# Interpretation of constants in ODE models

Full disclosure: I have asked a couple of questions over the last few days, but I'm still having some problems with describing parameters in ODE models in words.

In the model:

$$$$\frac{dX}{dt} = \alpha X - \beta XY$$$$

where $$X$$ is the prey population size, $$t$$ is time, $$\alpha$$ is the birth rate, $$\beta$$ is the predation rate, and $$Y$$ is a population of predators, $$\alpha$$ must have dimensions 1/time and $$\beta$$ must have dimensions 1/(predator x time) to make the RHS dimensionally consistent.

Can we, then, interpret $$r$$ as "the number of prey born per member of the population per time", which has units number of prey born / number of prey / time = 1/time?

And can we interpret $$\beta$$ as "the number of prey eaten per member of the prey population per each predator per time", which has dimensions number of prey eaten / number of prey / number of predators / time and reduces to 1/predator/time = 1/(predator x time). Could another way of describing this be "$$\beta$$ is the probability of being eaten by a predator per time point", or is strictly "the per-prey number of prey eaten per predator per time period".

Are these interpretations correct? I'm finding a lot of ODEs have 'hidden dimensions', in that the coefficients are not explained very well. This becomes a problem when you have to work in applied contexts and interpret the actual numbers.

Thanks a lot!

It would seem that it's prudent to interpret the terms a certain way here. Ignoring for a moment the literal meaning of each parameter, we can see what happens to the system when certain terms change. If $$X$$ remains constant, then we can interpret $$\beta$$ in a general sense. That is, it represents the change in $$dX/dt$$ when $$Y$$ changes by one unit. From this, the most useful description of the parameter would be the quantified impact of predation on the change in $$X$$ in number of predators over time.
Similarly, $$\alpha$$ can certainly be viewed as the intrinsic growth rate of $$X$$ and as such, you can view it as the change in the $$X$$ of the first term with respect to time. Both $$X$$ and $$Y$$ are functions of time and their units are simply the respective population number. I don't think that you need to convolute the interpretation of the second term due to $$X$$ and $$Y$$ referring to different species, generally. They are both populations. So, if I absolutely had to assign units to $$\alpha$$ and $$\beta$$, I'd say they are "1/time."