 Solution

ID: past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2024/paper-1/b2/b/solution

F^{'} (x) = \frac{r ( b - x )}{( b + x ) ^{3}} .

At the positive fixed point,

b + \overset{x}{ˉ} = r

F^{'} (\overset{x}{ˉ}) = \frac{2 b}{r} - 1.

Since

0 < b / r < 1

, we have

- 1 < F^{'} (\overset{x}{ˉ}) < 1

ε_{t}

obeys

ε_{t + 1} = F^{'} (\overset{x}{ˉ}) ε_{t} + O (ε_{t}^{2})

, and hence shrinks locally. Thus the positive fixed point is locally asymptotically stable by fixed point stability for an iteration.

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