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Coexistence and chaos in complex ecologies
J. C. Sprott, J. A. Vano, J. C. Wildenberg, M. B. Anderson & J. K. Noel
Departments of Physics and Mathematics, University of Wisconsin, Madison, Wisconsin
53706, USA
An ongoing debate1,2 among ecologists involves the observation that complex and
diverse ecologies are relatively stable3-5, whereas most models predict instability
and extinction of most species6-8. Furthermore, erratic fluctuations are common
and are usually attributed to random external influences, but they may be evidence
of chaos. While laboratory experiments with flour beetles do suggest chaos9,10,
ecologists have yet to document an unequivocal example of chaos in nature11,
perhaps because of dynamical complexity and measurement limitations. One
difficulty with most simple models is that chaos tends to occur over a relatively
narrow range of parameters, bounded on one side by stable behaviour and on the
other by extinction. Here we show that a simple model with realistic and purely
deterministic adaptation produces systems in which most species coexist with
weakly chaotic fluctuations independent of the initial conditions. The proposed
mechanism offers an explanation for the observed biodiversity and at least some of
the fluctuations and unpredictability, and it suggests why it may be difficult to
stabilize such systems by human intervention.
Many different mathematical models12 have been used to study the dynamics of
interacting species or agents in a variety of different contexts and systems. We choose
here a variant of the generalized Lotka-Volterra equations13,14 because our aim is to
show that the observed results are very general and because the chosen model can be
viewed as the first approximation in a Taylor series expansion for a wide class of
models15.
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We consider a system consisting of N competing species with population xi for i =
1 to N satisfying
N
dxi
 ri xi (1   aij x j )
dt
j 1
(1)
where all the biology (economics, sociology, etc.) is contained in the vector of growth
rates ri and the matrix of interactions aij, all elements of which are positive to indicate
competition. The element aij describes the average extent to which (members of) species
i competes with (members of) species j. Note that as any given species approaches
extinction, the averaging of the aij elements for this species will occur over smaller and
smaller populations and hence more variability becomes possible and the model breaks
down.
In general, one expects the linear growth rates ri to be different for each species,
but Coste et al.16 have shown that any N-dimensional Lotka-Volterra system with
unequal growth rates can be extended to an equivalent (N + 1)-dimensional system with
equal growth rates. Since we are concerned with high-dimensional systems, we take ri =
1 for 1 ≤ i ≤ N in the interest of simplicity. Without further loss of generality, we can
take the self-interaction terms aii equal to unity, which is equivalent to measuring xi in
units of its carrying capacity in the absence of the other species. Taking all the aij
positive amounts to ignoring mutualism and the effect of varying individual prey
populations on the predators, but it guarantees that the solutions remain bounded in the
range 0 to 1, and the results are not substantially altered if some of the aij are negative.
For competitive systems, chaos is not possible with fewer than four species. For
larger ecologies, to get a sense of the rarity of parameter values that lead to chaotic
solutions, note that choosing aij from a random exponential distribution (so as to have a
broad spectrum of positive values) with mean 1.0, with N = 4 leads to chaotic solutions
with all species coexisting in only about 1 in 105 cases for a sample of 106 cases, and for
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N = 5 only 1 in 4105 cases. Coexisting chaotic systems for realistically large N ( 100)
are vanishingly rare and almost impossible to find in such a random search, although
work of Smale17 guarantees their existence. The conditions for coexistence (an
equilibrium with all xi positive) and for chaos (the equilibrium being locally unstable)
are somewhat mutually exclusive and occur in very small regions of this vast space of
parameters.
Presumably nature does not choose randomly from all possible ecologies, but
instead individual species are adapting to their environment so as to enhance their
survival. While many popular models assume extinction when a species drops below a
critical level18,19 or modify the basic equations to prevent such extinction20, we believe
there is considerable justification to instead consider models in which adaptation occurs
at these points. The individuals in a nearly extinct species are presumably the most fit
and are those best able to survive by finding alternate resources and by evading their
predators. Thus, as a species approaches extinction, the increased variability of the aij
coefficients in Eq. (1) along with the effects of directional selection naturally lead to a
shift in the aij coefficients. Also, when the population of a species becomes too small its
predators may find it too inefficient to prey upon. Finally, the model could be
interpreted as species becoming extinct and being replaced with new similar (perhaps
mutated) species that are less susceptible to the prevailing competition21,22. We model
this process as follows:
Whenever a species drops below some preset value (say xi < 10–6), the aij
interactions corresponding to competition with that species are permanently reduced in
proportion to xj until the decline is arrested. In addition to this species-specific
adaptation, all the off-diagonal elements in the matrix are slowly and proportionately
increased to model general adaptation of the entire ecosystem over time toward
enhanced competition or to model a slowly increasing environmental stress such as
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climate change or human encroachment23. In this way, the system is guaranteed to have
all N species coexisting with near optimal fitness. The details of the procedure are not
critical but are described in the Methods section. Note that this model is purely
deterministic (no stochastic component) even while it is adapting, and thus any
persistent aperiodic fluctuations are evidence of deterministic chaos. In addition to its
ecological plausibility, the method provides a powerful numerical algorithm for finding
the rare chaotic solutions for large N.
Such a system slowly evolves into one that is weakly chaotic with a typical largest
Lyapunov exponent of 0.001 ± 0.001, whereupon it remains chaotic with all or most of
the species coexisting even if the adaptation mechanism described above is turned off.
The Lyapunov exponent is a measure of the sensitivity to initial conditions, with a
positive value signifying chaos. Abrupt or premature termination of the adaptation
mechanism, however, may result in the extinction a few species.
Figure 1 shows the evolution of the largest Lyapunov exponent along with the
total biomass and biodiversity (defined in the Methods section) for a typical highly
competitive case that starts with most of the 100 species on the verge of extinction and
strongly chaotic dynamics. As it evolves, the largest Lyapunov exponent generally
decreases but remains mostly positive, while the biomass and biodiversity increase.
Fluctuations in the biodiversity are suggestive of sporadic volatility and punctuated
equilibria24. A similar final state is reached if the initial aij values are small, giving a
high initial degree of coexistence and stability with an increasing Lyapunov exponent.
Figure 2 shows the chaotic fluctuations in the total biomass and biodiversity for a
typical such system with N = 100 after adaptation has been turned off. Note the very
different time scales for the adaptation in Fig. 1 and the fluctuations in Fig. 2. Figure 2
can be viewed as a slice of the dynamics in Fig. 1 over a short time scale where
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adaptation is negligible. Fluctuations in the population of individual species are much
larger than in the total biomass.
Figure 3 shows a portion of the strange attractor for the same system projected
onto the space of total biomass and biodiversity. The dimension of the attractor is about
6.3 with two positive Lyapunov exponents, making it hyperchaotic, although only
weakly so with a largest Lyapunov exponent of about 0.0021. This value implies that
the system has memory and predictability on the time scale of 500 growth times for a
typical species.
There does not appear to be a unique distribution of aij toward which the model
evolves, suggesting that a wide variety of weakly chaotic ecologies is possible, although
the mean value of the matrix elements for the case in Fig. 2 and 3 is about 0.38, which is
typical for cases with N = 100. Since these systems are fully connected with relatively
large connection strengths, they violate the May-Wigner stability condition25, which
states that a network whose stability matrix contains elements from a normal
distribution with mean zero and variance σ2 is almost certainly stable if NCσ2 < 1, where
the connectivity C is the probability that a matrix element is nonzero. We have
generated model ecologies with up to 400 surviving species by this method.
Presumably the evolution method would work for almost any network model
characterized by a matrix of interactions between its agents, and it is not restricted to
models of ecology. Other networks that involve competition for resources and that are
subject to crashes include financial markets, the electrical power grid, the Internet,
traffic flow, and the brain. It is likely that all these systems are pushed toward a weakly
chaotic dynamic by a mechanism similar to the one described here, and that such a state
is relatively robust to modest disturbances, either intentional or unintentional. This state
has been called “the edge of chaos”26, and it provides the condition under which
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complex systems exhibit adaptation, self-organization, scale invariance, self-organized
criticality, and similar features that characterize dissipative systems that are driven far
from equilibrium by the throughput of energy or some equivalent quantity. Thus models
of this type are ripe for further study and detailed comparison with the corresponding
natural systems.
Methods
Equation (1) is solved by a fourth-order Runge-Kutta method with a fixed step size of
0.05. Initial conditions are not critical, and they can be either random in the range 0 to 1
or purely deterministic. Initially the elements aij are taken from an exponential
distribution with mean 1, although any distribution of positive definite values will
suffice. The system converges fastest to the weakly chaotic state if the mean aij is close
to the desired final value, which is typically on the order of 10 / N . A time on the order
of t = 106 is typically required to reach a state that is chaotic in the absence of
adaptation.
At each iteration for which any xi falls below 10–6, xi is clamped at 10–6 and its
matrix elements aij are replaced with aij(1 – ε1xj) for j = 1 to N and j  i. Furthermore, all
the off-diagonal elements of the matrix are increased by the factor (1 + ε2) every 20
iterations. The values are not critical, but they should be small and are here taken as ε1 =
10–4 and ε2 = 10–6. There may be some advantage to starting with relatively large ε
values and slowly reducing them in the spirit of simulated annealing.
The largest Lyapunov exponent is calculated by the method27 in which an
arbitrary perturbation is advanced forward in time with a renormalization of the
perturbation size, but not direction, at each time step. The exponent is calculated from
the logarithm of the growth rate averaged along the orbit. The spectrum of N Lyapunov
exponents is calculated independently by integrating the Jacobian matrix of partial
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derivatives along the orbit with a modified Gram-Schmidt orthonormalization at each
time step28. The dimension of the strange attractor is determined from the spectrum of
Lyapunov exponents using the Kaplan-Yorke conjecture29.
The total biomass is defined as
M 
1
N
N
x
i 1
i
(2)
and the biodiversity is defined as
N
xi
1
D  1
1

2( N  1) i 1 M
(3)
so that each is a dimensionless number30 in the range of 0 to 1. The biomass is unity if
all species are at their carrying capacity in the absence of intra-species competition. The
biodiversity is zero if all but one species are extinct and unity if all species are equally
abundant.
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Supplementary Information accompanies the paper on Nature’s website (http://www.nature.com).
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Acknowledgments We thank D. J. Albers, R. S. Chapman, M. W. Hirsch, and J. W. Robbin for useful
discussion.
Competing interests statement The authors declare that they have no competing financial interests.
Correspondence and requests for materials should be addressed to J.C.S. (e-mail:
sprott@physics.wisc.edu).
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Figure 1 Evolution of the largest Lyapunov exponent (0.02 full scale), total
biomass (0.05 full scale), and biodiversity (1 full scale) showing how a typical
system with 100 species slowly evolves toward a weakly chaotic state with high
diversity and then remains there with fluctuations in the biodiversity suggestive
of sporadic volatility and punctuated equilibria.
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Figure 2 Typical chaotic fluctuations in the total biomass and biodiversity for a
system with 100 species after adaptation has been turned off.
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Figure 3 A portion of the strange attractor for the same typical system as in Fig.
2 with 100 species after adaptation has been turned off.