0
$\begingroup$

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:

\begin{equation} \frac{dX}{dt} = \alpha X - \beta XY \end{equation}

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!

$\endgroup$

1 Answer 1

1
$\begingroup$

Full disclosure: I may be corrected since I'm making an intuitive observation here.
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."

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .