satisfy
the condition 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
 =
constant = 
|
(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 long-term 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. Perez-Jimenez, A. Solis Magallanes, H. Steers, and C. Waterman. 2001. Investigations in commonness and rarity: a comparative analysis of co-occuring, 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.
is
three times 
and that 
