About the Reaction Front Simulation
The Fisher-KPP equation — named after Ronald Fisher (1937) and independently Kolmogorov, Petrovsky, and Piskunov — is one of the most fundamental partial differential equations in mathematical biology and chemistry. It combines two physical processes: diffusion, which spreads a substance through space, and logistic growth, which models autocatalytic amplification that saturates when reactant is exhausted. The result is a traveling wave that maintains its shape indefinitely while propagating at a constant speed determined entirely by the two parameters D and r.
This simulation solves the 1D Fisher-KPP PDE numerically using finite differences with an explicit Euler time-stepper. The spatial domain has N=600 cells with Neumann (zero-flux) boundary conditions. The timestep is adaptively chosen to satisfy the stability criterion dt ≤ dx²/(2D), ensuring the numerical solution remains bounded. Three substeps are taken per animation frame to improve temporal resolution while maintaining 60 fps rendering.
The main canvas displays the current concentration profile u(x) as a filled area plot with a line overlay. The inset kymograph in the top-right corner shows a space-time diagram: each horizontal row is a snapshot of u(x) captured at successive time steps, scrolling downward. The diagonal streak of the wave front across the kymograph directly encodes the wave speed — steeper diagonal means faster propagation. Compare the measured wave speed with the theoretical prediction v = 2√(D·r) to verify the Fisher-KPP formula.
Frequently Asked Questions
What is the Fisher-KPP equation?
The Fisher-KPP (Kolmogorov–Petrovsky–Piskunov) equation ∂u/∂t = D∂²u/∂x² + ru(1-u) describes the spread of an advantageous gene (Fisher, 1937) or an autocatalytic chemical reaction. The term D∂²u/∂x² models diffusion, while ru(1-u) is the logistic growth term that saturates at u=1.
What is a traveling wave front?
A traveling wave front is a solution u(x,t) = f(x - vt) that maintains its shape while moving at constant speed v. In the Fisher-KPP equation, any initial condition with compact support evolves into a wave moving at the minimum speed v = 2√(Dr), selected by the steepest-descent (or pulled-front) principle.
What does the wave speed formula v=2√(Dr) mean?
The wave speed v = 2√(Dr) shows that faster diffusion (larger D) and faster growth (larger r) both increase the propagation speed. This formula emerges from linearizing the equation around u=0: the front is 'pulled' by its leading edge where u is small and growth is approximately exponential.
What is autocatalysis?
Autocatalysis is a chemical reaction where the product catalyzes its own formation. If a small amount of product exists, it accelerates more production, leading to exponential growth until reactants are depleted — exactly the logistic ru(1-u) behavior modeled here.
How does the bistable wave differ from Fisher-KPP?
The bistable wave uses a cubic reaction term r·u·(u-a)·(1-u) with a threshold a. Unlike Fisher-KPP which spreads from any positive seed, bistable waves only propagate if the initial perturbation exceeds the threshold a — relevant to nerve impulse models and chemical switches.
What is the nerve impulse analogy?
The Hodgkin-Huxley nerve impulse is a traveling wave in a reaction-diffusion system where the membrane voltage u diffuses along the axon and the ionic gates provide a bistable nonlinearity. The Fisher-KPP model is a simplified version capturing the essential wave-propagation physics.
What is the steepest-descent principle for wave speed selection?
When the initial condition decays faster than e^(-λx) for λ = √(r/D), the Fisher-KPP wave settles at the minimum speed v_min = 2√(Dr). Steeper initial conditions (faster decay) produce faster waves; the 2√(Dr) speed is the slowest possible stable wave, selected as t→∞.
What is the space-time diagram (kymograph)?
The inset shows a kymograph: a 2D image where the horizontal axis is space x and the vertical axis is time t (scrolling downward). Each horizontal strip is a snapshot of u(x) at one moment. The diagonal streak is the traveling wave front, whose slope gives the wave speed directly.
Why does the reaction front maintain its shape?
The shape is maintained because diffusion tends to spread the front (making it wider) while the reaction term opposes this broadening (faster growth at the front steepens it). These two effects balance at a characteristic width ~√(D/r), giving the wave its stable profile.
Where do reaction-diffusion waves appear in nature?
Traveling chemical waves appear in the Belousov–Zhabotinsky reaction (spiral waves in dishes), calcium signaling in cells (fertilization waves), spreading of invasive species (Fisher's original motivation), action potentials in neurons, and epidemic spread across populations.