bursts are mediated by spatio-temporal maximal canards. The ODE model also exhibits delayed loss of stability due to slow passage through Hopf bifurcations. Numerical simulations of the PDE reveal that this delayed stability loss persists in the presence of diffusion. To quantify and predict the delayed loss of stability, we show that the Complex Ginzburg-Landau equation exhibits the same property, and derive a formula for the space-time boundary that acts as a buffer curve beyond which the delayed onset of oscillations must occur.