# Finding units for constants in ODE's

$$\begin{array}{r c l} \frac{dN}{dt} & = & rN\left(1-\frac{N}{K}\right)-\alpha \frac{NP}{\beta P + \gamma N} \\ \frac{dP}{dt} & = & \epsilon \frac{NP}{\beta P + \gamma N} - \delta P \end{array}$$

with $$N(t)$$ the number of prey and $$P(t)$$ the number of predators.

For the equation above I'm trying to find the units for the constants $$r$$, $$N$$,$$K$$,$$\alpha$$,$$\beta$$,$$\epsilon$$,$$\gamma$$ and $$\delta$$

$$N$$ and $$P$$ have the units $$\frac{population}{mile^2}$$

$$t$$ has the units $$days^-1$$

What I've done so far to find $$r$$ and $$K$$

$$\frac{dN}{dt}\ = \frac{population}{mile^2}\ days^-1$$

$$rN = \frac{population}{mile^2}\ days^-1\$$

$$r\frac{population}{mile^2}\ = \frac{population}{mile^2}\ days^-1\$$

This would make $$r = days^-1$$

To find $$K$$

$$\frac{rN^2}{K} = \frac{population}{mile^2}\ days^-1$$

$$\frac{days^-1 (\frac{population}{mile^2})^2}{K} = \frac{population}{mile^2}\ days^-1$$

I found $$K$$ to have the same units as $$N$$ and $$P$$

I found $$\delta$$ to have the unit $$days^-1$$

When trying to find $$\alpha$$ $$\beta$$ $$\gamma$$ and $$\epsilon$$ I get

$$\frac{\alpha}{\beta+\gamma} = days^-1$$

I know I've done something wrong I just can't see my error. Any help would be appreciated!

The structure of the equations and the units of the main variables do not tell you the units of $$\alpha,\beta,\gamma$$ and $$\epsilon$$ separately. This is because you can multiply all of them by the same constant and the equations remain invariant. All that you know is that $$\beta$$ and $$\gamma$$ have the same units and $$\frac{\alpha}{\beta}$$ and $$\frac{\epsilon}{\beta}$$ have $$\mathrm{day}^{-1}$$ units.
If you are made uneasy by this, you can rewrite the equations in terms of the variables $$\alpha^{-1} \beta,\alpha^{-1} \gamma,\epsilon^{-1} \beta,\epsilon^{-1} \gamma$$, which have well-defined units.