132 Egbert G. Leigh, Jr
of glacial cycles) and vice versa (when wet for-
est was expanding at the beginning of inter-
glacials) was frequent in the Neotropics during
the Pleistocene (Penningtonet al. 2004). Further
research on how tree species originate is urgently
needed.
Dangers of theory: analysis of a trade-off
What trade-offs enable pioneer tree species to
coexist with superior competitors? Answering this
question shows how simple theory can clar-
ify thinking, and how it can mislead. Two tree
species can coexist if each species can invade a
forest consisting only of the other. To invade a
forest of superior competitors, a pioneer species
that colonizes treefall gaps needs a steady supply
of colonizable gaps. Pioneers will spread from gap
to gap only if a pioneer in a new gap grows fast
enough to set seed before superior competitors
overtop it and crowd it out, and if, on the aver-
age, more than one of its seeds repeats this feat in
other gaps. Similarly, the superior competitor can
invade a forest of pioneers because its seeds can
germinate in the pioneers’ shade and its saplings
grow up to overtop them. It is a question for theo-
rists how big and how frequent treefall gaps must
be, how fast pioneers must grow, and how fast and
how densely they must scatter their seeds, to allow
a pioneer species to spread from gap to gap.
Using mathematical techniques of Horn and
MacArthur (1972), Tilman (1994) developed a
schematic model of how pioneers coexist with
superior competitors. If pioneers are absent, let a
proportionC(t)of th efor est’s spac eb eoccupi ed by
superior competitors in yeart. Following Tilman,
let mortality empty a proportionmof this occu-
pied space each year, and let superior competitors
tak eov er a proportionrC(t)of th e empty spac e.
Then,
C(t+ 1 )=C(t)+rC(t)[ 1 −C(t)]−mC(t)
At equilibrium, whenC(t+ 1 )=C(t)=C,1−C=
m/r. Now consider a pioneer species that does
not slow the superior competitor’s recruitment.
When can it invade? LetP(t)b eth eproportion of
the forest’s space occupied in yeartby the pioneer.
Each year, let the pioneer take over a proportion
krP(t)( 1 −rC)of the empty space, wherek>1,
and let superior competitors replace a proportion
rCof the pioneers. Then, if all pioneers die from
replacement by superior competitors,P(t+ 1 )=P(t)+krP(t)( 1 −rC)
×[ 1 −C−P(t)]−rCP(t)The pioneer invades if P(t+ 1 )>P(t) when
P(t)≈0. This happens whenkr( 1 −rC)( 1 −C)>rC,k>C/[( 1 −C)( 1 −rC)]
=(r−m)/[m( 1 −r+m)]In this model, pioneers can invade if they colonize
empty space over(r−m)/[m( 1 −r+m)]times
more rapidly than superior competitors.
This theory misled ecologists in two ways. First,
extrapolating this theory led Tilman (1994) to
conclude that the trade-off between highkand
competitive superiority would allow an indefinite
number of tree species to coexist, each species
jbeing competitively superior but having lower
k than all speciesi<j. This conclusion only
holds, however, if a speciesjreplaces any com-
petitively inferior speciesi<jas rapidly as if the
competitive inferior’s space were empty. If the
competitive advantage of a speciesjover a species
i<jdecreases withj−i, the trade-off between
kand competitive ability allows only a few species
to coexist (Adler and Mosquera 2000). In fact,
gap-creating disturbance cannot explain diversity
gradients in tropical forests (Conditet al. 2006).
Pasoh Reserve, Malaysia, is much more diverse
than Barro Colorado, with 815 species among
th e335,000 st ems≥1 cm dbh on 50 ha, com-
pared with Barro Colorado’s 305 species among
235,000 such stems on 50 ha (Conditetal. 1999).
At Pasoh, however, gaps are much smaller, less
frequent, and less varied in size (Putz and Appanah
1987). Accordingly, th ePasoh plot has only thr e e
pioneer species with sapling diameter increase
averaging more than 4 mm year−^1 among its
422 species of canopy tree, compared with 16
of 141 on Barro Colorado’s plot (Conditet al.
1999).
Tilman’s (1994) theory also led him to con-
clude that coexistence between pioneers and