Appendix A. The model: biological assumptions and mathematical development. A pdf file of this appendix is also available.
In a model in which two species compete directly and the recruitment parameters _{} satisfy the condition

(A.1)

the species with the smaller mean recruitment will have an age profile which fluctuates more about a smooth curve than that of the commoner. This is because the recruitment fluctuations are equal in magnitude and opposite in sign

and so if species 1 has the smaller mean recruitment

(A.2)

This is illustrated in Fig. A1.
If the two species system has long term stability,

(A.3)

(where the bar denotes longterm averages) and if lifetimes are not dissimilar, the species with smaller mean recruitment is the rarer.
Condition (A.1) is realized in the two component lottery model (Chesson and Warner 1981), exactly for the case of equal d parameters. From this, we devised a model to explain the results of Kelly et al. (2001), as follows.
Suppose that when a site that can support an adult becomes available through the death of an adult there is some probability that it is most suitable for species in class A (which might be a genus) and that such sites are at once colonised by species _{} which are members of class A. We have in mind spatiotemporal partitioning into such site classes, as in Pacala and Tilman (1994). Species _{} and _{} compete for such class A sites. In the absence of differential recruitment caused by environmental fluctuations, the system would relax to a configuration in which one species is wholly dominant. However, if _{} goes to zero on a quasiperiodic basis (as a result of environmental fluctuations) then _{} and _{} can coexist, provided _{} has sufficient competitive advantage to win over _{} during those periods in which young _{} are safe. The necessary assumptions are not extreme – merely that class A sites are colonised by class A faster than by species in another class and that _{} can colonise vacant A sites in a time short in comparison with the typical length of an _{} recruitment crisis.
In the two component lottery model a location suitable for a new adult becomes available at a rate equal to the rate at which adults die and the probability of species _{} taking it is

where _{} is the product of a reproduction rate and a competitive factor (Chesson and Warner 1981). The quantity _{} is the fraction of sites occupied by _{} and the site is at once taken by either _{} or _{}. Then

(A.4)

with analogous equations for _{} (a difference equation is given below). Addition of the equations for _{} and _{} demonstrates at once that _{} + _{} = 1 at all times in a mature forest.
The second term on the right hand side of Eq. A.4 is the recruitment _{}.
The sum _{} + _{} is given by

For fluctuations on a time scale much less than _{}, the fraction of sites occupied does not deviate much from _{} (although the recruitment fluctuations can be large) so

(A.5)

and this becomes exact for _{}. For the moment we treat the case _{}, which is not at odds with the data of Kelly et al. ( 2001) and Kelly and Bowler (2002). Then (A.4) becomes

(A.6)

Suppose that there are periods in which juveniles of _{} are taken out (by freezing or baking or being eaten). The variable _{} is zero (OFF) in such periods, _{} and _{} leaps to _{}. At other times _{} is ON and suppose it has a value _{} at such times. Suppose further that _{} _{} constant with value _{}; that is, _{} is taken as being relatively invulnerable to the climatic variations which directly or indirectly take out the young of _{}. The long term average of _{} is zero if the system is long term stable and the values of _{} in the denominator of (7) can be replaced by _{} for fluctuations on timescales much less than _{}. Then

and _{} for long term stability.
Then

and

(A.7)

where _{} is the fraction of time during which _{} ON. _{} and _{} are both restricted to the range 0 1. Equation A.7 is the same as Eq. 2 in the main body of the text. This equation only has an admissible solution provided that

and as _{} is decreased so _{}, _{}. If the competitive factor _{} does not sufficiently exceed _{} then species _{} drives species _{} to extinction. Conversely _{} can never exceed _{}. For admissible solutions to Eq. A.7 the two species coexist.
DISCRETE FORMULATIONS
Equation 1 may be presented in discrete form as

(where in t is an integer).
Equation A.4 may also be presented in discrete form, as

The equation was originally presented in this form, suitable for numerical simulations, by Chesson and Warner (1981).
The discrete equation equivalent to (A.4) is

with age profile

where t and g are integers. For long term storage the difference and differential formulations are equivalent.
Chesson, P. L., and R. R. Warner. 1981. Environmental variability promotes coexistence in lottery competitive systems. American Naturalist 117:923–943.
Kelly, C. K., H. Banyard Smith, Y. M. Buckley, R. Carter, M. Franco, W. Johnson, T. Jones, B. May, R. Perez Ishiwara, A. PerezJimenez, A. Solis Magallanes, H. Steers, and C. Waterman. 2001. Investigations in commonness and rarity: a comparative analysis of cooccuring, congeneric Mexican trees. Ecology Letters 4:618–627.
Kelly, C. K., and M. G. Bowler. 2002. Coexistence and relative abundance in forest tree species. Nature 417:437–440.
Pacala, S. W., and D. Tilman. 1994. Limiting similarity in mechanistic and spatial models of plant competition in heterogeneous environments. American Naturalist 143:222–257.