Mathematical Biology

Homework Assignment 5

2024–25

Please submit solutions to the following two questions as

Homework Assignment 5

by 16:00pm on Monday, November 25, 2024.

I.  Consider the reaction-diffusion equation

under the additional conditions that u(0, t) = 0 = ∂x/∂u (π, t), where a is some positive constant.

(a)  Assuming that Equation (1) admits separable solutions of the form u(x, t)  = X(x)T(t), show that X(x) and T(t) have to satisfy the differential equations

X''  = λX    and    T˙ = (λ - a)T,                                     (2)

where λ is a real constant.

(b)  Solve Equation (2) for the functions X(x) and T(t) under the given conditions.

(c)  Deduce that any function of the form.

with n = 0, 1, 2, . . . and Cn constant, is a solution for (1).

II.  The Burgers-Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) advection-reaction- diffusion equation can be written in rescaled form. as

ut + kuux  = uxx + u(1 - u),                                              (4)

where k > 0 is a real constant.

(a)  Determine the homogeneous – i.e. time- and space-independent – rest states of Equation (4)

(b)  Let z = x - ct, with c positive, and derive the travelling wave equation corre- sponding to (4) that is satisfied by U(z).

(c)  Rewrite that equation as the first-order system

U,  = V,                                                                   (5a)

V,  = -cV + kUV - U(1 - U);                                      (5b)

then, determine the equilibria thereof, and decide their stability.

(d)  Given your findings in item (c), deduce that monotonic front solutions to (4) only exist for c > 2.

(e)  Verify that, for c = 2/k + k/2 with k > 2, a heteroclinic connection between the equilibria of (4) is given explicitly by

V(U) = -2/KU(1 - U)                                                 (6)