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 $e^{\mathrm{m}}$?
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$^1$ $\alpha$ is strictly speaking called $\alpha'$ in the article.