 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2021/9/b/i/solution

N_{t} = N

, the discrete equation reduces to

N [\frac{r}{1 + b N ^{2}} - (δ + 1)] = 0.

Write

s = δ + 1 > 1

. For

r < s

, no nonzero real root is possible, so zero is the only real steady state. For

r > s

the real roots are

N = 0, N = \pm \frac{r - s}{b s} .

Thus there is exactly one strictly positive steady state,

N^{*} = (r - s) / (b s)

. There are three real steady states in total, two of them nonnegative. At

r = s

the only one is zero. This distinction keeps the positive-population statement separate from the count over all real numbers.

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