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

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!

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1 Answer 1

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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.

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