0

in ecological populations models, the standard variable is the density of individuals, i.e. the number of individuals per space unit (area, volume). The meaning of this is intuitively clear to me, but I'm having trouble putting this into rigorous terms:

Say the density $u(x)$ is the number of lions contained in a disk of radius r around x, divided by the disks's area $r^2 \pi$. But then clearly $u(x)$ depends on $r$: If I chose $r$ sufficiently small, $u$ will either be 1 or 0, depending whether or not there is a lion in $x$. Thus, it is not like there was a convergence for $r\to 0$. I checked a couple of books on mathematical population models, but didn't find a solution to this dilemma. Hence my question: What is the definition of a spatial population density?

kamui
  • 11

1 Answers1

1

You have to realise these are models and won't model extreme cases such as very small disks.

Models also make assumptions... one of these might be that density over a region is uniform. Then if you have 10 lions per 1000 m$^2$ then the absolute correct density is 0.01 lions per m$^2$.

You are possibly confusing the density with the method of calculating it for a specific area.

JP McCarthy
  • 8,420
  • Of course they model extreme cases, just take a chemical reaction-diffusion model, there the space unit is very small. Yet, no matter at which scale, all of these density-based models use the same notion of density, which seems odd to me. I don't understand what you mean by "one of these [assumptions] might be .. is uniform". Do you mean the assumption that the number of lions per 1000m$^2$ is the same as the 0.01$\cdot$ the number of lions per 100m$^2$ - but possibly not the same as 0.0001$\cdot$ the number of lions per 10m$^2$? Seems arbitrary to me... – kamui Feb 20 '14 at 16:38