The Fisher-KPP equation u_t = u_xx + u(1-u) models:
ASpread of an advantageous gene or population invasion with logistic growth
BHeat conduction in a metal bar
CVibration of a string
DElectrostatic potential
Fisher (1937) introduced this equation to model the spatial spread of a favorable allele in a population. The diffusion term u_xx models spatial migration, and u(1-u) is logistic growth. It admits traveling wave solutions u(x,t) = φ(x - ct) connecting the states u = 0 (uninvaded) and u = 1 (invaded).
Question 2 True / False
Turing instability occurs when diffusion destabilizes a spatially homogeneous steady state.
TTrue
FFalse
Answer: True
Turing's remarkable insight (1952) was that in a system of two reacting-diffusing species with different diffusion rates, a steady state that is stable without diffusion can become unstable when diffusion is added. The faster-diffusing inhibitor cannot keep up with the slower-diffusing activator, leading to spatial pattern formation.
Question 3 Short Answer
What is a traveling wave solution of a reaction-diffusion equation?
Think about your answer, then reveal below.
Model answer: A solution of the form u(x,t) = φ(x - ct) that maintains a fixed profile φ while moving at constant speed c
Substituting u = φ(ξ) with ξ = x - ct into u_t = u_xx + f(u) gives the ODE -cφ' = φ'' + f(φ). Traveling wave analysis reduces the PDE to an ODE boundary value problem, which can be studied using phase plane methods. The minimum wave speed is a key quantity.
Question 4 True / False
For the reaction-diffusion equation u_t = Δu + u^p with p > 1, solutions with large initial data can blow up in finite time.
TTrue
FFalse
Answer: True
The reaction term u^p with p > 1 is superlinear, and for large enough initial data, the reaction overwhelms diffusion and drives the solution to infinity in finite time. The critical exponent p = 1 + 2/n (Fujita exponent) separates regimes: for p below this threshold, ALL nontrivial positive solutions blow up.