Can this quantity defined by exponentiating lengths be meaningful?

Question

I just read Accurate predictions of coexistence in natural systems require the inclusion of facilitative interactions and environmental dependency in which they defined the following property:$^1$

$$\rho = \frac{e^{\alpha_{ij}}e^{\alpha_{ji}}}{e^{\alpha_{ii}}e^{\alpha_{jj}}}.\tag{11}$$

The problem is the $\alpha$ are (I think) not dimensionless. They do not explicitly talk about units in the article however the $\alpha$ come from a difference equation that is essentially equivalent to the time discrete Lotka Volterra model. I would say the unit of $\alpha$ is $\frac{m^2}{\text{individual}}$ (for this question we can just assume it is).

Is the $\rho$ than even defined? Can we argue that the unit of the denominator and the numerator will cancel themselves out and $\rho$ is dimensionless? Are there any attempts to define $e^{\mathrm{m}}$?

--

$^1$ $\alpha$ is strictly speaking called $\alpha'$ in the article.

Answer 1

One can write out the exponential as \begin{equation} e^m = \sum_k \frac{1}{k!} m^k, \end{equation} so this expression would yield a series of terms which all have different units (namely, a different power of $m$). Such a series would not make physical sense.

Quantities are often made dimensionless before being plugged into an exponential by dividing out some arbitrary reference value. In your case, the choice of reference value would only affect $\rho$ by raising it to some power. Such an effect can be irrelevant, depending on the context; in this case, I suppose it would be.