= Solution
Since $dN_i/dt=K_ir_1\,du_i/d\tau$, division of the equations by $K_ir_1$ gives
$$
\frac{du_1}{d\tau}=u_1\left(1-u_1-\frac{b_{12}K_2}{K_1}u_2\right),\qquad
\frac{du_2}{d\tau}=\frac{r_2}{r_1}u_2\left(1-\frac\delta{r_2}-u_2-\frac{b_{21}K_1}{K_2}u_1\right).
$$
Thus the dimensionless parameters are
$$
\boxed{\alpha_{12}=b_{12}K_2/K_1,\quad\alpha_{21}=b_{21}K_1/K_2,\quad\rho=r_2/r_1,\quad\gamma=1-\delta/r_2}.
$$
Each is a ratio of quantities with matching units. In particular $\gamma$ can be negative even though every original model parameter is positive.
Back to article page
