= Cubic fast-slow reaction-diffusion model
{title2=$u_t=Du_{xx}+u(1-u)(u-\alpha)-w+I,\quad w_t=\varepsilon(u-\beta w)$}
This nondimensional two-variable <reaction–diffusion system> combines a fast cubic local response with slow negative feedback. With $D>0$, $0<\alpha<1$, $\beta>0$ and $\varepsilon\ll1$, its spatially uniform rest state at $I=0$ is linearly stable: the Jacobian has <trace> $-\alpha-\varepsilon\beta$ and <determinant> $\varepsilon(1+\alpha\beta)$. For suitable parameters a finite perturbation can nevertheless cause a large excursion followed by recovery. Frozen recovery gives a <bistable cubic reaction-diffusion equation>; the full coupled travelling-wave boundary-value problem determines whether front pairs form a pulse. This is a generic excitable-medium formulation of the <FitzHugh-Nagumo model>, without assigning any biological variable or application.
Back to article page
