Why not teach ecology using the simpler Verhulst form of the logistic equation rather than the Gause-Witt form?
The Verhulst form for per capita population growth is:
Of course, it’s the same equation really, because K = r/α.
However, the simpler form above:
1) Is more historically correct. Verhulst, Lotka, and Volterra all introduced the first form; it was Pearl and then Gause who first popularized the r-K form. K was equated with “carrying capacity” only later, only after WWII, as far as I can tell. Today, the r-K form has come to seem intuitively correct, in spite of its many issues given below (in my paper I trace the history).
2) Makes it easier to understand the relation with Lotka-Volterra predation equations.
3) Makes clear why Andrewartha and Birch were insistent that “density – independent” factors control population density. In a sense they’re right; all the density independence is contained in r = (b-d), birth rate – death rate, since if you alter, say the density-independent death rate d, you alter the equilibrium r/α proportionately.
4) Explains “Levins’ Paradox:” that with the r-K formulation, N shoots up to +infinity in finite time given a negative r and N > K initially.
5) Simplifies understanding of the Gause-Witt isoclines in two-species competition, and explains why r is indeed involved in the stability of competitive interactions. The analysis is exactly the same, but about 1000x clearer, once you get used to it.
The simpler, historically correct Lotka-Volterra competition equations look like this:
Here is Fig. 2 from the paper cited below, showing the isoclines and equilibria:
The isoclines hit the N1 axis at r1/α11 (where α11 is the intraspecific competition coefficient, i.e. α in the single species equation above) and r2/α21, and the N2 axis at r2/α22 and r1/α12. Note, r1 and r2 are very much involved in competition stability, as they should be! (But aren’t using the Gause-Witt equations).
6) Clarifies the relationship of natural selection to competition. Natural selection is indeed competition within species. And constant-selection models commonly used in population genetics are simply equivalent to alterations of b or d between genotypes in r with constant α values.
7) Helps to re-unify evolution and population ecology; this work was started by Lotka and Volterra, but became very confused after WWII with ideas such as “r– and K-selection.”
8) Can demonstrate that the actual renewable resources produced in the environment will not in fact be completely used up at equilibrium. In other words, the actual “carrying capacity” is not equal to the equilibrium (r/α) density.
9) Can readily be derived from a consideration of a variety of considerations of consumption of limited renewable resources.
Many mathematical biologists have understood this issue all along (e.g. RA Fisher, FB Christiansen, N Barton), and others are increasingly drawn to this understanding now. However, this simplification has yet to enter any introductory textbook on ecology or evolution.
When I started out on this journey, I thought I was doing original research. Eventually, on sabbatical in 2008-9, I discovered Smouse’s little-cited paper (1976) in Am Nat and realized that everything I knew about the topic had been discovered again and again, and that my new understanding was doomed to be merely a kind of historical overview. Nonetheless I hope it is useful.
Instead of citing me, please do cite the people I cite! (And probably others I don’t know about?) We have here a case of poorly connected-up salami science; we (collectively, at least) understand this issue, but nobody has put it all together yet in our textbooks.
You could be the first to do so…
To read my detailed defense of these opinions, you can find my paper online:
Mallet, J. 2012. The struggle for existence: how the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution and speciation. Evolutionary Ecology Research 14: 627-665. http://www.ucl.ac.uk/taxome/jim/pap/mallet_the_struggle_2012.pdf