I have collected here some comments I have seen or heard about the no-interaction model, together with my response. Some of these questions and some answers can be found in Journal of Ecology, 85, 95-102 (1997) which has some articles discussing the no-interaction model.
Yes, this is actually true. But the circularity is in using data on species-richness and biomass: These are circularly dependent. That is the reason, why I called the hump so produced an artefact. If you are using quadrats of constant size for plants of variable sizes, the result will be that you will have variable number of individuals in your quadrats, and this alone is sufficient to produce the hump.
The model does not criticize the humped-back shape. Actually, no-interaction model predicts just that shape. It only says that this shape is a geometric circularity, and tells us very little about ecological interactions.
No. The model has nothing to do with self-thinning process. It does not describe growing plants, but plants that are of different size.
Perhaps the figure 1 [jpeg, 89K] of my article was misleading: Bill Gates did not provide a wider range of plant figures, and so I used only one and same.
The only connection to self-thinning theory is that the so-called -3/2 power law is derived using simple geometric reasoning which relates plant (cube) volume to the surface area occupied by plants in closed stand. I used the same geometric analogue in deriving the inverse quadratic decrease of plant number per fixed area in closed stands. No functional relation to the self-thinning law was assumed.
The model deals with plants which are of different size in different quadrats. It makes no reference to development within a single stand.
It may be that the figure [jpeg, 89K] is misleading again. The model does not change if every plant in the figure is of different species.
Species number increases statistically with increasing number of plants (sampling units). Naturally, species number cannot decrease when you increase number of sampling units, and there is always a possibility to get new species by continuing sampling. Therefore I find it surprising that some people seem to find this point (my assumption number three) hard to swallow. Moreover, any textbook of ecology will tell you the same, if not in chapter dealing with diversity, then when dealing with species-area relations in island biogeography or within stand.
I based the model fitting in no-interaction model to Fisher's model [Fig: jpeg, 9K] of dependence between species number (S) and number of sampling units (N). This is a traditional choice, but fairly similar shapes are used in species-area curves.
No-interaction model says that you can get humped-back response as an artefact when you assume that plant shape is constant and diversity is constant. It says that if you observe the humped-back response, you don't need to assume anything more: You cannot conclude anything on diversity or ecological interactions based on observations of humped-back response.
The model does not say there are no interactions, or that diversity is constant: It only says that you cannot conclude anything about them based on the humped-back curve. Conclusion is that instead of species richness and biomass, you should study directly interactions, or diversity, or plant shape.
Life is not walking across the field, and science is not walking across the Cairngorms: You have to study systematically ecologically meaningful parameters. There is no need to try to show that the no-interaction model does not work: The model is only a mirror which shows you problems in parameters used in data. It seems that the predictions of the model do not change with relaxing some of the assumptions, as the simulations indicate.
[ front page] [ assumptions and model] [ simulations] [ what to do?] [ fitting] [ case study ]