The basic no-interaction model was based on simple geometric reasoning without explicit reference to variation in plant size or spatial constraints. Here I study the validity of its prediction using a simple stand simulation model. Since we are using constant diversity in a model (= the same number of plants is expected to give the same number of species), a scale-independent measure should be independent of quadrat size. If the same measure shows a pattern with real data, we have some support to claim that it is not an artefact. That is the reason why we use a perverse looking way of trying to find a good indicator by trying to find something that does not have relation to biomass: Only deviations from the null-model are worth explaining.
The following questions are studied:
The stands were simulated following the reasoning in the no-interaction model. It was assumed that plants have a minimum expected size (real sizes vary) of 0.1%, vegetation becomes closed at 160%, and after that biomass can be higher only if plants have a larger expected size. So below 160% cover, there should be linearly increasing species richness, and above 160% there should be decreasing species richness with inverse squared biomass. The stands below 160% had expected total covers of 40%, 80%, 120%. The expected average sizes of species starting from 160% (where it was still 0.1%) were 0.12%, 0.17%, 0.25%, 0.5%, 1%, 2%. This gives us 10 different stands. Two replicates were used for each stand size. Observed biomass differred clearly between replicates: It was used to plot the results.
Some examples of simulated stands, directly from the output screen of the simulation program:
The results reported here are for "high diversity" stands where the most abundant species is expected to comprise 10% of all plants. This parameter was constant in all models. For the same sample size (number of plants) we expect the same number of species. Scale independent measure of diversity should thus be independent of biomass. Any systematic departure from horizontal line implies a scale-dependent artefact.
Results are displayed for the smallest (3.1%) and largest (50%) sample plot.
Even with variable plant size in multi-species stand, species richness displayed the hump artefact [Fig: jpeg, 24K], but the hump was less pronounced with larger plots. The number of plants per area is independent of quadrat size, and so we can expect the hump at all quadrat sizes. However, in large quadrats with large number of individuals we are at the flattening phase of the species number - sample size curve [Fig: jpeg, 9K], and so the hump is not as pronounced.
Diversity indices are based on species abundance relations, and so they are expected to be less dependent on sample size than species richness. Recent studies on real stands have found that this is indeed the case (Condit et al. (1996) Journal of Ecology, 84, 549-562). Even with small quadrat size the hump is less pronounced [Fig: jpeg, 21K] than in species richness. With large quadrat size, the diversity is independent of biomass. In low diversity communities, even the small quadrat diversities were independent of biomass. So it seems that diversity is a more robust measure of richness than species number.
This indeed seems to be a(n almost) linear relation. The linear relation is proposed by species-area models in island biogeography as well. Rapson et al. (1997), Journal of Ecology, 85, 99-100 have proposed that this slope could be used as a scale-independent estimate of diversity. This is indeed confirmed by stand simulation [Fig: jpeg, 12K].
[ front page] [ assumptions and model] [ misconceptions] [ what to do?] [ fitting] [ case study ]