Well I think it’s about time we got down to the theory of population dynamics . To be of any use, a theory must be applicable to real data. So I would like to get some data into this discussion. I will begin with the best data set I know of, the larch bud moth (Zeiraphera diniana) in the Swiss Alps (and elsewhere). Two sets of data, one from Sils and the other from the Engadine (covering 40 years), are presented in Turchin et al. (2003), and can be seen at http://www2.bren.ucsb.edu/~kendall/pubs/2003Ecology.pdf (see Figure 1). Measurements of population numbers per kilogram of branches and foliage were made once a year when the larvae were in the 3rd instar. Extreme cycles of the bud moth can be seen to oscillate with a period of 8-9 years and about 100, 000 fold increase between trough and peak numbers. At the maximum numbers, the moth completely defoliates the larch trees (the trees are completely brown and the foliage entirely destroyed). It is therefore restricted to this density (i.e., it cannot increase any further). Now the paper by Turchin et al. (2003) pursues a mathematical argument, but we can see what happens without any mathematics. First, the bud moth grows approximately linearly on a logarithmic scale. It is then prevented from growing further by shortages of food; i.e., all the needles of larch trees are destroyed; however, the trees can reproduce new needles after the moths have pupated and so do not usually die. Thus, the growth of the moth population is first stopped by a lack of food (whether the quality of the food changes or not). Then the population of insect parasitoids builds up at a faster rate (see Figure 3 in Turchin et al.). Note that, prior to the bud moth peak, percentage parasitism is increasing at an insufficient rate to stop the bud moth. This can be seen by the slope of the parasitoid growth rate which is generally less than the slope of the bud moth prior to the peak (see Figure 3). Thus, growth is stopped by running out of food, not parasitism. However, after the population ceases growing, percentage parasitism quickly builds up to suppress the bud moth to a very sparse density. Note from Figure 3 that the rate of change of percentage parasitism increases significantly after peak density, reaching a maximum in the second or third year following the peak. Thus, growth is stopped by a shortage of food and synchronized to an 8-9 year periodicity by the interaction with parasitoids; without parasitoids the population of moths would not reach such low numbers or have such a consistent periodicity.

It is interesting that the bud moth peak of 1989 (not shown in the Figures) was curtailed by several years of unfavorable weather in winter and spring (Baltensweiler 1993). The bud moth did not reach the usual peak density and no widespread defoliation occurred. Presumably parasitoids were able to suppress this population prematurely because of the lower growth rates.

N.B. Lars Witting has an alternative explanation of larch bud moth population cycles. Perhaps he will be interested in explaining this hypothesis?

Now what does all this example say about the theory (see Berryman 1999)?
1. The population of bud moths grows from a very low level approximately linearly on a logarithmic scale (1st principle).
2. The population may grow exponentially on a logarithmic scale for a short time due to cooperative processes operating at very low densities (2nd principle; see Figure 4.4 in Berryman 1999). However, sampling error is very high at low densities so we cannot say anything definite about this.
3. Growth is terminated by competition for food, either because of changes in food quality or because all food is consumed at peak densities (3rd principle).
4. Cyclic dynamics are induced by time delays in the numerical feedback with parasitoids (4th principle) which take over the dynamics after the peak.
5. The 5th principle does not come into play because the numbers of bud moth vary so dramatically (from about 1 to 100,000).

Baltensweiler, W. 1993. Why the larch bud-moth cycle collapsed in the subalpine larch–cembran pine forests in the year 1990 for the first time since 1850. Oecologia 94:62–66.

Berryman, A. A. 1999. Principles of population dynamics and their application. Stanley Thornes Ltd (available at Taylor and Francis, http://www.taylorandfrancis.co.uk/ )

Turchin, P., et al. 1983. Dynamical effects of plant quality and parasitism on population cycles of larch budmoth. Ecology 84: 1207–1214.