86 SPECIES NUMBERS GAMES: THE MACROECOLOGY OF ISLAND BIOTAS
The equation for equilibrium is:where Sis the number of species at time t,St 1 is the
number at time t 1,Idenotes additions through
immigration,Vdenotes additions through evolu-
tion (where applicable), and Edenotes losses by
extinction (it will be noted that as species are inte-
gers, continuous representations are approxima-
tions). Integration of the rates of I andE as
they vary over time produces the colonization
curve (Box 4.2). This rises steeply initially, but ever
more slowly as eventual equilibrium is neared
(Fig. 4.4).
MacArthur and Wilson (1967) recognized that
biologists can rarely, if ever, be certain of recording
all immigration and extinction events in real-world
systems. But they provided a little-used way around
this, reasoning that the colonization curve might be
used to back-calculate indirect estimates of what
the real IandErates might be at two stages of the
colonization process. The first point is near the
beginning of colonization, where Imay be assumed
to be close to the rate of colonization and Eto be
insignificant (although both assumptions appear
rather bold when successional factors are
considered: Rey 1985; Chapter 5). The second point
is near the equilibrium condition. In this case they
developed a proof for derivation of the extinction
rate at equilibrium as a function of the species num-
bers on other similar islands already at equilibrium
and the time taken to reach 90% of the equilibrial
number of species on the target island. They
St 1 StIVEdetailed a number of studies in apparent support of
this line of reasoning. The first of which was for
bird recolonization of the Krakatau islands, which
appeared to have reached a very dynamic equilib-
rium by the 1920s (MacArthur and Wilson 1963).
This example is still cited in support of the theory
(e.g. Rosenzweig 1995), but as MacArthur and
Wilson (1963, p. 384) speculated might be the case,
this was in fact a premature conclusion; indeed data
had already been published which showed a further
increase in species number (Hoogerwerf 1953).
The above constitutes the simplest manifestation
of their theorizing, the basic dynamic equilibrium
model (the EMIB), which occupies the second and
third chapters of MacArthur and Wilson’s (1967)
monograph. It provides an essentially stochastic
macroecological model of biological processes on
islands, in which the properties of individual
species get little attention. Its great virtues were
first, that it was a dynamic model invoking univer-
sal ecological and population processes, and sec-
ond, its apparent testability. The predictive nature
of the equilibrium theory is important. Although it
may have heuristic value without it, the contribu-
tion of the theory to biogeography was in large part
to point the way towards a more rigorous scientific
approach to the geography of nature. As Brown
and Gibson (1983, p. 449) put it:... like any good theory, the model goes beyond what is
already known to make additional predictions that can
only be tested with new observations and experiments.
Specifically, it predicts the following order of turnover
rates at equilibrium: TSN TSFTLN TLF.
Number of species presentRateTime TimeColonization
curveIE Figure 4.4Integration of immigration and
extinction curves (left) should, theoretically, produce
the colonization curve as shown (right). (Redrawn
from MacArthur and Wilson 1967, Fig. 20.)