vol. 157, no. 5 the american naturalist may 2001
Fundamental Unpredictability in Multispecies Competition
Jef Huisman
1,
*
and Franz J. Weissing
2,†
1. Aquatic Microbiology, Institute for Biodiversity and Ecosystem
Dynamics, University of Amsterdam, Nieuwe Achtergracht 127,
1018 WS Amsterdam, The Netherlands;
2. Department of Genetics, University of Groningen, P.O. Box 14,
9750 AA Haren, The Netherlands
Submitted May 30, 2000; Accepted December 26, 2000
abstract: One of the central goals of ecology is to predict the
distribution and abundance of organisms. Here, we show that, in
ecosystems of high biodiversity, the outcome of multispecies com-
petition can be fundamentally unpredictable. We consider a com-
petition model widely applied in phytoplankton ecology and plant
ecology in which multiple species compete for three resources. We
show that this competition model may have several alternative out-
comes, that the dynamics leading to these alternative outcomes may
exhibit transient chaos, and that the basins of attraction of these
alternative outcomes may have an intermingled fractal geometry. As
a consequence of this fractal geometry, it is impossible to predict the
winners of multispecies competition in advance.
Keywords: biodiversity, chaos, resource competition, fractal basin
boundaries, phytoplankton, prediction.
More individuals are born than can possibly survive. A grain
in the balance will determine which individual shall live and
which shall die, which variety or species shall increase in number,
and which shall decrease, or finally become extinct. (Darwin
1859, p. 467)
Darwin, as this famous quote testifies, was well aware that
the outcome of competition may depend on tiny differ-
ences. Here, we show that Darwin’s “grain in the balance”
can have a fractal structure. This fine-grained fractal struc-
ture makes it impossible to predict the winners of multi-
species competition.
We consider a resource competition model widely used
in plankton ecology and plant ecology. Theory (Leo´n and
* E-mail: jef.huisman@chem.uva.nl.
†
E-mail: weissing@biol.rug.nl.
Am. Nat. 2001. Vol. 157, pp. 488 494.
2001 by The University of Chicago.
0003-0147/2001/15705-0002$03.00. All rights reserved.
Tumpson 1975; Tilman 1982, 1988; Huisman and Weissing
1995; Grover 1997) and experiments (Tilman 1977, 1982;
Sommer 1986; van Donk and Kilham 1990; Rothhaupt
1996; Huisman et al. 1999) based on this model reveal that
competition for one or two resources leads to a stable and
predictable species composition. Motivated by the early
findings of Gilpin (1975), May and Leonard (1975), Smale
(1976), and Armstrong and McGehee (1980), we recently
discovered that competition for three or more resourcesmay
generate oscillations and chaotic fluctuations in species
abundances (Huisman and Weissing 1999, 2000, 2001).
Here, we show that the predictions of resource competition
models can become even more complicated. For this pur-
pose, we make a distinction between the “time course” of
competition and the “outcome” of competition. Chaos im-
plies that the time course of competition shows sensitive
dependence on initial conditions. That is, the long-term
dynamics of the species are unpredictable. However, in case
of a single chaotic attractor, it is still possible to predict
which of the species will persist and within which bounds
these species will fluctuate. In this article, we demonstrate
that multispecies competition can also become unpredict-
able in a more surprising sense: it may be impossible to
predict the outcome of competition. It may be impossible
to foretell which of the species will be excluded and which
will remain.
Competition Model
We consider n species competing for three abiotic re-
sources. Let N
i
denote the abundance of species i, and let
R
j
denote the availability of resource j. The dynamics of
the species depend on the availability of the resources. The
dynamics of the resources depend on the rates of resource
supply and the amounts of resources consumed by the
organisms. The model reads (Leo´n and Tumpson 1975;
Tilman 1977, 1982; Huisman and Weissing 1999)
dN
i
p N [m (R , R , R )
m ],
ii123 i
dt
i p 1,… , n (1a)

Fundamental Unpredictability 489
Figure 1: Chaos on three resources. A, Time course of competition.
1, 2, 3, 4,Black p species red p species bluep species green p species
and 5. B, The corresponding chaotic attractor. The at-yellow p species
tractor is plotted using three of the five species, for the period from
to d. For parameter values, see appendix.tp 1,000 tp 4,000
n
dR
j
p D(S
R )
c m (R , R , R )N ,


jj jii123i
dt
ip1
j p 1,… , 3. (1b)
Here, is the specific growth rate of species im (R , R , R )
i 123
as a function of the resource availabilities, m
i
is the specific
mortality rate of species i, D is the resource turnover rate,
S
j
is the supply of resource j, and c
ji
is the content of
resource j in species i. We assume that the specific growth
rates follow a Monod equation (Monod 1950) and are
determined by the resource that is most limiting, as in
Von Liebig’s (1840) “Law of the Minimum”:
rR rR rR
i 1 i 2 i 3
m (R , R , R )p min , , ,
i 123
()
K RKRK R
1i 12i 23i 3
(2)
where r
i
is the maximum specific growth rate of species i,
K
ji
is the half-saturation constant for resource j of species
i, and “ ” is the minimum function. This model for-min
mulation is widely used and particularly suited for primary
producers like phytoplankton (Leo´n and Tumpson 1975;
Tilman 1977, 1982; Sommer 1986; van Donk and Kilham
1990; Rothhaupt 1996; Grover 1997; Huisman andWeissing
1999). The model also provides a conceptual framework for
competitive interactions among terrestrial plants (Tilman
1982, 1988).
Chaos on Three Resources
Previous work based on this competition model revealed
periodic oscillations on three resources and chaos on five
resources (Huisman and Weissing 1999, 2001). Here, we
start by noting that competition for three resources is ac-
tually sufficient to generate chaos (fig. 1). We consider five
species. These five species form a complicated system that
can best be described by two competing cycles. Species 1–3
form one cycle. Here, species 1 is a strong competitor for
resource 3 but becomes limited by resource 1. Species 2 is
a strong competitor for resource 1 but becomes limited by
resource 2. Species 3 is a strong competitor for resource 2
but becomes limited by resource 3, and so on. This generates
cyclic dynamics (Huisman and Weissing 1999, 2001). The
second cycle has a similar structure but is now based on
species 1, 4, and 5. The two cycles are connected via species
1, and the system switches chaotically back and forth be-
tween the two cycles. This structure is clearly visible in figure
1B. Simulations reveal that the time course of competition
shows sensitive dependence on initial conditions, one of the
characteristic features of chaos. But the outcome of com-
petition is independent of the initial conditions. Whatever
the initial conditions, the system always ends up with the
same five species on the same chaotic attractor (fig. 1B).
Fractal Basin Boundaries
For slightly different parameter combinations, the chaotic
attractor of figure 1 breaks down and the two cycles become
disconnected. Thus, now there are two attractors, two limit
cycles to be precise. Simulations show that there is still a
period of transient chaos during which the dynamics switch
back and forth between the two limit cycles (fig. 2). The
duration of this transient period is highly variable; it may
last from !50 d to 11,500 d. In the end, the dynamics always