Consider the reaction-diffusion equation ut u1 - uq 2ux2

Question

Questions -

Q1. Consider the reaction-diffusion equation

∂u/∂t = u(1 - u^q) + ∂²u/∂x², (1)

where q > 0 is a constant, and look for a traveling wave solution u(x, t) = U(z), z = x - ct.

(a) Show that U satisfies the differential equation

U'' + cU' + U(1 - U^q) = 0 (2)

(b) Rewrite the ODE (2) as a first-order system for (U, V), where V = U'. Next, find all physically meaningful equilibria of the system and assess their local stability properties. Here u represents population. What restrictions are there on c for a physically meaningful set of critical points for the traveling wave?

(c) Based on your answer in (b), find appropriate conditions for the traveling wave, U, as z → ±∞.

(d) Find an exact solution to (2) (and hence (1)), in the form

U(z) = 1/(1 + ae^bz)^s (3)

for appropriate positive constants s, b and c, and subject to appropriate conditions at ±∞ based on (c). (Hint: Consider separately the cases 2 - sq = 0, 1 and 2, ruling out two of these three possible options and note that a is left arbitrary for the case that works)

(e) Based on your result in (d), what is the speed c of the traveling wave of (1)? Discuss whether or not c meets the conditions you found in (b) for physically meaningful critical points.

(f) For q = 1, find a so that U(0) = 1/2, and then use Matlab to graph the traveling wave solution at t = 0, t = 1 and t = 2.

Q2. Suppose a population, p, satisfies the reaction-diffusion equation

∂p/∂t = D ∂²p/∂x² + rp(α - p)(p - β).

where D, r, α and β are positive constants and 0 < α < β.

(a) Show that in the absence of diffusion, p = 0, and p = β are locally asymtotically stable equilibrium points, while p = α is unstable.

(b) Look for a traveling wave solution p(x, t) = P(z), z = x - ct and express the resulting ODE as a first-order system for P and P'. Find and classify the critical points of that system.

(c) Comment on whether or not traveling wave solutions are possible, comment on any restrictions on the wave speed c for (α, 0) to be a stable node, and finally comment on possible conditions for the traveling wave P as z → ±∞ in this (stable node) case.