 Solution

ID: past-exam-of-the-further-mathematical-biology-course-of-the-university-of-oxford/2023/3/b/solution

Use

K

as the population scale,

N_{0}

r^{- 1}

L

B = K b, N = N_{0} n, T = t / r, X = Lx .

For example, the chemotaxis term becomes

- χ_{B} \partial_{X} (\frac{B}{( N _{0} + N ) ^{2}} N_{X}) = - \frac{χ _{B} K}{N _{0} L ^{2}} \partial_{x} (\frac{b}{( 1 + n ) ^{2}} n_{x}) .

rK

r N_{0}

gives the requested system with the following six dimensionless quantities:

d_{b} = \frac{D _{B}}{r L ^{2}}, d_{n} = \frac{D _{N}}{r L ^{2}}, χ_{b} = \frac{χ _{B}}{r L ^{2} N _{0}}, λ = \frac{a}{r N _{0}}, b_{0} = \frac{B _{0}}{K}, μ = \frac{d K}{r} .

The rescaled domain is

0 < x < 1

t > 0

b_{x} (0, t) = b_{x} (1, t) = n_{x} (0, t) = n_{x} (1, t) = 0,

b (x, 0) = \frac{B _{initial} ( Lx )}{K}, n (x, 0) = \frac{N _{initial} ( Lx )}{N _{0}}, 0 \leq x \leq 1.

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