Modelling coevolution in multispecies communities

Abstract
We introduce the Webworld model, which links together the ecological modelling of food web structure with the evolutionary modelling of speciation and extinction events. The model describes dynamics of ecological communities on an evolutionary timescale. Species are defined as sets of characteristic features, and these features are used to determine interaction scores between species. A simple rule is used to transfer resources from the external environment through the food web to each of the species, and to determine mean population sizes. A time step in the model represents a speciation event. A new species is added with features similar to those of one of the existing species and a new food web structure is then calculated. The new species may (i) add stably to the web, (ii) become extinct immediately because it is poorly adapted, or (iii) cause one or more other species to become extinct due to competition for resources. We measure various properties of the model webs and compare these with data on real food webs. These properties include the proportions of basal, intermediate and top species, the number of links per species and the number of trophic levels. We also study the evolutionary dynamics of the model ecosystem by following the fluctuations in the total number of species in the web. Extinction avalanches occur when novel organisms arise which are significantly better adapted than existing ones. We discuss these results in relation to the observed extinction events in the fossil record, and to the theory of self-organized criticality.

arXiv:adap-org/9801003v2 25 Jan 1998
Modelling coevolution in multispecies communities
Guido Caldarelli 0,1,2, Paul G. Higgs 2 and Alan J. McKane 1
1 Department of Theoretical Physics, University of Manchester,
Manchester M13 9PL, UK
2School of Biological Sciences, University of Manchester,
Manchester M13 9PT, UK
Abstract
We introduce the Webworld model, which links together the ec ological modelling of
food web structure with the evolutionary modelling of speci ation and extinction events.
The model describes dynamics of ecological communities on a n evolutionary timescale.
Species are deﬁned as sets of characteristic features, and t hese features are used to
determine interaction scores between species. A simple rul e is used to transfer resources
from the external environment through the food web to each of the species, and to
determine mean population sizes. A time step in the model rep resents a speciation
event. A new species is added with features similar to those o f one of the existing
species and a new food web structure is then calculated. The n ew species may (i) add
stably to the web, (ii) become extinct immediately because i t is poorly adapted, or (iii)
cause one or more other species to become extinct due to compe tition for resources.
We measure various properties of the model webs and compare t hese with data on
real food webs. These properties include the proportions of basal, intermediate and top
species, the number of links per species and the number of tro phic levels. We also study
the evolutionary dynamics of the model ecosystem by followi ng the ﬂuctuations in the
total number of species in the web. Extinction avalanches oc cur when novel organisms
arise which are signiﬁcantly better adapted than existing o nes. We discuss these results
in relation to the observed extinction events in the fossil r ecord, and to the theory of
self-organized criticality.
0Present Address TCM, Cavendish Laboratory Madingley Road, Cam bridge CB3 0HE UK

1 Introduction
Our aim in this paper is to introduce a model for the evolution of many in teracting species in
an ecosystem. A great deal of information has been assembled on t he properties of food webs
for naturally occurring ecosystems (Cohen, 1989; Martinez, 199 1; Polis, 1991; Hall & Raﬀaelli,
1991; Goldwasser & Roughgarden, 1993), and a variety of models d escribing the structure of
food webs have been studied (Pimm, 1982; Briand & Cohen, 1984; Co hen et al., 1990; Cohen,
1990; Pimm et al., 1991, Morin & Lawler, 1995). Most of these models c oncentrate on static
properties of the webs, such as the proportions of species at eac h of the trophic levels, and
the lengths of food chains. There have also been models for the ass embly of food webs by
gradual addition of species (Luh & Pimm, 1993; Morton & Law, 1997) . One essential property
of real food webs is that they have evolved. Species diversity will ac cumulate over time as
new species arise due to speciation. Species may also become extinct due to competition
from better adapted species. Here we study a model which has the evolutionary dynamics of
speciation and extinction built into it. We will analyse the properties of the food webs which
arise in the model and compare these with ecological data.
The variation in the diversity of species seen in the fossil record is we ll documented (Sep-
koski, 1993). Attention has focused on the large-scale extinction events in which very large
fractions of the existing species apparently become extinct almost simultaneously on a geolog-
ical timescale. Some of these large scale extinctions may be caused b y catastrophic external
events, or by major climate changes. However it has been shown (S neppen et al., 1995; Sol´ e
et al., 1997) that both the distribution of sizes of extinction events and the distribution of
lifetimes of species in the fossil record are very broad and approxim ate to power-laws. It has
been argued that the large ﬂuctuations in species diversity which ar e seen may result from
the internal dynamics of the ecosystem. A newly-evolved species m ay cause extinction of its
immediate competitors. The extinction of one species may lead to the extinction of other
species which prey upon it, and these extinctions may lead to furthe r extinctions, etc. Hence
there is the possibility of avalanches of extinction events spreading through an ecosystem.
A link has been made between macroevolutionary dynamics and the th eory of self-organized
criticality (SOC). This theory was originally formulated in the language of sandpiles (Bak et
al., 1988) and has since been applied to many other dynamical system s including earthquakes
and forest ﬁres. In the sandpile model one grain of sand is added at a time onto a pile of
sand. This may either add stably to the pile, or may cause an avalanch e of one or more grains
to tumble from the pile. The evolutionary analogy is clear: evolution lea ds to new species
being added slowly to an ecosystem, and these may sometimes add st ably to the system or
may sometimes cause extinction events.
There is now a huge variety of evolutionary models inspired by the idea of SOC (Bak &
Sneppen, 1993; Paczuski et al., 1996; Kramer et al., 1996; Sol´ e & Bascompte, 1996; Sol´ e et
al., 1996; Roberts & Newman, 1996). These models are deliberately e xtremely simple, and
they focus on the dynamical properties of extinction avalanches a nd species lifetimes. They
do not tell us anything about ecosystem structure, because the couplings between species
are introduced either at random or in an ad-hoc manner (e.g. specie s are arranged on a line
and neighbours interact). In our view it is essential to have a realist ic model of ecosystem
structure in order to study the way extinction events will propaga te through a food web.
By linking the ecological modelling of food web structure with the evolu tionary modelling of
speciation and extinction events we believe that the Webworld model described in this paper

makes a signiﬁcant advance in both ﬁelds.
At this stage it is useful to introduce some notation used in the desc ription of food webs.
If one species preys on another then they are said to be linked. Basal species are those
with predators but with no prey and top species are those with prey but with no predators.
Intermediate species have both predators and prey. We will refer to the percentages o f basal,
intermediate and top species as B, I, and T . Two species are said to belong to the same
trophic species if they share the same set of prey and the same set of predators ( see Figure
1). It is customary to group the original species in a web into trophic species and to analyse
the properties of the trophic species web. This goes some way towa rd standardizing webs
obtained by diﬀerent researchers using diﬀerent criteria for inclus ion of species and diﬀerent
degrees of resolution. The statistics presented in this paper will be for trophic species webs.
One food web statistic which we quote for the model webs is the ratio of prey to predator
species, deﬁned as ( B + I)/ (T + I). Another characteristic property of webs is the average
number of links per species, deﬁned simply as the total number of link s divided by the total
number of species. It is also useful to classify the links according to the type of species which
they connect, i.e. we can measure the proportion of links between t op and basal ( T B), top
and intermediate ( T I), intermediate and intermediate ( II ) and intermediate and basal ( IB )
species.
Ecosystems are reliant on the input of resources from the extern al, non-living environment.
Within the model the environment is treated explicitly as a node in the f ood web. Species
which are linked to the environment node are primary producers, i.e. they may survive in
absence of any other species. It follows that basal species (thos e having no prey) must be
primary producers, otherwise they would have no means of surviva l. However not all primary
producers are basal species, since a primary producer may also be a predator of other species.
We will use the following deﬁnition of trophic levels. All species linked to the environment
node (i.e. primary producers) are deﬁned as level 1 species. All spe cies which have at least one
level 1 prey are deﬁned as level 2 species. All species with at least on e level 2 prey are deﬁned
as level 3, and so on. Hence, the level of a species is the length of th e shortest food chain
from the external environment to that species. We could also have chosen to characterize
species by the length of the longest chain linking them to the environm ent, or the average
length of all the possible chains, however these quantities are time c onsuming to calculate
for large webs, and also special rules are required to deal with the p ossibility of loops in the
web. The deﬁnition we use is rapid to calculate, and is unambiguous, ev en when loops are
present in the web. Also the majority of resources obtained by a sp ecies are likely to come
via the shortest route, since transfer of resources at each link o f the food chain is relatively
ineﬃcient. Therefore it seems that the deﬁnition of trophic level wh ich we use is meaningful
from an ecological point of view.
2 Deﬁnition of the Webworld Model
Features and Species
A species is deﬁned in terms of a set of L characteristic features. Such features may
be either morphological or behavioral characteristics possessed by all the individuals of that
species, e.g. sharp teeth, binocular vision, webbed feet, being noc turnal, forming social groups
etc. The L features for each species are picked from a total number of K possible features.

In the simulations presented here L = 10 and K = 500. The species are labelled by integers
n, n ′ = 1, 2, ... .
The predator-prey relationships between species are determined by the features possessed
by those species. The matrix, m, is a K × K matrix of scores representing the usefulness
of any feature i of one species for predation against any feature j of another species. We
suppose that m is anti-symmetric, and that the elements are assigned randomly, su ch that
mi,j is a random Gaussian variable with mean zero and variance 1 if i > j , mj,i = − mi,j, and
the diagonal elements mi,i are zero. This matrix of the scores is chosen at the beginning of a
simulation run and does not change during the run. The score Sn,n′ of one species n against
another species n′ is then deﬁned as
Sn,n′ = max{0, 1
L
∑
i∈n
∑
j∈n′
mi,j} (1)
where i runs over all the features of species n and j runs over all the features of species n′.
The max operator ensures that all scores are positive or zero. A positive s core indicates that
n is adapted to be a predator of n′, and a zero score indicates that there is no interaction.
The factor of 1 /L has been chosen so that the scores have a root mean square value of order
1.
The external environment is represented as an additional species 0 which is assigned a set
of L features randomly at the beginning of a run and which does not chan ge. If Sn,0 > 0 the
species n may be a primary producer, provided it is not out-competed (see be low).
Transfer of Resources
A total amount, R, of external resources is distributed amongst the primary produ cers, as
a function of the scores Sn,0, according to the rules for competition described below. These
resources determine the population size of the primary producers . The population of species n
is denoted N(n). For simplicity, we measure resources and population size in the sam e units,
so that N(n) is equal to the amount of resources obtained by species n. Predator species
obtain resources from their prey. We assume that a fraction λN (n) of the resources of a given
species n is available to be passed on to predators of n. Thus λ is a parameter of the model
which determines the relative sizes of predator and prey population s.
Competition for Resources
The resources obtained from species n are distributed between the predators of n in the
following way. (The same rules are used for the case n = 0, where the resources are the
external resources R, and the ‘predators’ are the primary producers). The better ad apted the
predator, the more resources it gets. We therefore deﬁne the main predator of n as the one
with the best score against n:
SM
n = max{Sn′,n} where n′ is a predator of n (2)
The predators of n will obtain a share of the available resources proportional to the qu antity:
Fn′,n = max
{
0,
(
1 − SM
n − Sn′,n
δ
)}
(3)
where δ is a parameter of the model which determines the strength of comp etition. The
smaller δ the stronger the competition. We also deﬁne Fn′,n to be zero if Sn′,n = 0, so that

only species for which Sn′,n > S M
n − δ, and Sn′,n > 0 successfully obtain any resources from
n. The actual fraction of resources obtained is found by normalizing the Fn′,n:
γn′,n = Fn′,n
∑
m Fm,n
(4)
Population Sizes
We will calculate the population sizes of the species by iteration of a se t of equations
representing transfer of resources between the species. Each iteration represents a small time
period of order one generation time. If N(n, t ) is the population of species n at iteration t
then we may write
N(n, t + 1) = γn,0R +
∑
n′
γn,n′λN (n′, t ) + γn,nλN (n, t ). (5)
The ﬁrst term represents external resources obtained by n, which may be zero if γn,0 = 0.
The second term represents resources obtained from prey ( n′ runs over all the prey of n). The
third term represents resources lost to predators. Here we hav e deﬁned
γn,n =
{
− 1, if n has at least one predator species
0, if n has no predators (6)
After several iterations the population sizes will converge to stat ionary values. We suppose
that the stationary population sizes determined by this method rep resent mean population
sizes over many generations. The iteration procedure can be viewe d as an algorithm which
determines the solution of the following set of linear equations for th e stationary values:
N(n) =
∑
n′
γn,n′λN (n′). (7)
Here n′ runs over all the species, including n and 0, and for convenience we have deﬁned
N(0) = R/λ .
There have been many studies of population dynamics at short times cales using either
diﬀerence equations like (5), or diﬀerential equations like the Lotka -Volterra equations, or
generalizations of them (Vida et al., 1990; Berryman et al., 1995; Ard iti & Michalski, 1996;
Zheng et al., 1997). Many of these equations have interesting dyna mical behaviour, such as
periodic solutions or chaos which may be relevant to real ecological p opulation dynamics. In
fact we are not interested in the population dynamics at the scale of a few generations. We are
only interested in evolutionary time. We have deliberately chosen diﬀe rence equations which
are as simple as possible, and which have only one stationary state. T hese equations converge
rapidly, and always reach the same stationary state irrespective o f the initial conditions. It
is a property of our equations that the amount of resources pass ed from a prey to a predator
species is proportional to the prey population size only. This situatio n is usually called donor
control (Zheng et al., 1997). We will be interested in the way the pop ulation sizes change
as the ecosystem evolves. Before describing the long timescale evo lutionary dynamics of
our model it is useful to consider the following simple example which illust rates competition
between species and transfer of resources between levels.
Suppose there are two primary producers, 1 and 2, having scores S1,0 = 1. 0 and S2,0 = 0. 95.
In addition, species 3 is a predator of species 1, i.e. S3,1 > 0. Let the competition parameter

δ be 0.1. From the above rules γ1,0 = 2 / 3 and γ2,0 = 1 / 3, and since 3 is the sole predator of
1, γ3,1 = 1. The iterative equations are therefore
N(1, t + 1) = 2 R/ 3 − λN (1, t )
N(2, t + 1) = R/ 3 (8)
N(3, t + 1) = λN (1, t )
and the stationary states are N(1) = 2 R/ 3(1 + λ), N(2) = R/ 3, and N(3) = 2 λR/ 3(1 + λ).
Notice that the sum of the populations is equal to R. It is a property of the model that
the sum of the populations at the stationary state is always equal t o the amount of external
resources put in.
Given the lists of features representing any set of species it is poss ible to calculate the
scores Sn,n′ and hence the steady state population sizes. We suppose that the re is a minimum
population size necessary for survival of the species, and we set t his limit to 1. Any species
for which N(n) < 1 becomes extinct and is deleted from the list of species. The resultin g
ecosystem is then stable, and properties of the web can be measur ed.
For the purposes of deﬁning the food web structure, a link is assum ed to be present
between n and n′ if n successfully obtains resources from n′, i.e. if γn,n′ > 0. Cannibalism
has been excluded from the model ( Sn,n is deﬁned to be zero for all n). Also, since the mij
matrix is antisymmetric, then for any pair of species it is impossible for both Sn,n′ and Sn′,n
to be non zero, i.e. there are no reciprocal pairs of links. Loops of t hree or more species can
occur in the model, however.
Evolutionary Dynamics
Evolutionary time in the model proceeds in timesteps such that one s peciation event occurs
in every timestep. At each step an existing species is chosen to unde rgo speciation with a
probability proportional to its population size. A new species is creat ed by copying the list
of features of the parent species, randomly picking one these fea tures, and replacing it by
another randomly chosen feature from the complete list of possible features. The new species
thus diﬀers by only one feature from the old one. The stationary po pulation sizes of all the
species are now recalculated, taking account of the presence of t he new species. There are
several possible outcomes: (i) the new species may add to the web in a stable fashion, so that
the total number of species increases by one; (ii) the new species m ay be poorly adapted, and
may become extinct immediately; (iii) the new species may survive, and cause one or more
other species to become extinct. A new species will often be in direct competition with the
parent species, therefore a special case of outcome (iii) is that th e new species simply replaces
its parent in the ecosystem.
Having eliminated any species which become extinct due to the new add ition, the web
is once again in a stable state. This is the end of one complete timestep . This process is
repeated many times and the properties of the webs are recorded at the end of each step and
subsequently averaged. When choosing the new feature for each new species a restriction is
made that no feature may appear more than once in the list of featu res possessed by any one
species. Also, when the new feature list is created, we check that it is not identical to the
feature list of any other species, and that it is not simply a permutat ion of any other list.
This prevents the occurrence of multiple copies of identical species .
The only remaining thing to be speciﬁed is the initial state of the web. O ne possibility is to
consider an ‘origin of life’ scenario, where the web begins with exactly one primary producer

species. Another possibility is to begin with a small number of randomly chosen species. We
carried out runs where the initial state was composed of 1, 10 and 2 0 diﬀerent species, and
found very little diﬀerence between these. All the results present ed here were started with a
set of 10 random species. Since the features of these species are chosen at random, the species
are not well adapted to survive together, hence there is a tenden cy for several extinctions to
occur immediately on the ﬁrst timestep.
3 Properties of the Model
In this section we discuss some of the important properties of the m odel which are relevant to
understanding the numerical results in the next section. The mode l contains three parameters,
R, λ and δ, and these will aﬀect the shapes of the food webs in various ways.
We would intuitively expect that increasing the amount of resources available should
increase the diversity of the ecosystem. In the model the sum of t he population sizes is
equal to R, and since every species must have a population of at least one, the re can never
be more than R species. In fact the mean number of species observed is always muc h less
than R because resources are not distributed evenly between species. S pecies at lower levels
have much larger populations. The ratio of population sizes between predator and prey is
controlled by λ. Together R and λ determine the maximum possible number of trophic levels
in the system.
Consider a web consisting of a single food chain of k species. The stationary population
sizes obtained from the model for each level j except the top level are N(j) = λ j−1R/ (1 +λ)j,
and the population of the top level is N(k) = λ k−1R/ (1 + λ)k−1. A chain of length k can only
be supported if N(k) > 1. This gives the condition
k < 1 + log (R)
log ((1 + λ)/λ ) (9)
and, since k is an integer, the maximum level is actually the largest integer below th is limit.
Thus the maximum food chain length only increases logarithmically with R. Although this
limit has been calculated assuming the web is a single chain, we believe tha t this is a strict
limit to the number of levels for any possible web generated by the mod el. This is because
adding extra links and more species to the web always decreases the fraction of resources
reaching the top level in comparison to the single chain calculation.
The number of levels observed in the model food webs is strongly sen sitive to λ. We have
chosen to set λ = 0 . 1 in all the simulations reported here. This gives realistic numbers of
trophic levels in the model food webs and is also consistent with measu red values of predator
prey population ratios, and estimates of the ecological eﬃciency (P imm, 1982).
The value of the parameter δ aﬀects the properties of the web considerably. In order to
choose a sensible value of δ we note that the scores Sn,n′ are all of order 1, and that when
a single feature is changed, the score changes by an amount of ord er 1 /L . In other words,
species competing for the same resources should have scores whic h diﬀer by an amount of
order 1/L . We should therefore set δ to to be roughly of this size. If we make δ >> 1/L , even
very uncompetitive species will be allocated some resources. Hence the number of predator
species per prey species will be large, and this will lead to a highly conne cted web with a large
number of links per species. As δ → 0, only the main predator will be allocated resources,

and so in the limit the web will become a single food chain. For most of the results given in
this paper δ is in the range 0.05 - 0.2, which is comparable with 1 /L = 0. 1.
It is useful to mention several features which have deliberately be en excluded from the
model. Firstly, there is no variation between individuals of a given spec ies, and there is
no genetics. Species are simply represented by a list of phenotypic f eatures, which represent
average properties for all members of that species. A model which included both many species
and many individuals per species, each with its own genotype and/or p henotype would require
enormous computational resources. Secondly, even though spe ciation is an essential part of
the model, we do not attempt to consider the mechanism by which spe ciation occurs. We
simply suppose that species have an inherent tendency to diversify . Speciation involves the
establishment of reproductive isolation by some means or another, and we cannot deal with
this in the absence of a genetic description of the species. When a ne w species is created it
diﬀers by only one feature from the parent species. However, this is intended to represent
a major change to the phenotype, which would probably involve chan ges in more than one
gene sequence, and possibly some considerable alteration to the de velopmental biology of the
organism. Thus changing a single feature does not represent a sing le mutation, but is the
result of many changes at the genetic level. In addition we make no dis tinction between
sexual and asexual methods of reproduction, and the model is int ended to apply equally well
to either case.
4 Results - Structure of Model Food Webs
The ﬁgures presented in the tables of results represent average s taken over many webs. For
each set of parameters several independent simulation runs were performed using diﬀerent
randomly generated m matrices and diﬀerent random feature sets for the environment ( species
0). Each run was for 250,000 timesteps, with the exception of the r uns with R = 10 10, which
were for 500,000 timesteps. In each case, average quantities wer e measured during the second
half the run. The diﬀerent runs for each parameter set were then averaged. Fluctuations in
the number of species and the number of links between diﬀerent run s with the same parameter
values were surprisingly large, with standard deviations being up to 5 0% of the mean. Since
these quantities increase in relation to one another, the ﬂuctuatio ns in the number of links per
species are much smaller (typically ± 0.2). Standard deviations in the percentages of B and
I species are typically 7% − 10%. We have omitted error estimates from the tables for clarity.
We are principally interested in the main trends in the web properties a s the parameters are
varied, rather than in precise values. We have checked that these trends are signiﬁcant. The
quoted ﬁgures in the tables apply to webs of trophic species.
Tables 1 and 2 show results for R = 10 3, 10 6 and 10 10, for δ in the range 0.05-0.2. In all
cases λ = 0 . 1. There is a clear tendency for the mean number of species to incre ase with R,
for ﬁxed values of δ and λ. However, it should be noted that the number of species is very
much less than the theoretical maximum of R. In fact the number of species only increases
very slowly with R: a change in R by seven orders of magnitude only causes a ﬁvefold increase
in the number of species. In contrast, the number of species incre ases rapidly with δ at ﬁxed
R. Also, changing δ has a large eﬀect on the total number of links. The number of links
increases more rapidly than the number of species, so that the num ber of links per species
increases signiﬁcantly with δ. The number of links per species increases slightly with R for
ﬁxed δ.

R = 10 3
δ = 0. 05 δ = 0. 1 δ = 0. 15 δ = 0. 2
no. species 40 80 290 320
no. links 55 150 800 1250
links per species 1. 4 1. 9 2. 7 3. 8
average level 1. 7 1. 4 1. 2 1. 1
max level 3. 0 3. 0 3. 0 3. 0
B species (%) 40 56 74 78
I species (%) 59 44 26 22
T species (%) 1 0 0 0
IB links (%) 53 73 83 88
II links (%) 43 27 17 12
TI links (%) 4 0 0 0
prey/predators 1. 6 2. 2 3. 8 4. 6
Table 1: Results of a simulation of the model with λ = 0. 1 and R = 10 3 for four values of the
competition parameter δ.
R = 10 6 R = 10 10
δ = 0. 05 δ = 0. 1 δ = 0. 15 δ = 0. 2 δ = 0. 05
no. species 150 200 410 810 220
no. links 240 510 1420 5930 350
links per species 1. 6 2. 5 3. 5 7. 3 1.6
average level 2. 5 2. 4 2. 3 1. 8 3.2
max level 5. 2 5. 2 5. 3 5. 5 7.2
B species (%) 10 25 32 60 3
I species (%) 90 75 68 40 97
T species (%) 0 0 0 0 0
IB links (%) 22 47 51 70 10
II links (%) 78 53 49 30 90
TI links (%) 0 0 0 0 0
prey/predators 1. 1 1. 3 1. 5 2. 5 1.0
Table 2: Results of a simulation of the model with λ = 0 . 1, R = 10 6 or 10 10 and various
values of δ.

These results imply that competition is a signiﬁcant factor in determin ing the number
of species and the number of links in the web. The competition parame ter δ determines the
cut-oﬀ in the function Fn,n′ in equation (3), and hence controls the number of predator specie s
which can successfully obtain resources from each prey. It is ther efore to be expected that
the number of links per species will increase with δ. Also, when the competition is weaker,
it is easier for a species to ﬁnd at least one prey for which it is suﬃcient ly well adapted
to be a predator, therefore the total number of species should a lso increase with δ, as is
observed. The importance of δ in controlling species numbers also provides an explanation of
why the number of species only increases very slowly with R. The number of level 1 species
is limited by competition for external resources. Increasing R at ﬁxed δ tends to increase
the populations of all level 1 species proportionately, rather than increasing the number of
level 1 species. The same is also true of intermediate levels. However at the higher trophic
levels species numbers are limited by the criterion of the minimum popula tion size necessary
for viability ( N(n) > 1). The principal eﬀect of increasing R is therefore to increase the
maximum number of trophic levels possible in the web, and to allow a few h igh level species
to survive, without changing the number of low level species very mu ch. We have already
shown that the maximum number of levels increases logarithmically with R, and this therefore
suggests that the total number of species should increase as log R, which is consistent with
our results. Figure 2 shows a histogram of the mean number of spec ies at each level, and the
way that this depends on R. This conﬁrms the fact that increasing R leads principally to an
increase in species at higher trophic levels.
Tables 1 and 2 also show the change in the number of levels in the web wit h R and δ. The
average level is deﬁned by calculating the level of each species, ave raging these for all species
in each web, and then averaging these web averages over many web s for each parameter set.
The maximum level is deﬁned by calculating the maximum level present in each web, and
averaging these maximum values over many webs for each paramete r set. Both average and
maximum levels increase approximately logarithmically with R as expected. For R = 10 3, 106
and 10 10 the maximum possible number of levels from equation (9) are 3, 6 and 1 0. For
R = 10 3 the limiting value is always achieved. For R = 10 6 the mean value of the maximum
level is only slightly below the limit, whereas for R = 10 10 it is considerably below. Two
factors contribute to this. Firstly, the theoretical limit was based on a single food chain since
this is the most eﬃcient way of transferring resources to high levels . The webs generated
by the model are multiply connected and contain omnivorous links and loops, hence the
maximum attainable number of levels is reduced. Secondly, the calcula ted limit takes only
ecological eﬃciency into account, but ignores evolutionary constr aints, which may also limit
the number of levels in the Webworld model. Well adapted level 1 specie s may begin to evolve
from the start of a simulation run, since the environment is ﬁxed. Hig h level species depend
on lower level ones for their resources, hence well adapted high lev el species tend to evolve
at later stages of the simulation when the properties of the lower lev el species are changing
less rapidly. The properties of the runs with R = 10 10 still appeared to be changing after
250,000 timesteps, hence they were continued for 500,000 timeste ps. This led to a signiﬁcant
increase in the number of species and the number of links, and a slight increase in the number
of levels. Most of the late evolving species are on higher levels, hence the proportion of basal
species was found to decrease signiﬁcantly between 250,000 and 50 0,000 timesteps.
Also from the tables it can be seen that the average level decrease s with increasing δ.
This is understandable, because the level is deﬁned as the shortes t possible chain to the

external environment node. If δ is large, then a species with a low score Sn,0, but a high
score Sn,n′ against another species n′ may successfully compete for the external resources,
and will therefore count as level 1 (the link to n′ would not aﬀect the calculation of the level).
If δ is smaller, the same species would only survive by being a predator of n′, and would
therefore count as a higher level species. The maximum number of le vels does not change
much with δ. The apparent slight increase in the maximum level with δ in table 2 is probably
not signiﬁcant.
The proportions of basal, intermediate and top species are given in t ables 1 and 2. In-
creasing δ at ﬁxed R leads to an increase in B and a decrease in I and T . This is for the same
reason that the average level decreases with δ. (However, in the following section we consider
a case with extremely large δ where this trend is no longer true). Increasing R at ﬁxed δ
leads to a decrease in B and an increase in I. Again this is because increasing R increases
the number of levels, and thus the proportion of species in level 1 go es down. The number of
top species is always very small: T ≤ 1. 5% when R = 10 3, and no top species were observed
at all for R = 10 6 and R = 10 10. Thus increasing the number of levels does not mean that
the number of top species increases. In the model webs, for the p arameters discussed so far,
almost all species have predator species, even if they are at high tr ophic levels. This implies
the presence of large numbers of loops in the food web and large num bers of omnivores. The
behaviour of T is an important property which has been remarked upon in the study of real
food webs, and we will return to it in the following section.
The tables also give the fractions of links between top, basal, and int ermediate species,
and the ratio of prey species to predator species. These quantitie s are given primarily for
comparison with the real food web data. Trends in these quantities can be understood in
terms of the trends discussed above in B, I, and T . For example, as δ increases at ﬁxed R,
or R decreases at ﬁxed δ, the percentage of species which are intermediate decreases and as a
consequence the fraction of the links which are between intermedia te species also decreases.
5 Comparison with Real Food Webs
The problems in getting reliable data on real food webs are readily ack nowledged by ecologists
(Cohen et al. 1993). These problems centre around the tremendo us amount of eﬀort required
to observe all species and all predator-prey interactions in a given ecosystem. Data from a
large number of food web studies has been assembled in the EcoWeb d atabase (Cohen, 1989).
Our analysis of the average properties of all the 181 webs in this dat abase is given in Table 3.
In addition we give data from several well studied webs which have be en published recently
(see caption to Table 3).
Before these ﬁgures can be compared with the model it is necessar y to note several points.
For both the model and the real data, all statistics are for troph ic species. Many webs contain
detritus as a ‘species’, and we have followed the convention of treat ing this as a single basal
species. Plants are often treated in a very aggregated way, e.g. t he St. Martin Web divides
plants into fruit, nectar, leaves, wood and roots, each of which is t reated as a ‘species’. Several
webs split taxa into adult and juvenile ‘species’ where these have diﬀerent diets. Again we have
followed the conventions of the source article in these cases. In th e model webs the external
resources are treated explicitly as a node in the web, and basal spe cies must be linked to this
node. Real webs have no such external resources node (althoug h evidently external resources
do enter the real ecosystems). Therefore, when counting the n umber of links in the model

Experimental Data
ECOWEB Lovinkhoeve Coachella St. Martin Ythan Little Rock
no. species 16 15 29 42 83 93
no. links 33 30 262 203 398 1033
links per species 2. 0 2. 0 9. 0 4. 8 4.8 11. 1
average level 2. 1 2. 5 2. 0 2. 1 2.5 2. 2
max level 3. 2 3 3 4 4 3
B species (%) 21 13 10 14 5 13
I species (%) 49 74 90 69 59 86
T species (%) 30 13 0 17 36 1
TB links (%) 10 3 0 3 1 0
IB links (%) 29 10 13 19 10 9
II links (%) 29 57 87 53 51 91
TI links (%) 32 30 0 25 38 0
prey/predators 0. 89 1. 0 1. 11 0. 97 0.67 1. 13
Table 3: Data from experimentally studied food webs: the ECOWEB da tabase - average
properties of all webs, (Cohen, 1989); the Lovinkhoeve Experime ntal Farm soil food web (De
Ruiter et al, 1995); the Coachella Valley desert (Polis, 1991); St. Ma rtin Island (Goldwasser
& Roughgarden, 1993); the Ythan Estuary (Hall & Raﬀaelli, 1991; H uxham et al., 1996); and
Little Rock Lake (Martinez, 1991).
webs, links to the external resources node were omitted, in order to allow comparison with the
real webs. The real webs contain some cannibalistic links (particular ly the Coachella web).
For simplicity we included these in the count of links, whereas cannibalis m does not occur in
the model webs. The statistics in the table diﬀer very little according to whether one counts
or discounts cannibalistic links.
Most of the webs in the EcoWeb compilation are small, and they have be en criticised as
being incomplete, and biased due to observational methods (Martin ez, 1991; Polis, 1991).
We have chosen the individual webs since in these cases attempts ha ve been made to be
systematic and inclusive. Nevertheless, one suspects that the lar ge diﬀerence in the number
of trophic species between these data sets reﬂects the degree o f aggregation chosen by the
diﬀerent workers, rather than the actual diversity of the ecosy stems. The ﬁve individual webs
are listed in order of increasing number of trophic species. There ar e some trends apparent
when comparing these webs with each other, although in many ways e ach must be considered
as a special case. There is a general trend for the number of links p er species to be higher
in larger webs. This would suggest that systematic study of a commu nity over a long period
of time leads to a greater increase in the number of links observed th an in the number of
species. It may be that the number of links is severely underestimat ed in some of the Ecoweb
food webs. The Coachella web breaks this trend, since it has a very la rge number of links per
species and is only fairly small.
There is a strong similarity between the ﬁgures for the Coachella and Little Rock webs,
despite the diﬀerence in the number of species. They both have a hig h number of links per
species, a low value of B, a high value of I and very low T . Martinez & Lawton (1995) have
suggested that, in the limit of extremely large webs which represent large geographical areas,
I will increase to about 95%, B will decrease to about 5%, and T will decrease to zero. Our

data in tables 1 and 2 conﬁrm that as R increases, the number of species increases and the
fraction of intermediates becomes larger ( I = 97% in the run with R = 10 10, for which the
average number of trophic species is 200). The data from the Ytha n web appear to go against
this trend, however, since there is a very high value of T (36%), even though the web is almost
as large as the Little Rock web. We have used the version of the Ytha n web without parasites
(Huxham et al. 1996). Many of the top species become intermediate species when parasites
are included, however the parasites themselves form a new top leve l which are not themselves
preyed on. The number of top species is still very large in the Ythan w eb when parasites
are included, hence this cannot account for the large diﬀerence be tween the Ythan and Little
Rock webs.
When comparing the model results with the real data we view R as a parameter which
could change between ecosystems in diﬀerent places, whereas δ is a fundamental property of
the competition interactions between species which should presuma bly not vary much between
locations. Therefore we would wish to choose a value of δ which best ﬁts a range of real data.
The model results for δ = 0 . 05 have high I, low B and very low T , which seems to be
characteristic of the well-characterised real webs. All the real w ebs have a maximum level of
3 or 4. We obtain this number of levels in the model if R = 10 3 or 10 4. The very high value
of 7 for the maximum level in the runs with R = 10 10 does not appear to be consistent with
the data, however it should be noted that all the model data is for λ = 0 . 1. If we reduce λ
then the number of levels in the model will reduce, and we will require la rger values of R in
order to obtain 3 or 4 trophic levels. Low values of δ seem to be preferred for matching the
values of B and I in the real data, however in this case the model predicts less than 2 links
per species. The number of links per species in the real webs tend to be higher (as much as
11 in Little Rock). It is possible to get high numbers of links per species in the model by
increasing δ (see e.g. R = 10 6, δ = 0. 2). However in this case we get much higher values of B
than are seen in real webs. We conclude that the properties of the model webs are generally
of similar orders of magnitude to those of the real webs, but that w e have a not found a
single set of parameters R, λ and δ which accurately match all the properties of the real data
simultaneously. We have certainly not done an exhaustive search of parameter space yet for
the model: we did not change λ at all. So it is possible that the agreement between the model
and the data could be improved. Also, there are many reﬁnements w hich we could make to
the model, such as changing the rules by which we distribute resourc es between species, or
changing the way the scores are calculated. Consideration of all th ese possibilities would be
justiﬁed if there were an excellent set of data on real food web str ucture against which to
test the models. However, there are still problems with even the be st experimental data, and
in view of this, it does not seem worthwhile worrying too much about th e precise values of
the food web statistics from the model. The model may in fact make s ome useful predictions
which are diﬃcult to test in real life. For example, we noted above tha t if δ is increased, so
as to give high numbers of links per species, the value of B in the model appeared too high.
Real webs tend to apply much higher degree of aggregation to low lev el species than high
level ones, so maybe the values of B in the real webs should really be higher than they are.
Over the range of δ of order 1 /L considered in the tables, I is a decreasing function of δ
and B is an increasing function. However, as a limiting case, we also carried o ut a simulation
with R = 100 and δ = 25. With this very large δ there were almost no extinctions due to
competition. After 500 speciations there were approximately 90 diﬀ erent species which had
average population sizes just greater than 1. Extinctions occurr ed only due to the rule that

a minimum population size of 1 is necessary for viability. In this run we fo und that I = 99%
and B = 1%. This implies that there is a reversal of the trend in B and I for very large values
of δ. We do not think that this very large δ value is a reasonable one for comparison with
real webs, however this simulation does illustrate a limiting case of our model: in absence
of competition very large highly connected webs arise which are almos t entirely intermediate
species.
We wish to comment further on the meaning of trophic levels in the Web world model. The
simple picture of ecosystems which is often envisaged supposes tha t organisms can be grouped
uniquely into trophic levels representing plants, herbivores, carniv ores, secondary carnivores
etc. This picture supposes that each level only interacts with the le vels immediately above
and below it. Although it was argued from some early food web data th at omnivores (i.e.
species feeding on more than one trophic level) were rare (Cohen et al., 1990), it has since
been argued that omnivory in well-characterized webs is frequent, that organisms apparently
assigned to the same level are far from equivalent in their dynamics a nd interactions, and
hence that the concept of trophic levels is of little use (Polis & Strong , 1996). We found that
omnivory is frequent in the model, and that it is not possible to assign s pecies to levels such
that interactions are only with the levels immediately above and below. However, we maintain
that the idea of level which we use here is a useful one, since it is a mea sure of food chain
length: the level of a species is the length of the shortest chain fro m the external resources
to the species (including the link to the external node). This is a diﬀer ent deﬁnition of chain
length and trophic level from that used in many other studies. For e xample, Martinez (1991)
calculates the average length of all food chains leading to each spec ies in the Little Rock web
using two algorithms which diﬀer in the way they deal with loops (it is nec essary to exclude
loops in some way or else chain length is inﬁnite). He then deﬁnes the tr ophic level as the
closest integer to the average chain length + 1. Since there are hug e numbers of chains in
large webs, algorithms which involve averaging all possible chains are s low. Martinez was
limited by computer time to looking at aggregated versions of his web u sing his algorithm. In
analysing the model data we have thousands of webs, each with hun dreds of species, rather
than just one real web, therefore we must use a rapid algorithm fo r practical reasons. The
maximum level is 3 in the Little Rock trophic species web using our deﬁnit ion, whereas
using Martinez’s algorithm the maximum level is 9. It is clear that the Lit tle Rock web is
extremely complex and that long chains occur frequently, however , this is a statement about
the combinatorial properties of highly connected graphs, and may not be signiﬁcant from an
ecological point of view. Our algorithm shows that there is no species in the Little Rock web
which does not have a chain of length 3 or shorter. We believe that sh orter chains are much
more important energetically than longer ones. This is certainly true in the model webs, since
there is an eﬃciency factor of λ = 0 . 1 associated with every additional link. It is probably
true also in real webs, and could be tested using data which measure s energy ﬂow along each
link. Therefore we believe that our algorithm for chain length and tro phic levels is justiﬁed
ecologically as well as on grounds of practicality.
Since real webs are likely to be incomplete due to observational diﬃcu lties, Goldwasser &
Roughgarden (1997) investigated the eﬀect of incompleteness of data by deliberately omitting
links from the original St. Martin island web. They have shown that mo st web properties vary
strongly as links are removed. We carried out the following procedur e in order to investigate
the eﬀect that incompleteness of data would have on the food webs in our model. Beginning
with a set of webs generated from the model with R = 10 6 and δ = 0. 05, we deleted a certain

δ = 0. 05, R = 10 6
all links 10% removed 30% removed 50% removed
no. species 150 150 140 130
no. links 240 220 170 125
links per species 1. 6 1. 5 1. 2 1. 0
average level 2. 5 2. 3 2. 3 2. 1
max level 5. 2 4. 5 4. 2 4. 0
B species (%) 10 16 24 30
I species (%) 90 79 57 43
T species (%) 0 5 19 25
TB links (%) 0 2 8 16
IB links (%) 22 25 29 31
II links (%) 78 68 47 31
TI links (%) 0 5 15 21
prey/predators 1. 1 1. 2 1. 1 1. 0
Table 4: The eﬀect on the web properties of removing a fraction of t he links at random.
proportion of links randomly. Deleted links represent real predato r-prey interactions which
were not observed due to poor experimental resolution. After re moval of the links, if there
were any species which were left unconnected to the rest of the we b these species were also
removed from the web. The deletion of the links was carried out on th e original species web,
and the trophic species web for the new set of original species inter actions was then found.
As with all the tables in this article, ﬁgures refer to the trophic spec ies webs. Table 4 shows
the results of this process. T increases rapidly as links are removed, I decreases rapidly, and
B increases more slowly. There are consequent changes in the propo rtions of T B, IB , II
and T I links. The number of links per species, the average level and the max imum level all
decrease as links are removed. All these trends are the same as th ose observed when links
are removed from the St. Martin web (Goldwasser & Roughgarden, 1997). The proportions
of B, I and T species after removal of a substantial fraction of links are similar t o those in
the EcoWeb data, whereas before removal of the links they are clo se to the values of the
more highly resolved webs like Little Rock. This supports the hypothe sis that many of the
webs in the EcoWeb collection suﬀer substantially from incompletenes s. Real webs are also
inﬂuenced by the degree to which species are aggregated into clust ers. Martinez (1991) has
shown that aggregation of species within the original Little Rock web results in smaller webs
with statistics which are closer to those of the EcoWeb compilation th an the full web, which
suggests that the smaller webs are also inﬂuenced signiﬁcantly by ag gregation. We have not
yet considered aggregation eﬀects in our model webs, but we expe ct that the changes will be
similar to those observed with the Little Rock web.
6 Results - Evolutionary Dynamics
Figure 3 shows the number of species as a function of evolutionary t ime for three runs of
the simulation with the same parameter values. The numbers of spec ies diﬀer signiﬁcantly
between diﬀerent random assignments of the mij matrix. In each case the number of species

tends to increase fairly rapidly at ﬁrst, but after a certain time the system tends to a steady
state with fairly constant species number. Extinction avalanches a re visible as sudden drops
in the species number. The ﬁgure shows the number of original spec ies, but the curves for
the number of trophic species follow the same pattern with slightly sm aller numbers, and
show extinction events occurring in the same places. In previous mo dels of macroevolution,
attention has focused on the dynamics of extinction events when t he model is in a stable state.
The avalanche size distribution has been measured and has been obs erved to have a power
law shape for some of these models (Bak & Sneppen, 1993; Sol´ e & B ascompte, 1996). In the
Webworld model we may deﬁne the avalanche size at a given timestep a s the number of species
which become extinct due to the addition of one new species (if the ne w species adds stably to
the web this is an avalanche of size zero). We intend to discuss the siz es of extinction events
in our model in more detail in a subsequent paper. However, our initia l results suggest that
the size of avalanches tends to decrease with time in the Webworld mo del. Once the system
has reached a steady state, ﬂuctuations in species number tend t o be small, whilst in the
initial stages of the simulation large avalanches can occur. Within the model large avalanches
are associated with evolutionary progress. Initially there are relat ively few species, and these
are relatively poorly adapted. If we consider only level 1 species initia lly, then we expect that
the scores Sn,0 of species will gradually increase as better adapted primary produc ers evolve.
In the early stages of the simulation there is a reasonable chance th at a new species will be
substantially better than existing ones, and an extinction avalanch e may occur when the new
species evolves. As time goes on it will become increasingly more diﬃcult to evolve species
which are better adapted, and when improvements do occur, scor es are likely to increase by
smaller amounts. Hence avalanches are likely to decrease in size and in frequency as time
goes on. Since the level 1 species tend to change less rapidly as they become better adapted
it follows that conditions for higher level species become more stable . Hence level 2 species
can become increasingly better predators of the existing level 1 sp ecies, and so on throughout
the web. Therefore evolutionary changes on all levels of the web ar e likely to slow down in
the same way.
Self-organized models of evolution are designed in such a way that th e stationary state is
critical and has interesting dynamics. In these models the avalanch e size distribution and the
species lifetime distribution are power laws. We believe that after a ve ry long time our model
would reach an absolutely stationary state where it would be impossib le for any new species
to evolve. None of our simulations ever reached this point entirely, a lthough the probability
that a new species survives on the ﬁrst timestep in which it evolved be came very small toward
the end of the runs. The time to reach this point of evolutionary ‘sta gnation’ should depend
on the total number of possible species in the model. The number of c ombinations of 10
features out of 500 possible ones is approximately 2 . 5 × 1020, which is large but still ﬁnite. We
intend to investigate the eﬀect of changing the total number of po ssible features K in future
work.
The initial period of evolutionary progress and large avalanche sizes is an interesting feature
of our model, and we believe it may tell us something about the real wo rld. Is the real world
in a stationary state? Are the properties of ecosystems today st atistically equivalent to those
in earlier geological periods? We do not know how to answer these que stions, but we believe
they are important questions to ask. It is clear that the model has a ﬁnite number of possible
species, but it is not clear whether this is true for the real world. Th e earth is obviously much
more complex than any computer model, however it is still ﬁnite in term s of the available

space, energy and raw materials. If the earth were unchanging, it does not seem unreasonable
to suppose that real ecosystems would gradually perfect themse lves to the external conditions,
and that the consequent rate of evolutionary change would decre ase, just as in the model.
The origin of life probably occurred about 3 . 5 − 3. 8 × 109 years ago. It is not clear whether
this is a long or a short period measured on the timescale of what is evo lutionarily possible.
Measures of diversity in the fossil record (Sepkoski, 1993) show a general increasing trend in
species numbers between the Cambrian and the present, despite s everal very large extinction
events, and there is no real indication of any levelling oﬀ in diversity.
Of course, the conditions on earth are not ﬁxed: climatic change oc curs on a wide range
of timescales. If we view the non-living world as continually changing, t hen the living world
must also continually change, and never has the opportunity to rea ch evolutionary stagna-
tion. There is also the possibility that the external conditions might c hange by sudden rare
cataclysmic events, such as meteorite strikes, rather than by sm ooth gradual change. Such a
large scale external event might cause a large scale extinction, and might change conditions
suﬃciently that the evolutionary clock would eﬀectively be set back t o an early stage where
ecosystems were poorly adapted to the environment. This would en able a new burst of evo-
lution, with many novel species arising. This suggests a picture of th e real world where there
is a considerable rate of inherent evolutionary change and consider able ﬂuctuation in species
numbers, and where external events causing major changes hap pen suﬃciently often to pre-
vent evolutionary stagnation from occurring. The data in the foss il record may some-day be
complete enough to give some answers these questions. In the mea ntime there are still many
useful issues which can be addressed by studying models such as ou rs.
7 Conclusions
The Webworld model describes the interactions of coevolving specie s. The model makes
predictions concerning the structure of food webs which can be co mpared with data on real
webs. There are only three parameters in the model — R, λ, and δ — hence, the number of
testable predictions is much larger than the number of parameters . The measured quantities
such as the number of links per species, the number of trophic levels , and the proportions of
top, basal and intermediate species, are not far from the values o bserved in real food webs in
most cases. It would be possible to choose parameters so that the results match a particular
set of real food web data as closely as possible. However, given the uncertainties in most of
the present food web data, we have not tried to match these data too closely. Instead we have
tried to point out the major qualitative trends in food web propertie s which occur when the
parameters are changed. These trends make sense from an ecolo gical point of view. We would
like to contrast our model with the cascade model of food web stru cture (Cohen, 1990; Cohen
et al., 1990). Even though the cascade model successfully describ es many food web properties,
it is basically just a set of probabilistic rules for assigning links between nodes in a graph.
The justiﬁcation of these rules comes entirely from comparison mod el webs with real data.
In contrast, the parameters in our model have an ecological mean ing: the available external
resources, the fraction of resources transferred from prey t o predator, and the strength of
competition are all meaningful quantities in real webs.
An important question in food web theory which has had considerable attention recently
is the issue of food web assembly (Luh & Pimm, 1993; Morton & Law, 19 97). In assembly
models species are invading the ecological community from an extern al species pool and are

therefore unrelated to existing species, whereas in the Webworld m odel new species are arising
by evolution and are therefore similar to the species from which they evolve. We intend to
develop the model to compare properties of webs generated by inv asion and by evolution. One
observation in assembly models is that the probability of successful invasion of a community
tends to decrease with time. This seems to have a parallel in Webworld , where the survival
probability of a newly generated species tends to decrease with time .
This work was motivated in part by the wish to explore recent claims th at models of
evolution may have a tendency to “self-organise” into a critical non -equilibrium state which
has avalanches on all scales. We approached the problem by trying t o design a realistic model
for co-evolution in which the evolutionary dynamics can be studied, r ather than by deliberately
designing a very simple model with interesting dynamics, but which is mo re diﬃcult to relate
to real evolutionary and ecological phenomena. This article has con centrated on the ecological
properties of the food webs, and we are currently investigating th e evolutionary properties of
Webworld in more detail.
Acknowledgements
We thank Mark Huxham for clarifying some details regarding the Ytha n web. This work
was supported in part by a grant from the University of Mancheste r and by EPSRC grant
GR/K/79307.
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level 2
level 3
level 1
Figure 1: An illustrative example of a food web. Arrows indicate the dir ection of ﬂow
of resources. Input of external resources is speciﬁcally indicate d. Basal species are black,
intermediate species are white, and top species are patterned. Bo xed species form part of the
same trophic species. The trophic level of a species is the length of t he shortest food chain
from the external resources to that species.
0 1 2 3 4 5 6 7 8 9 10
Level
0.00
20.00
40.00
60.00
80.00
Number of species
R=10
R=10
R=10
Figure 2: The mean number of species on each trophic level with λ = 0 . 1, δ = 0 . 05 and
R = 10 3, 106 and 10 10.

0.0 50000.0 100000.0 150000.0 200000.0
time
0.0
100.0
200.0
300.0
number of species
Figure 3: The number of species in the web is shown as a function of tim e for three diﬀerent
runs of the Webworld model with the same parameters R = 10 6, δ = 0. 05 and λ = 0. 1.


Invariant closure for the Fokker-Planck equation

Abstract
We develop the principle of dynamic invariance to obtain closed moment equations from the Fokker-Planck kinetic equation. The analysis is carried out to explicit formulae for computation of the lowest eigenvalue and of the corresponding eigenfunction for arbitrary potentials.

arXiv:adap-org/9801004v3 4 Feb 1998
Invariant closures for the Fokker–Planck equation
Iliya V. Karlin ∗
Computing Center RAS, Krasnoyarsk 660036, Russia
V. B. Zmievskii †
LMF/DGM, Swiss Federal Institute of Technology, CH-1015 Lausa nne, Switzerland
We develop the principle of dynamic invariance to obtain clo sed moment equations from the
Fokker–Planck kinetic equation. The analysis is carried ou t to explicit formulae for computation
of the lowest eigenvalue and of the corresponding eigenfunc tion for arbitrary potentials.
05.20.Dd, 05.70.Ln, 83.10.-y
The Fokker–Planck equation (FPE) is a familiar model in various proble ms of nonequilibrium statistical
physics [1]. In this paper we consider the FPE of the form
∂tW = ∂x ·{D · [W ∂xU + ∂xW ]} . (1)
Here W (x, t ) is the probability density over the conﬁguration space x, at the time t, while U (x) and D(x) are
the potential and the positively semi-deﬁnite ( y · D · y ≥ 0) diﬀusion matrix. The dot denotes convolution in
the conﬁguration space. The FPE (1) is particularly important in stu dies of polymer solutions [2]. Let us recall
the two properties of the FPE (1), important to what will follow: (i). Conservation of the total probability:∫
W (x, t )dx = 1 . (ii). Dissipation: The equilibrium distribution, Weq ∝ exp(−U ), is the unique stationary
solution to the FPE (1). The entropy,
S[W ] = −
∫
W (x, t ) ln
[ W (x, t )
Weq(x)
]
dx, (2)
is a monotonically growing function due to the FPE (1), and it arrives a t the global maximum in the equilibrium.
These properties are most apparent when the FPE (1) is rewritten as follows:
∂tW (x, t ) = ˆMW
δS[W ]
δW (x, t ) , (3)
where ˆMW = −∂x · [W (x, t )D(x) · ∂x] is a positive semi–deﬁnite symmetric operator with kernel 1. The fo rm
(3) (the dissipative vector ﬁeld is a metric transform of the entrop y gradient) is an example of the dissipative
part of a structure termed GENERIC in a recent series of papers [3 ].
Usually one is interested in dynamics of moments of the distribution fu nction W rather than in the dynamics
of the W itself. Except for simplest potentials U and diﬀusion matrices D, the moment equations, as they
follow from the FPE (1), are not closed. Therefore, closure proce dures are required.
In this paper we address the problem of closure for the FPE (1) in a g eneral setting. First, we review the
maximum entropy principle (MEP) as a source of suitable initial approx imations for the closures. We also
discuss a version of the MEP, valid for a near–equilibrium dynamics, an d which results in explicit formulae for
arbitrary U and D.
The MEP closures are almost never invariants of the true moment dy namics, and corrections to the MEP
closures is the central issue of this paper. For this purpose, we ap ply the method of invariant manifold [4],
which is carried out (subject to certain approximations explained be low) to explicit recurrence formulae for
one–moment near–equilibrium closures for arbitrary U and D. These formulae give a method for computing
the lowest eigenvalue of the problem, and which dominates the near– equilibrium FPE dynamics.
MEP closures [5]. Let M = {M0, M 1, . . . , M k} be linearly independent moments of interest, Mi[W ] =∫
mi(x)W (x)dx, and where m0 = 1. We assume existence a function W ∗ (M, x ) which extremizes the entropy
S (2) under the constraints of ﬁxed M . This MEP distribution function may be written
W ∗ = Weq exp
[ k∑
i=0
Λ imi(x) − 1
]
,
where Λ = {Λ 0, Λ 1, . . . , Λ k} are Lagrange multipliers. Closed equations for moments M are derived in two
steps. First, the MEP distribution is substituted into the FPE (1) or (3) to give a formal expression: ∂tW ∗ =
ˆMW ∗ (δS/δW )
⏐
⏐
W =W ∗ . Second, applying a projector Π ∗ ,
Π ∗ • =
k∑
i=0
(∂W ∗ /∂M i)
∫
mi(x) • dx,

on both sides of this formal expression, we derive closed equations for M in the MEP approximation. Further
processing requires an explicit solution to the constraints, ∫ W ∗ (Λ , x )mi(x)dx = Mi, to get the dependence of
Lagrange multipliers Λ on the moments M . Though typically the functions Λ( M ) are not known explicitly,
one general remark about the moment equations is readily available. Speciﬁcally, the moment equations in the
MEP approximation have the form:
˙Mi =
k∑
j=0
M ∗
ij(M ) ∂S ∗(M )
∂M j
, (4)
where S∗ (M ) = S[W ∗ (M )] is the macroscopic entropy, and where M ∗
ij is an M -dependent ( k + 1) × (k + 1)
matrix:
M ∗
ij =
∫
W ∗ (M, x )[∂xmi(x)] · D(x) · [∂xmj(x)]dx.
The matrix M ∗
ij is symmetric, positive semi–deﬁnite, and its kernel is the vector δ0i. Thus, the MEP closure
reproduces the GENERIC structure on the macroscopic level , the vector ﬁeld of macroscopic equations (4) is a
metric transform of the gradient of the macroscopic entropy.
Triangle MEP closures [6]. The following version of the MEP makes it possible to derive more exp licit results
in a general setting: In many cases, one can split the set of moment s M in two parts, MI = {M0, M 1, . . . , M l}
and MII = {Ml+1, . . . , M k}, in such a way that the MEP distribution can be constructed explicitly for MI
as W ∗
I (MI , x ). The full MEP problem for M = {MI, M II } in the ”shifted” formulation reads: extremize the
functional S[W ∗
I + ∆ W ] with respect to ∆ W , subject to the constraints MI [W ∗
I + ∆ W ] = MI and MII [W ∗
I +
∆ W ] = MII . Let us denote as ∆ MII deviations of the moments MII from their values in the state W ∗
I . For
small deviations, the entropy is well approximated with a quadratic f unctional ∆ S[∆ W ] which is an expansion
of the functional (2) in the state W ∗
I up to the terms of the order ∆ W 2. With MI [W ∗
I ] = MI , we come
to the following problem: extremize the functional ∆ S[∆ W ], subject to the constraints MI [∆ W ] = 0, and
MII [∆ W ] = ∆ MII . The solution to the latter problem is always explicitly found from a ( k + 1)× (k + 1) system
of linear algebraic equations for Lagrange multipliers.
In the remainder of this paper we deal with one–moment near–equilib rium closures: MI = M0, (i. e. W ∗
I =
Weq), and the set MII contains a single moment M =
∫
mW dx, m(x) ̸= 1. We will specify notations for the
near–equilibrium FPE , writing the distribution function as W = Weq(1 + Ψ), where the function Ψ satisﬁes an
equation:
∂tΨ = W − 1
eq ˆJΨ , (5)
where ˆJ = ∂x · [WeqD · ∂x]. The triangle one–moment MEP function reads:
W (0) = Weq
[
1 + ∆ M m(0)
]
(6)
where ∆ M = M − ⟨m⟩, and
m(0) = [⟨mm⟩ − ⟨m⟩2]− 1[m − ⟨m⟩]. (7)
Brackets ⟨. . . ⟩ =
∫
Weq . . . dx denote equilibrium averaging. The superscript (0) indicates that th e triangle MEP
function (6) will be considered as an initial approximation to a proced ure which we address below. Projector
for the approximation (6) has the form
Π (0)• = Weq
m(0)
⟨m(0)m(0)⟩
∫
m(0) • dx. (8)
substituting the function (6) into the FPE (5), and applying the pro jector (8) on both the sides of the resulting
formal expression, we derive an equation for M : ˙M = −λ 0∆ M , where 1 /λ 0 is the inverse eﬀective time of
relaxation of the moment M to its equilibrium value, in the MEP approximation (6):
λ 0 = ⟨m(0)m(0)⟩− 1⟨∂xm(0) · D · ∂xm(0)⟩. (9)
Invariant closures. Both the MEP and the triangle MEP closures are almost never invarian ts of the FPE
dynamics. That is, the moments M of solutions to the FPE (1) vary in time diﬀerently from the solutions t o the
closed moment equations like (4), and these variations are generally signiﬁcant even for the near–equilibrium
dynamics. Therefore, we ask for corrections to the MEP closures to ﬁnish with the invariant closures [4].
First, the invariant one–moment closure is given by an unknown distr ibution function W (∞ ) = Weq[1 +
∆ M m(∞ )(x)] which satisﬁes an equation

[1 − Π (∞ )] ˆJ m(∞ ) = 0. (10)
Here Π (∞ ) is a projector, associated with an unknown function m(∞ ), and which is also yet unknown. Eq. (10)
is a formal expression of the invariance principle for a one–moment n ear–equilibrium closure: considering W (∞ )
as a manifold in the space of distribution functions, parameterized w ith the values of the moment M , we require
that the microscopic vector ﬁeld ˆJ m(∞ ) be equal to its projection, Π (∞ ) ˆJ m(∞ ), onto the tangent space of the
manifold W (∞ ).
Now we turn our attention to solving the invariance equation (10) ite ratively, beginning with the triangle
one–moment MEP approximation W (0) (6). We apply the following iteration process to the Eq. (10):
[1 − Π (k)] ˆJ m(k+1) = 0, (11)
where k = 0, 1, . . . , and where m(k+1) = m(k)+µ (k+1), and the correction satisﬁes the condition ⟨µ (k+1)m(k)⟩ = 0.
Projector is updated after each iteration, and it has the form
Π (k+1)• = Weq
m(k+1)
⟨m(k+1)m(k+1)⟩
∫
m(k+1)(x) • dx. (12)
Applying Π (k+1) to the formal expression, Weqm(k+1) ˙M = ∆ M [1 − Π (k+1)] ˆJ m(k+1), we derive the macroscopic
equation, ˙M = −λ k+1∆ M , where λ k+1 is the ( k + 1)th update of the inverse eﬀective time (9):
λ k+1 = ⟨∂xm(k+1) · D · ∂xm(k+1)⟩
⟨m(k+1)m(k+1)⟩ . (13)
Specializing to the one–moment near–equilibrium closures, and followin g a general argument [4], solutions to
the invariance equation (10) are eigenfunctions of the operator ˆJ, while the formal limit of the iteration process
(11) is the eigenfunction which corresponds to the eigenvalue with t he minimal nonzero absolute value.
Diagonal approximation. To obtain more explicit results in the iteration process (11), we intro duce an
approximate solution on each iteration . The correction µ (k+1) satisﬁes the condition ⟨m(k)µ (k+1)⟩ = 0, and can
be decomposed as follows: µ (k+1) = α ke(k) + e(k)
ort. Here e(k) = W − 1
eq [1 − Π (k)] ˆJ m(k) = λ km(k) + R(k) is the
variance of the kth approximation, where
R(k) = ∂x · [D · ∂xm(k)] − ∂xU · D · ∂xm(k). (14)
The function e(k)
ort is orthogonal to both e(k) and m(k): ⟨e(k)e(k)
ort⟩ = 0, and ⟨m(k)e(k)
ort⟩ = 0. Our diagonal
approximation (DA) consists in disregarding the part e(k)
ort. Speciﬁcally, we consider the following ansatz at the
kth iteration:
m(k+1) = m(k) + α ke(k). (15)
Substituting the ansatz (15) into the Eq. (11), and integrating th e latter expression with the function e(k), we
evaluate the coeﬃcient α k:
α k = Ak − λ 2
k
λ 3
k − 2λ kAk + Bk
, (16)
where parameters Ak and Bk represent the following equilibrium averages:
Ak = ⟨m(k)m(k)⟩− 1⟨R(k)R(k)⟩ (17)
Bk = ⟨m(k)m(k)⟩− 1⟨∂xR(k) · D · ∂xR(k)⟩.
Finally, putting together Eqs. (13), (14), (15), (16), and (17), we arrive at the following DA recurrency
solution, and which is our main result:
m(k+1) = m(k) + α k[λ km(k) + R(k)], (18a)
λ k+1 = λ k − (Ak − λ 2
k)α k
1 + (Ak − λ 2
k)α 2
k
. (18b)
To test the convergency of the DA process (18) we have consider ed two potentials U in the FPE (1) with a
constant diﬀusion matrix D. The ﬁrst test was with the square potential U = x2/ 2, in the three–dimensional
conﬁguration space, since for this potential the detail structur e of the spectrum is well known. We have
considered two examples of initial one–moment MEP closures with m(0) = x1 + 100(x2 − 3) (example 1), and

m(0) = x1 + 100x6x2 (example 2), in the Eq. (7). The result of performance of the DA fo r λ k (18b) is presented
in the Table I, together with the error δk which was estimated as the norm of the variance at each iteration:
δk = ⟨e(k)e(k)⟩/ ⟨m(k)m(k)⟩. In both examples, we see a good monotonic convergency to the min imal eigenvalue
λ ∞ = 1, corresponding to the eigenfunction x1. This convergency is even striking in the example 1, where the
initial choice was very close to a diﬀerent eigenfunction x2 − 3, and which can be seen in the non–monotonic
behavior of the variance. Thus, we have an example to trust the DA as converging to the stationary point of
the original iteration procedure (11).
For the second test, we have taken a one–dimensional potential U = −50 ln(1 − x2), the conﬁguration
space is the segment |x| ≤ 1. Potentials of this type (so–called FENE potential) are used in applic ations of
the FPE to models of polymer solutions [2]. Results are given in the Table II for the two initial functions,
m(0) = x2 + 10x4 − ⟨x2 + 10x4⟩ (example 3), and m(0) = x2 + 10x8 − ⟨x2 + 10x8⟩ (example 4). Both the examples
demonstrate a stabilization of the λ k at the same value after some ten iterations.
In conclusion, we have developed the principle of invariance to obtain moment closures for the Fokker–Planck
equation (1), and have derived explicit results for the one–moment near–equilibrium closures, particularly
important to get information about the spectrum of the FP operat or.
I. V. K. is thankful to M. Grmela for providing a preprint of their pap er [3], and to H. C. ¨Ottinger for
stimulating discussions. V. B. Z. is thankful to J.–C. Badoux and M. D eville for the possibility of the research
stay at the EPFL. The work of I. V. K. was supported by CNR, and b y RFBR through grant No. 95-02-03836-a.
∗ The author to whom correspondence should be sent. Present ad dress: Istituto Applicazioni Calcolo CNR, V. del
Policlinico, 137, Roma I-00161, Italy. E-mail: iliya@momi x.iac.rm.cnr.it
† Permanent address: Computing Center RAS, Krasnoyarsk 6600 36, Russia
[1] N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam 1981); H. Risken,
The Fokker–Planck Equation (Springer, Berlin 1984).
[2] R. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids , 2nd edn., (Wiley, New
York, 1987); M. Doi and S. F. Edwards, The Theory of Polymeric Dynamics (Clarendon Press, Oxford, 1986); H. C.
¨Ottinger, Stochastic Processes in Polymeric Fluids (Springer, Berlin, 1996).
[3] M. Grmela and H. C. ¨Ottinger, Phys. Rev. E (1997) to appear.
[4] A. N. Gorban and I. V. Karlin, Transp. Theory Stat. Phys. 23, 559 (1994).
[5] The MEP was used by many authors for various systems, e. g. for the Boltzmann equation: A. M. Kogan, Prikl.
Math. Mech. 29, 122 (1965), for a recent thorough study see C. D. Levermore, J. Stat. Phys. 83, 1021 (1996); for
Markov chains: A. N. Gorban, Equilibrium Encircling. Equations of Chemical Kinetics an d their Thermodynamic
Analysis (Nauka, Novosibirsk, 1984); for the BBGKY hierarchy: J. Kar kheck and G. Stell, Phys. Rev. A 25, 3302
(1984); A very general discussion of the MEP is given in: R. Ba lian, Y. Alhassid and H. Reinhardt, Phys. Rep. 131,
1 (1986). We, however, failed to ﬁnd a reference for a direct a pplication of the MEP to the FPE.
[6] I. V. Karlin, in: Mathematical Problems of Chemical Kinetics , eds. K. I. Zamaraev and G. S. Yablonskii, (Nauka,
Novosibirsk, 1989), p. 7; A. N. Gorban and I. V. Karlin, Phys. Rev. E 54, R3109 (1996).
TABLE I. Iterations λ k and the error δk for U = x2/ 2.
0 1 4 8 12 16 20
Ex. 1 λ 1.99998 1.99993 1.99575 1.47795 1.00356 1.00001 1.00000
δ 0. 16 ·10− 4 0. 66 ·10− 4 0. 42 ·10− 2 0. 24 0. 35 ·10− 2 0. 13 ·10− 4 0. 54 ·10− 7
0 1 2 3 4 5 6
Ex. 2 λ 3.399 2.437 1.586 1.088 1.010 1.001 1.0002
δ 1. 99 1. 42 0. 83 0. 16 0. 29 ·10− 1 0. 27 ·10− 2 0. 57 ·10− 3
TABLE II. Iterations λ k for U = − 50 ln(1 − x2).
0 1 2 3 4 5 6 7 8
Ex. 3 λ 213.1774 212.1864 211.9148 211.8619 211.8499 211.8453 211.8433 211.8422 211.8417
Ex. 4 λ 216.5856 213.1350 212.2123 211.9984 211.9295 211.8989 211.8838 211.8757 211.8713