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Stabilization of Large Generalized Lotka-Volterra Foodwebs By Evolutionary Feedback
G. J. Ackland* and I. D. Gallagher
School of Physics, The University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
(Received 25 June 2004; published 8 October 2004)
Conventional ecological models show that complexity destabilizes foodwebs, suggesting that
foodwebs should have neither large numbers of species nor a large number of interactions. However,
in nature the opposite appears to be the case. Here we show that if the interactions between species are
allowed to evolve within a generalized Lotka-Volterra model such stabilizing feedbacks and weak
interactions emerge automatically. Moreover, we show that trophic levels also emerge spontaneously
from the evolutionary approach, and the efficiency of the unperturbed ecosystem increases with time.
The key to stability in large foodwebs appears to arise not from complexity per se but from evolution at
the level of the ecosystem which favors stabilizing (negative) feedbacks.
DOI: 10.1103/PhysRevLett.93.158701
Ecosystems are a classic example of complexity [1–7],
being formed from a myriad of interactions between
various species. The mathematical study of ecosystems
has a long history, dating back to the work of Lotka and of
Volterra [8,9]. Such models tread a delicate balance between including so much detail that they lose the capability to make qualitative predictions, and being so
simple as to be wholly wrong. Striking features of ecosystems are their tendency to be arranged into a hierarchical structure with different trophic levels, their
development of many complex interactions, and their
chaotic population dynamics. In setting up a model, it
is necessary to decide which of these observed qualities
will be built into the model, and which (one hopes) will
emerge from solving the model. For example, May’s early
work based on random matrices did not assume trophic
structure, but did assume the number of interactions and
their distribution of strengths. These simple ecosystem
models exhibit chaotic population dynamics and stability
[2,3]. Further work by Pimm and co-workers [1,7,10,11]
showed that webs with an imposed hierarchical trophic
structure (i.e., the absence of ‘‘trophic cycles,’’ formally
defined by loops in the directed graph) were more stable
than random webs. However, large or highly connected
model foodwebs tended towards instability in both
models.
Subsequently, investigation has been done on webs
created by continuous introduction and extinction of species of preset trophic level and interactions, with success
depending on population dynamics [12,13]. These foodwebs evolve [14] to contain large numbers of species and
are persistent even though species are continually being
introduced and going extinct. However, in all of these
models increasing connectivity (the number of interactions per species) leads to instability. McCann et al. addressed this latter issue by investigating simple webs with
weak interactions added based on studies of interaction
strengths in real foodwebs [6]. This work showed that
weak interactions act to dampen oscillations and stabilize
highly connected systems.
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PACS numbers: 87.23.Kg, 05.65.+b, 89.75.Hc
It has been suggested [15] that nontrophic effects such
as cooperation or competition may be required to stabilize large systems. These effects are undoubtedly important in real ecosystems, but here we show that large,
stable, highly connected trophic foodwebs with chaotic
dynamics and many weak interactions arise from a generalized Lotka-Volterra (GLV) model with evolution of
the interaction strengths [16,17]. Such evolution may
arise from various biological phenomena, including genetic and behavioral evolution, predator choice, and
changes in the spatial overlap of populations [14]. The
interaction strength represents the balance of power between the species. Species tend to evolve [14] more effective means of dealing with the other, but the
interaction strengths, representing differences in effectiveness may increase or decrease. Interactions cannot, of
course, change sign as the role of predator and prey
cannot be reversed. We assume that the overall effect is
that large populations of predators become less well
adapted to capturing rare prey, since other food sources
are available, while small populations of prey become
better able to avoid their major predators. Thus we simulate coupled population and evolutionary dynamics, starting with a pool of species and eliminating any whose
population drops below a minimum threshhold
(extinction).
Lotka-Volterra-type models [1] consider populations of
species as their basic objects, modeling the interactions
between the various populations. The Lotka-Volterra
model [8,9] considers two species with populations x1
(prey/autotroph) and x2 (predator/heterotroph). x2 has a
constant death rate c and per capita reproduction proportional to the amount of prey, effectiveness of predation
(M), and birth efficiency (b). x1 has a per capita death rate
due to predation by x2 and a regeneration rate constrained
by environmental resources as described by the logistic
map. This gives the following equations for two species:
dx1
gx1 gx21 =K Mx2 x1 ;
dt
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(1)
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dx2
Mbx1 cx2 :
dt
(2)
This model can be readily generalized to N species: for
autotrophs, with x0 setting the limit on the population and
Mij always negative:
X
dxi
xi x2i =x0 Mij xi xj
(3)
dt
j
for heterotrophs, which may predate on some species and
be predated on by others so that Mij may be positive or
negative:
X
dxi
cxi Mij xi xj :
(4)
dt
j
We take death rate c 0:01 and draw the initial Mij
randomly from a flat distribution between 0 and 1.
Predation is not efficient, and following Lotka-Volterra
we assume that for positive Mij , Mji bMij with a
birth efficiency b, typically 0.1. This foodweb forms a
directed graph where the species form nodes connected
by interactions of strength Mij .
We define the resource flow (Fig. 1) into the network as
the sum of positive terms in Eqs. (3) and (4) and flow out
as sum of negative terms. Ensemble averaged values of
these quantities must balance in the steady state, but their
absolute value is not determined.
We also allow for evolution of the link strengths Mij
themselves. The Mij represent the probability of predation
between individuals of two species and its change therefore incorporates all genetic or behavioral changes which
affect this probability. The driving force for change in Mij
FIG. 1. Time series of the populations of typical species in a
typical evolving foodweb. Autotrophs are shown by thick lines,
heterotrophs by thin lines. Units of population (N) are arbitrary; time is related to the death rate of 0.01 for heterotrophs
(i.e., 100 is a mean life-span). Both population dynamics and
the dynamics of interaction strengths (not shown) are chaotic.
The dynamics can be explored using an applet [18].
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is proportional to the number and strength of interactions
per individual of the species. Since Mij affects two species, its actual change is the difference in evolution of the
two species: if both predator and prey are becoming
better adapted at the same rate, no change in Mij occurs.
dMij
Ni Nj Mij ;
dt
(5)
where sets the rate of change. We set an upper limit of 1
on the efficiency Mij —without this it is possible for
species to evolve so as to exist on vanishingly small
amounts of prey. In our calculations we iterate Eqs. (3) –
(5) in time, eliminating any species for which Ni < 0.
Preliminary calculations with constant interaction
strengths showed that this strategy produces large, feasible, viable foodwebs, but that the complexity of the
interaction network remains low, on average just one
link per species, independent of web size. This is consistent with previous numerical work on evolved webs
[12,13] and the exact result for random webs [1]. Hence,
as with all previous models, selection by extinction of
failing species does not reproduce the observed
complexity.
In our main calculations where the interactions evolve,
we introduce one new parameter and eliminate two
assumptions— neither the number of interactions per
species nor the distribution of interaction strengths need
be determined a priori—the system is able to evolve its
interactions or make them so weak as to effectively remove them altogether. The model also does not preassign
trophic levels to heterotrophs or (almost equivalently)
preclude trophic cycles.
We find that in our evolving-interaction model, when
very weak interactions (less than 0.0001 of the maximum) are neglected, a cycle-free trophic structure with
chaotic dynamics almost invariably emerges (Fig. 2). The
emergent absence of effective trophic cycles (e.g., A eats
B, B eats C, C eats A) is a particularly striking result —
the random allocation of Mij to autotrophs means that
such cycles are present initially, but the dynamics is always such as either to eliminate one of the species in the
cycle or to make one of the Mij negligibly small (eliminating it is not possible in the model). Consideration of the
flow of resource helps us to understand this: a trophic
cycle would have a flow of resource around it, dissipated
by b at each interaction, but cannot have any generation of
resource.
Moreover, the distribution of interaction strengths
(Fig. 3) is a power law with an exponent of 1 (within
error), independent of web size for large webs. This
means that at any one time the predators are obtaining
their resources both from a few strong interactions and
from many weak interactions, which also serve to stabilize the web, particularly by becoming stronger in times
of declining population.
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The behavior of our foodwebs can be monitored by the
flow of resources through the system over time [1]. We
monitored this during our simulations and found a remarkable result —the total flow of resource (and hence
total biomass) increases with time reaching a plateau
after many thousands of steps —the steady-state linkstrength ensemble distribution appears to be the one
which maximizes the use of resource. This type of optimization is consistent with what has been observed in
other ecological models [19,20]. If the model is recast in
terms of flow and dissipation, the maximization principle
is equivalent to maximum entropy production [21]: the
mathematical equivalent of ‘‘entropy production’’ is just
the total death rate, and hence the flow out. Unfortunately,
as with many systems described by maximum entropy, it
is not possible to determine analytically what the maximum flow actually is, and the principle can only be
inferred by the increase with time as the steady state is
approached.
Most results described here are robust against changing
initial conditions, varying b and c, or making them
species dependent. However, the power-law distribution
of link strengths depends on the existence of chaotic
population dynamics, which in general arises from either
finite time step or a sharp upper limit on Mij . The switch
from a power law to an exponential distribution is also
understandable in terms of the central limit theorem —in
both cases the distribution of average values for individual populations over time is distributed normally—in the
fixed point case this is also the instantaneous average. The
presence of instantaneous power-law distributions is consistent with the maximum entropy production [21].
If we take an alternate evolution of the interactions for
which Mij is self-limiting, replacing Eq. (5) with the
more complex form
and use an infinitesimal time step, then the dynamics tend
to one of many fixed points rather than the chaotic state.
Trophic level structure and high numbers of weak interactions still occur in this model, but curiously, the
evolved link strengths are now exponential rather than
power-law distributed. Eliminating the maximum allowed value of Mij has a similar mathematical effect,
although it makes little biological sense to allow a predator to extract unlimited resource from a given prey. The
switch between chaotic and steady behavior when going
from discrete to continuous dynamics could be anticipated, being typical of many systems, including the logistic map.
It is possible to construct different, more complex,
models with alternative relations between Mij and Mji
reflecting mutualism (both positive) or competition (both
negative); however, these are not essential to stability. For
example, Kondoh [17] recently considered restricting the
evolution to the predator only, essentially replacing
Eq. (5) with
dMij =dt Nj Mij
(7)
for Mij > 0. In this model the feedback also stabilized
realistic highly connected webs.
Generalized Lotka-Volterra models are widely used in
economics as well as in ecology, and it is worth noting
that the fixed point of our system does not represent a
Nash equilibrium. Mij contains both evolution of the
predator to catch its prey and that of the prey to avoid
its predator. The fixed point means that each species is
becoming better adapted to combat the other at the same
rate: the so-called ‘‘red queen’’ effect.
8
dMij =dt 1=Ni dNi =dt 1=Nj dNj =dtMij (6)
6
log(Number of Links)
Population [log]
10
1
0.1
0.01
4
2
0
0.001
0
2000
4000
6000
8000
Time
10000
12000
14000
FIG. 2 (color online). Graph of total population as a function
of time for a typical evolving web with 100 species, and flow of
resources in and out of the web (see the text for definition). The
graph of inward flow in broadened for clarity: it lies close to
(and leads) the outward flow throughout.
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−2
−7
−6
−5
−4
−3
−2
−1
0
log(link strength)
FIG. 3. Plot of the (log) number of links against their (log)
strength. The histogram is averaged over many snapshots from
many stable webs of size 100 species. In each stable web the
link strength varies with time —the interaction strengths are
instantaneous, not time-averaged values.
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In sum, we have shown that simply by allowing the
strength of interactions to evolve in a GLV model, several
features of observed foodwebs emerge spontaneously.
These include chaotic dynamics, maximal use of resources, stability engendered by many weak links, and the
absence of trophic cycles. While previous models have
shown some of these phenomena, in general one or more
have been be assumed in the formulation of the models.
This work uses a minimal number of assumptions to
show the powerful effect of evolution in structuring the
foodweb patterns of nature, focusing the system onto
stable states via a feedback process.
*Electronic address: [email protected]
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