Recently I posted a question to Ecolog about an interpretation of Levin's simpe model. Here is the orignal question and the responses I got.
Alan Griffith __________________________ I've been struggling to come up with a simple and intuitive explanation for the equilibrium result of Levin's simple metapopulation model. This is the model with internal colonization only. No rescue is possible. That result is p = 1 - e/c or the equilibrium proportion of patches occupied is determined solely by the ratio of extinction chance and colonization chance of individual patches. The particular situation I wish to describe is when e = c and the metapopulation goes extinct. This is always puzzling to students because they reason that if extinction probability just balances colonization probability, the metapopulation could just hang on as patches are just replaced as they go extinct. Does anyone have an easy explanation for this clash between the math and the mind? Alan Griffith Responses 1. Would it help to remind you students that the model is a differential equation, which means that everything happens instantaneously and at the same time? They cannot balance each other because at the instant that dispersal occurs, the population goes extinct, so you have colonization from a zero population. It sounds like they are looking at it from the destination patch's point of view, and forgetting that the source is also changing - I had a similar difficulty with the concept when I first encountered it. This would also be a good chance to discuss the difference between models and reality. They'll never encounter a real world example where e and c match exactly. Hope this helps, Nate -- Nathan Lichti Graduate Research Assistant Forestry and Natural Resources Purdue University 715 W State Street West Lafayette IN 47907-2061 2. This is a bit convoluted to explain, but basically is a consequence of the fact that the actual colonization rate of vacant patches within the landscape is a function of a constant colonization rate and the number of occupied patches. So colonization is dependent on occupancy. Extinction of occupied patches is independent though. So increased extinction (relative to c) reduces the number of occupied patch (all else equal) which reduces the 'production' of propagules that can colonize vacant patches. When e = c there simply aren't enough (any, actually) occupied patches left that can generate propagules fast enough to keep up. Maybe it helps to reassess the actual model in different terms: Levins model just describes the change in patch occupancy over time, which can be written as: Occupied-patches * patch-colonization rate - Occupied-patches * patch extinction, or dp/dt = p * m - p * e The equilibrium is where patch colonization equals patch extinction: m = e This is the equilibrium you and your students are thinking of. But the colonization rate depends on the number of occupied patches: m = c * (1 - p) So at the equilibrium a certain proportion (p) of patches is occupied: e = c * (1 - p) When e = c, that equilibrium proportion of occupied patches is 0. Extinction catches up with 'production'. I'll put a model on the web if you're interested. It is a simulation that might make things clearer. Or not... http://life.uiuc.edu/~sstoddard/Levins.html Steven T. Stoddard Program in Ecology and Evolutionary Biology University of Illinois at Urbana-Champaign www-u.life.uiuc.edu/~sstoddar [EMAIL PROTECTED] (217) 333 - 2235 3. My students have struggled with this as well (as have I). They invariably assume that at equilibrium e=c. I'm not a modeler, but I think one way to explain it is to remind the students that Levins' model is really just another way of expressing the logistic growth model dN/dt = rN(1-N/K), where ln(c/e) is analogous to r and N/K is analogous to P. We don't assume that r is equal to 0 at equilibrium. In fact, r > 0, but dN/dt approaches 0 as N approaches K. r represents sort of a biotic potential for growth in the absence of density dependence. Likewise, I think you could argue that c/e represents a biotic potential for population expansion throughout a region. Another point that might be relevant is that in order for the equilibrium metapopulation to persist when perturbed (i.e., when P is reduced below the equilibrium value), it must be able to increase when rare and return to that equilibrium. If e/c = 1, there is no way for that to happen. Steve Steve Brewer Associate Professor Department of Biology PO Box 1848 University, MS 38677-1848 telephone: (662) 915-1077 FAX: (662) 915-5144 4. / \ // \\ Alan B. Griffith, Ph.D. \\ | | // Department of Biological Sciences \ \`|()|'/ / University of Mary Washington \_`( )'_/ 1301 College Avenue /( )\ Fredericksburg, VA 22401-5300 / ^^ \ 540-654-1422 | | [EMAIL PROTECTED] | |
