 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2023/9/a/ii/solution

Since

d N_{i} / d t = K_{i} r_{1} d u_{i} / d τ

K_{i} r_{1}

gives

\frac{d u _{1}}{d τ} = u_{1} (1 - u_{1} - \frac{b _{12} K _{2}}{K _{1}} u_{2}), \frac{d u _{2}}{d τ} = \frac{r _{2}}{r _{1}} u_{2} (1 - \frac{δ}{r _{2}} - u_{2} - \frac{b _{21} K _{1}}{K _{2}} u_{1}) .

Thus the dimensionless parameters are

α_{12} = b_{12} K_{2} / K_{1}, α_{21} = b_{21} K_{1} / K_{2}, ρ = r_{2} / r_{1}, γ = 1 - δ / r_{2} .

Each is a ratio of quantities with matching units. In particular

γ

can be negative even though every original model parameter is positive.

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