 Solution

ID: past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2008/a/3/d/solution

Substitute

g b = c g^{'}

into the bacterial profile equation. This gives

b^{''} + c b^{'} + c g^{'} = 0,

which integrates once to

b^{'} + c (b + g) = C .

At the asymptotic constant states the profile derivative tends to zero. The state ahead of the travelling wave, where

b \to 0

and

g \to 1

, therefore gives

C = c

. The state behind it, where

g \to 0

and

b \to b_{0}

, gives

C = c b_{0}

. Since

c > 0

b_{0} = 1, b^{'} + c (b + g) = c .

This first integral expresses the matching between biomass produced and biological growth factor consumed after the chosen nondimensionalization.

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