Appendix G. More on caveats on the use of the simple model. A pdf file of this appendix is also available.
In the strict two component lottery, the fluctuation measure is inversely proportional to the square of the population density, for two species with equal lifetimes. Since

and from (A.3) _{}



(G.1)

where _{} is relative abundance. This prediction and that of strict antiphase are affected by the considerations below.
Age effects – There are in these data no trees whose ages are known with precision: the conversion of size profiles to age profiles is inevitably imprecise. The lack of precise age data has implications for working out strict antiphase. It may have some importance for the relation between _{} and _{}. An incorrect scaling factor would have no effect on _{}. However, the relation between age and size must have dispersion and this dispersion increases with age. Such dispersion will reduce both _{} and _{} but is not likely to affect their ratio.
_{} is thus unlikely to be affected greatly by lack of accurate age information. The relation between _{} and _{} may however be affected, for if the mean lifetimes of two species are different, (DA7.1) becomes

(G.2)

Finally, the measure of fluctuations has for each species been calculated about an exponential with mean lifetime equal to that of the sample. Deviations from any other exponential would be greater.
Statistical effects – In a strict two component lottery statistical fluctuations in recruitment would be binomial and because either one species or the other is recruited to each newly vacant site statistical fluctuations which occur when both species have recruitment ON would preserve strict antiphase. This is no longer necessarily true when size profiles are drawn from samples with different numbers of (potential) homes. Fluctuations would then not necessarily exhibit strict bin by bin antiphase, but one would expect the overall pattern to be preserved.
Even in a strict two component lottery, statistical fluctuations can affect the value of the measure _{}. For a rare species, such fluctuations have a Poisson variance; for two species of relatively similar abundance the binomial fluctuations are correlated. For the species in Fig. 2 of the main text, the square root of the variance V(_{}) estimated in Appendix E, Eq. E.7 is ~ 0.5 (larger in the case of J. chamelensis), where the values of _{} span the range 0.5 – 5.0.
Temporal variation of the number of homes – If the number of homes available varies with time then (i) an inphase variation in recruitment is superimposed on the antiphase behaviour and (ii) both species suffer a component of fluctuation in recruitment (by number) which is the same fraction of mean recruitment for each.
Equation (A.4) would be replaced by

(G.3)

where A(t) is the modulation imposed on the two component lottery through the variation in the number of homes available as a function of time.
Write recruitments _{} _{} where (for equal lifetimes) the sum _{} is a constant, so that



(G.4a)


(G.4b)

The recruitment deviations are a mixture of an in phase component (_{}) and a strictly out of phase component (Fig. G1).
If _{} and _{} are not correlated, then

(G.5a)


(G.5b)

The ratio _{} lies between 1 (when _{} dominates) and _{} (DA7.1) (when _{} dominates). Since a violently fluctuating species will scarcely have _{}, when _{}, _{} will be small and easily dominated by _{}. It is no surprise that relative abundances of 10:1 do not yield _{} ratios of 100, but the species with lower mean recruitment still has larger fractional deviation. Finally, there remains the possibility of a correlation between _{} and _{}. As we have defined it, _{} is the increase in recruitment of _{} over _{}. In our specific model (section 3), environmental effects reduce _{} quasiperiodically to ~ 0. It might well be that the same environmental effects depress the number of homes available, which would introduce a positive correlation between _{} and _{}. Then _{} would be increased over the ratio of (G.4a) to (G.5b).