species positions and extinction dynamics_ferenc.j.theo.biol

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J. theor. Biol. (2002) 215, 441448

doi:10.1006/jtbi.2001.2523, available online at http://www.idealibrary.com on

Species Positions and Extinction Dynamics in Simple Food Webs

Ferenc Jordan*wz, Istvan Scheuringy and GaborVidaw

nCollegium Budapest, Institute for Advanced Study, Budapest H-1014, Szenth !aroms !ag u. 2., Hungary

wDepartment of Genetics, Evolutionary Genetics Research Group, E.otv .os University, Budapest H-1117,P !azm !any P. s. 1/c, Hungary and y Department of Plant Taxonomy and Ecology, Research Group of

Ecology and Theoretical Biology, E.otv .os University, Budapest H-1117, P !azm !any P. s. 1/c, Hungary

(Received on 22 June 2001, Accepted in revised form on 18 December 2001)

Recent investigations on the structure of complex networks have provided interesting resultsfor ecologists. Being inspired by these studies, we analyse a well-defined set of small modelfood webs. The extinction probability caused by internal LotkaVolterra dynamics iscompared to the position of species. Simulations have revealed that some global properties ofthese food webs (e.g. the homogeneity of connectedness) and the positions of species therein(e.g. interaction pattern) make them prone to modelled biotic extinction caused bypopulation dynamical effects. We found that: (a) homogeneity in the connectedness structureincreases the probability of extinction events; (b) in addition to the number of interactions,their orientations also influence the future of species in a web. Since species in characteristicnetwork positions are prone to extinction, results could also be interpreted as describing the

properties of preferred states of food webs during community assembly. Our results maycontribute to understanding the intimate relationship between pattern and process inecology.

r 2002 Elsevier Science Ltd. All rights reserved.

Introduction

Interesting ideas in ecology frequently come

from distant fields of science. The conclusions of

recent studies on the structure of complex

networks (Watts & Strogatz, 1998; Barab!asi &

Albert, 1999; Albert et al., 2000; Jeong et al.,2000; Amaral et al ., 2000; Williams et al .,

submitted), giving information about how meta-

bolic and neuronal networks seem to be orga-

nized, may also be useful for ecologists. An

interesting result is that metabolic networksfollow a power-law connectivity distribution

indicating robust functional insensitivity against

random errors (Jeong et al., 2000).

The dynamical behaviour of a well-defined set

of small sink webs (Cohen, 1978) has been

studied (Jord!an & Moln!ar, 1999; Jord!an et al.,

submitted), giving rise to the problem of how thenetwork position of frequently extinct species

can be characterized. Since the large-scale

structure of biological networks depends highly

on internal dynamical processes and constraints

(Sugihara, 1984), we analyse how the internal

extinction dynamics (based on LotkaVolterra

models) in these webs is related to the topology

of trophic interactions. We think that these basic

equations, even if their applicability is limited,

are good as a starting point for further analyses.zAuthor to whom correspondence should be addressed.

E-mail: [email protected]

0022-5193/02/$35.00/0 r 2002 Elsevier Science Ltd. All rights reserved.


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Methods

We compared the extinction dynamics of

species constituting 25 model sink webs (i.e. food

webs with a single top-predator, see Fig. 1). This

set of webs contains all of the 25 topologicallydifferent model sink food webs possibly built up

from five species (graph nodes) and five trophic

interactions (links). Since the connectedness of

these webs equals uniformly C 0.5, we are able

to analyse the effect of link pattern on population

dynamics, distinguished from the effects of

changing connectedness. Food webs and indivi-

dual species are characterized by the following

structural and dynamical properties.

The degree (D) of a graph node is the number

of its neighbours (both prey and predators in the

food web graph). Because all these webs contain

five links, the sum of D values (sumD) equals

10 for every web. However, the D values of

individual species differ; this difference is mea-

sured by the standard deviation of D within a

web (stdD).

The orientation of neighbours is characterized

by D0DinDout (where Din is the in-degree: the

number of links directed to a particular node,

and Dout is the out-degree: the number of links

coming from a particular node). Low and high

values of D0 characterize multiple exploitedproducers and generalist consumers, respec-

tively, in the directed graph.

The concept of net status (Harary, 1959,

1961) has been modified slightly in order to be

ecologically more reasonable (see Jord!an et al.,

1999; Jord!an, 2000, 2001). The modified index

was constructed to give a quantitative measure

of the importance of species (species of

outstanding importance are called keystone

species, Paine, 1969; Power et al., 1996). This

is the positional keystone index (K) which

considers not only neighbours in the web (like

D) but their neighbours as well, thus providing

more information about network structure. If

bottom-up and top-down forces are considered

to be of equal importance, then let Ki Kib

Kit ; where K

ib and K

it refer to the bottom-up

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Fig. 1. Twenty-five model sink webs containing five species (nodes) and five trophic links. Both species and webs arenumbered for further reference. Higher species consume lower ones (species #1 is always the top-predator; the direction oflinks is not shown for simplicity). For data on structural (D, D0, K) and dynamical (E) characterization of species, consulthttp://falco.elte.hu/Bjordanf/data2.prn.

F. JORDA N ET AL.442


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and top-down effects. These indices count the

number of disconnected species after the re-

moval of species i. If species i is in an important

network position it has a rich system of trophic

interactions, then deletion will lead to more

disconnections in both directions. The effects ofspecies deletion on the disconnection of both

bottom-up material flows and top-down flows of

trophic regulatory effects are quantified by these

indices of positional importance. The Kb value of

the i-th species can be calculated as

Kib

Xnc1

1

dc

1

dcK

cb

; 1

where n is the number of its predators, dc is the

number of prey eaten by its c-th predator, andKc

b is the bottom-up keystone index of its c-th

predator. Thus, the bottom-up index should be

calculated first for top species. Kt is calculated

similarly after turning the web upside down. It

should be emphasized that K refers to keystone

species only in a trophic sense (i.e. importance in

maintaining trophic network flows). K charac-

terizes food web pattern at an intermediate scale,

between the local (measured by D and D0) and

global (measured by stdD) properties of webs.

Nevertheless, in our small webs K seems to be

a rather global index. Both the sum and thestandard deviation of K-values within webs

(sumK and stdK) were also calculated for

analyses.

We are interested in whether these structural

indices are possibly as informative and interest-

ing as traditional food web statistics having been

used extensively in the literature (e.g. trophic

height, compartmentalization, the level of

omnivory, etc., see Pimm, 1980).

The behaviour of species is characterized by

dynamical LotkaVolterra models. Population

dynamics is described by the following set of

equations:

xi aiXnj1

bijxj

!xi; i 1;y; n: 2

Here, xi is the population density of species i,

ai is the per capita rate of reproduction (for

growth, ai40, for decay, aio0). The parameter

bij indicates the per capita effect of species

j on the per capita reproduction rate of species

i. Here, ai is a uniformly distributed random

variable in the range (0 1) if species i is a basal

species, while ai is selected randomly from (1 0)

if species i is intermediate or top species.

Similarly, bij falls into (0 1) if i is the pre-dator of j, and bji is selected from (1 0) if i

eats j. If ij, bij is selected from (1 0) in case

of producers but equals zero for higher species

(May, 1973; Pimm, 1982). The dynamics of

system (2) is simulated with 2000 random

parameter sets selected in the way described

above. We considered a species to be extinct

if its density fell below a critical level (here,

d1015; results do not differ qualitatively

for other values of d) after a given time (here,t 200 steps). The extinction value (E) o f a

species gives the number of extinctions out of

2000 simulations (we suggest that our approach

to stability is more general than to study stable

fix points).

Results

The distribution of the basic structural indices

(D and K) of species in these model webs is

shown in Fig. 2: the majority of species have two

neighbours (D 2), while K is distributed evenly

(especially if sink species are not considered).The most frequently extinct nodes can be

characterized by an intermediate degree (D 2,

see Fig. 3). The keystone index itself does not

seem to be in correlation with the probability of

extinction (Fig. 3; but seemingly extinctions are

not frequent with very low K). However, the sum

of extinction events within a web is clearly higher

if the sum of keystone indices is higher (Fig. 3).

The fact that Kcorrelates with Eonly at the level

of whole webs reflects how keystone index is

calculated: it also quantifies indirect effects,which are of more global nature than locally

understandable direct interactions. In order to

study the effects of both D and K on extinction

dynamics, they are plotted together (Fig. 4). If

D 1, species go extinct with high probability if

K 4; however, the probability of extinction is

low at smaller K indices. If D 2, extinction can

be very probable only with low Kvalues. In case

of a dangerous number of neighbours (i.e. if

D 2), additional indirect effects (e.g. K 4) may

EXTINCTION DYNAMICS IN FOOD WEBS 443


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decrease the probability of dynamical extinction.

If D 3 or 4, extinction is less probable and

is less sensitive to K. Thus, measuring both

exclusively direct (D) and direct plus indirect (K)

effects gives an interesting extinction probability

landscape.

Fig. 2. The frequency distribution for the degree of nodes is a discrete, unimodal distribution (on the left). K values areevenly distributed (on the right; note that for all nodes of the last column K 4: 25 out of 47 nodes represent the sink species;thus, basal and intermediate species are distributed much more evenly).

Fig. 3. Relationships between structural and dynamical properties: top left, the number of extinction events concerningeach species (E, out of 2000 simulations) is plotted against the number of their neighbours (D, ( ) show the average; theMannWhitney test on a distribution of 105 randomly generated samples shows significance at po0.0003, po0.0114, and

po0.0025 for differences between [D 1, 2], [D 2, 3], and [D 3, 4] pairs of categories, respectively); top right, the numberof extinction events (E) is plotted against species keystone indices (K); bottom, the sum of extinction events (sumE) withineach web is plotted against the sum of their species keystone indices (sumK).



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nodes). These positional properties of nodes may

characterize either extinction risk (as far as

randomly chosen interaction parameters are

regarded as perturbations) or preferences in

community assembly (as far as possible struc-

tures created by invaders are compared). If

positional properties (D, D0, K) influence extinc-

tion dynamics (E), species do not die out

randomly, but rather in an internally directed

way (alike attacks are defined as directed node

deletions in large networks, see Albert et al.,

2000). For this set of small sink webs, it can

be concluded that the presence of species in key

positions can be dynamically either advanta-

geous [if only direct interactions are considered

(see stdD in Fig. 6)] or disadvantageous [if

both direct and indirect interactions are

taken into account (see Fig. 3)] against these

internal attacks [in contrast with Albert et al.

(2000), where non-random deletions are more

Fig. 5. The number of species extinction events (E) is plotted against species D0 indices ( ) show the average; seeexplanation in text): not only the number but also the orientation of direct trophic links influence extinction dynamics. The

MannWhitney test on a distribution of 105

randomly generated samples shows significant difference between [D0

1, 0] atpo0.00001. [D00, 1] and [D01, 2] show different average E at po 0.1. D02 and 3 differ significantly at po0.003, while[D0 2, 1] and [D03, 4] do not differ significantly.

Fig. 6. The sum of extinction events (sumE) for each web is plotted against the standard deviation of degrees ( stdD,( )) and the standard deviation of keystone indices (stdK, ( )) of species. Living in a homogeneous web is moredangerous.



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dangerous in heterogeneous networks]. The

description of extinction-prone network posi-

tions is suggested to identify unpreferred struc-

tural arrangements during web assembly

(cf. Sugihara, 1984): the success of invasion

may depend characteristically on the particularposition of the invader within the community

network (and this may hold also in case of

speciation).

The main mechanisms behind the simulated

network dynamics include direct (e.g. predation,

food supply) and indirect (e.g. exploitative

competition, trophic cascade) trophic interac-

tions among species (see a catalogue of indirect

interactions in Menge, 1995). Given the topology

of a trophic network, one may predict either the

nature of a particular interaction or the fate of a

particular species, but the dynamics of whole

webs remain unclear until simulated. Only joint

dynamical equations can give higher-level pre-

dictions (e.g. the fate of a whole web). Our

simulation results show typical patterns in the

dynamics of food webs and their species. For

example, let us consider food web #12 in Fig. 1.

This web can be considered typical as for the

frequency of extinctions (1760 extinction events

occurred out of 2000 simulations). The extinc-

tion frequency of the top-predator (#1, E482)

is close to the average. Its position is neither toorisky nor very safe. One of its prey (#2, E832)

is prone to extinction, as are species with D00

in general. They have a single source to consume

and their fate depends strongly on their pre-

dators behaviour. Its other prey (#4, E241)

has a high extinction probability compared to

other producers. It is caused by the following: if

the top-predators density increases, it exploits

species #4 both directly (by predation) and

indirectly (by having a negative effect on species

#2, which is good for species #3, which is badagain for species #4). Species #3 (E 87) is well

saved by topology: it has a single predator and

two preys. Finally, there is a producer in this

community (#5, E 118) which goes to

extinction less frequently than the other one

(#4), because its position is better as for the

number of predators.

We can conclude that dynamically caused

species extinctions are less frequent if the

following conditions are satisfied: at the local

scale, (1) species have either a single or many

(3 or 4) direct trophic links (see, for example,

species #5 and #1 of web #1, Fig. 1), (2) the

neighbours are clustered in either bottom-up or

top-down direction (D0a0, see species #2 in web

#3); at an intermediate scale, (3) they have eithera single neighbour and a low K-index or two

neighbours and a high K-index (see species #5 in

web #7 and species #5 in web #21), (4) the sum

of K-indices within a web is low (see web #3),

and, at the global scale, (5) whole webs have

differently connected nodes (heterogeneity, stdD,

see web #2). We have to emphasize that the

simplest dynamical model was analysed: asym-

metries in interaction strength (bij parameters)

may lead to qualitative differences in our

conclusions (Jord!an et al., submitted).

If trophic control in a particular community is

considered of primary importance, i.e. food web

studies have strong predictive power, then

linking trophic structure to species dynamicscould provide useful information for conserva-

tionists. The quest for keystone species (Poweret al., 1996) is a big challenge for community

ecologists. Quantitative approaches to identify-

ing keystone species in ecosystems (e.g. based on

food web position) could increase the predictive

power of studies on the relative importance of

species within communities. In addition, if key-stones really make communities less prone to

extinction, we can get closer to explaining their

outstanding importance. Thus, the prediction

of extinction dynamics by food web analysis

could give a useful tool for conservationists

(cf. Simberloff, 1998). If biological diversity can

be expressed in terms of positional diversity of

species in trophic networks, then the diverse

population dynamics of species can also be

approached quantitatively.

We focused only on a particular but well-defined set of small food webs. Thus, possible

effects of scale- and context-dependence were

neglected. First, it should be clarified whether

our results based on the analysis of small webs

would also be applicable for large networks (see

Bersier & Sugihara, 1997). Second, the problem

of how the dynamics of community webs differs

from that of sink webs is to be studied. The

behaviour of persistent nodes following pertur-

bations is to be analysed in detail (some

EXTINCTION DYNAMICS IN FOOD WEBS 447


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preliminary results on whole webs are given in

Jord!an et al., submitted). Further, the manner in

which certain network properties affect basic

organizational principles, similar to the ageing

and cost of nodes in the study ofAmaral et al.

(2000), should be considered. Nevertheless, wethink that this study can contribute to the

understanding of the pattern and process

problems in ecology. Future studies on the

homogeneity of ecological networks may reveal

important aspects of the organization of food

webs, and, in general, the ecological principles of

community assembly and biotic extinctions.

We are grateful to J!anos Podani for his help instatistics and correcting English. F. J. also thanksProfessor E .ors Szathm!ary for enabling a fruitful stay

at Collegium Budapest, Mauro Santos and EliasZintzaras for some methodical help. Two anonymousreviewers are kindly acknowledged for helpful com-ments on the manuscript. F. J. is supported by twogrants of the Hungarian Scientific Research Fund,OTKA F 029800 and OTKA F 035092. I. Sch. issupported by the grant OTKA T 029789. Both F. J.and I. Sch. are recipients of the Bolyai ResearchGrant of the Hungarian Academy of Sciences.

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