Global Stability, Local Stability and Permanence in Model...

J. theor. Biol. (2001) 212, 223}235doi:10.1006/jtbi.2001.2370, available online at http://www.idealibrary.com on

Global Stability, Local Stability and Permanence in Model Food Webs

XIN CHEN*- AND JOEL E. COHEN?

*¸aboratory of Populations, ¹he Rockefeller ;niversity, 1230 >ork Avenue, Box 20, New >ork,N> 10021-6399, ;.S.A. ?¸aboratory of Populations, ¹he Rockefeller ;niversity andColumbia ;niversity, 1230 >ork Avenue, Box 20, New >ork, N> 10021-6399, ;.S.A.

(Received on 12 March 2001, Accepted in revised form on 10 June 2001)

The dynamical theory of food webs has been based typically on local stability analysis. Therelevance of local stability to food web properties has been questioned because local stabilityholds only in the immediate vicinity of the equilibrium and provides no information about thesize of the basin of attraction. Local stability does not guarantee persistence of food webs instochastic environments. Moreover, local stability excludes more complex dynamics such asperiodic and chaotic behaviors, which may allow persistence. Global stability and permanencecould be better criteria of community persistence. Our simulation analysis suggests that thesethree stability measures are qualitatively consistent in that all three predict decreasing stabilitywith increasing complexity. Some new predictions on how stability depends on food webcon"gurations are generated here: a consumer}victim link has a smaller e!ect on the probabil-ities of stability, as measured by all three stability criteria, than a pair of recipient-controlledand donor-controlled links; a recipient-controlled link has a larger e!ect on the probabilities oflocal stability and permanence than a donor-controlled link, while they have the same e!ect onthe probability of global stability; food webs with equal proportions of donor-controlled andrecipient-controlled links are less stable than those with di!erent proportions.

( 2001 Academic Press

Introduction

The relationship between stability and complex-ity of food webs has been a central issue in theor-etical ecology. An in#uential theory, based on thelocal asymptotic stability (LAS) analysis of ran-domly assembled Lotka}Volterra model foodwebs, suggested that complexity reduces stability(May, 1972). This theory has been challenged fortwo reasons.

First, LAS may not be an appropriate criterionfor food web persistence. LAS holds only in anarbitrarily small neighborhood of the equilib-rium (Lewontin, 1969; Haydon, 1994) and

-Author to whom correspondence should be addressed.E-mail: [email protected]

0022}5193/01/180223#13 $35.00/0

depends on the interaction coe$cients which couldvary considerably in real food webs. Moreover,LAS excludes complex dynamical behaviors suchas periodic and chaotic solutions, which may beconsistent with community persistence (Hutson &Law, 1985; Anderson et al., 1992; Law & Blackford,1992; Law & Morton, 1993; Hastings, 1988,1996).

Second, randomly generated communitymatrices may allow biologically unrealistic struc-tures such as an absence of autotrophs (Lawlor,1978; Lawton, 1989; Hall & Ra!aelli, 1993).Many e!orts have been made towards incorpor-ating structural features of real food webs intothe pool of community matrices of dynamic mod-els (DeAngelis, 1975; Yodzis, 1981; Pimm, 1982),

( 2001 Academic Press

224 X. CHEN AND J. E. COHEN

including the &&Lotka}Volterra cascade model''(LVCM, Cohen et al., 1990b). The LVCM com-bines the trophic structure of the cascade modelwith the population dynamics of the Lotka}Vol-terra model. The model distinguishes among fourtypes of dynamical e!ects caused by feeding links,namely, consumer}victim, donor-controlled andrecipient-controlled interactions and links withno dynamical e!ects. The dynamical e!ects ofconsumer}victim interactions received much at-tention in previous studies. Some authors (Pimm,1982; Hawkins, 1992) suggested that donor-con-trolled interactions are quite common in natureand that they have normally less destabilizinge!ects than consumer}victim interactions. Moredetailed work has yet to be done on the e!ects ofthese di!erent types of interactions on stability ina more general context of complex food webs.

Qualitative global asymptotic stability (QGAS)and permanence are two alternative measures ofstability. QGAS depends only on the sign patternof the community matrix, but is independent ofthe values of the interaction coe$cients as well asof the initial states (Cohen et al., 1990b). Perma-nence measures the boundedness of the trajectoryof a system within the region of state space whereall species have positive abundances. Perma-nence includes complex dynamical behaviorssuch as periodic and chaotic motions (Law& Blackford, 1992; Law & Morton, 1993;Morton et al. 1995). Previous investigations ofthe permanence of Lotka}Volterra systems havemainly focused on community assembly (Law& Blackford, 1992; Law & Morton, 1993, 1996).They did not provide an explicit and systematicmeasurement of the probability of permanence inrelation to the complexity of food webs.

The purposes of this paper are to examine,"rst, whether these three criteria of communitypersistence, QGAS, LAS, and permanence,predict qualitatively consistent relationshipsbetween food web stability and complexity, andsecond, how the stability determined by thesethree criteria are a!ected by the con"guration ofdi!erent types of trophic links. We use theLVCM with a "nite number of species as themodel system for this investigation. Untilrecently (Chen & Cohen, 2001), the LVCM wasstudied only in the limit of an in"nite number ofspecies (Cohen et al., 1990b).

The Model and Methods of Analysis

THE LOTKA}VOLTERRA CASCADE MODEL

In the Lotka}Volterra model,

xRi"x

iAei#n+j/1

pijxjB, x

i(0)'0, i"1,2,n,

(1)

where xiis the abundance or biomass of species i,

xRiis the derivative with respect to time, e

iis the

intrinsic rate of increase or decrease of species i inthe absence of all other species, p

ijis the inter-

speci"c interaction coe$cient between speciesi and species j. All the p

ijmake up the community

matrix P"(pij)ni, j/1

. We assume that the system[eqn (1)] has a positive equilibrium, i.e., a con-stant n]1 vector Q"(q

i)ni

such that ei#

+nj/1

pijqj"0, with q

i'0 for all i.

A LVCM system is a Lotka}Volterra modelwith its trophic structure de"ned by rules of thecascade model: for each pair of species i, j"1,2,nwith i(j, species i has a probability of zero ofeating species j, while species j has a probabilityof c/n (0(c(n) of eating species i (Cohen &Newman, 1985; Cohen et al., 1990a, b).

The LVCM supposes that one of the fourdynamical e!ects occurs between each pair ofspecies, independently for each pair of speciesi, j"1,2,n with i(j, with probabilities:

(i) recipient-controlled interaction, PrMpij(0

and pji"0N"r/n;

(ii) donor-controlled interaction, PrMpij"0

and pji'0N"s/n;

(iii) consumer}victim interaction, PrMpij(0

and pji'0N"t/n;

(vi) neither species has a dynamical e!ect onthe other, or there is no trophic link betweenspecies i and j, PrMp

ij"0 and p

ji"0N"1!

(r#s#t)/n.

Here r, s, t*0 and r#s#t)n. The LVCMalso assumes that all the species are self-limited(Cohen et al., 1990b), i.e., PrMp

ii(0N"1,

i"1,2,n. Biologically, a recipient-controlledlink means that the predator diminishes the rateof increase of the prey population but does notenjoy any augmentation in the rate of increase ofits own population as a result of feeding on the

GLOBAL STABILITY, LOCAL STABILITY AND PERMANENCE 225

prey. This situation is typically seen in the rela-tionship between a generalist predator speciesand a prey species. A donor-controlled linkmeans that the predator bene"ts from feeding ona prey, but the prey su!ers little damage fromthe predator. This relation is quite common inplant}herbivore and host}parasite interactions.A consumer}victim link helps the consumer andhurts the victim; this is the common reciprocalrelationship between a predator and its prey.

In this study, all sample food webs are stochas-tically assembled. Hence we de"ne a food webcon"guration as the vector of probabilities (r/n,s/n, t/n) and not as the particular realization oftrophic links in a simulation. We denote Cr"r/n,Cs"s/n and Ct"t/n as the partial connectanceof r, s and t links, respectively.

There are two ways to count links (¸) and twocorresponding de"nitions of connectance (C) inthe LVCM (Cohen et al., 1990b). Connectancerefers to the expected fraction of all possible linksthat actually occur. If food webs are lookedupon as undirected graphs, then only a single un-directed link is recognized between any twointeracting species and, hence, the undirectedconnectance is de"ned by C

u"E(¸

u)/[n(n!1)/2]

(¸uis the number of undirected links and E(¸

u) is

the expected number of undirected links). A t linkis counted as a single undirected link, equivalentto an r or s link (r&s&t), which gives C

u"

(r#s#t)/n, E(¸u)"n (n!1)C

u/2"(n!1)

(r#s#t)/2. If food webs are regarded as di-rected graphs and cannibalism is ignored, thenthe directed connectance is de"ned by C

d"

E (¸d)/[n(n!1)]. A t link is counted as two

directed links, equivalent to an r link plus ans link (t&r#s), which gives C

d"(r#s#2t)/(2n),

E (¸d)"n (n!1)C

d"(n!1) (r#s#2t)/2.

SIMULATION

Numerical simulations are conducted on ran-domly constructed LVCM food webs withn"10 species and varying connectance to deter-mine the probability of stability, as measured byQGAS, LAS and permanence, respectively, inrelation to connectance and con"guration ofmodel food webs. The probabilities r/10, s/10 andt/10 are varied systematically. For each combi-nation of the probability values, a random

community matrix P"(pij)ni, j/1

is generated.Each non-zero element of P"(p

ij) is assigned

a uniformly distributed random value within theinterval of (0, 1) for each non-zero p

jiand a value

within the interval of (!1, 0) for each non-zeropij

and each piifor all i, j with i(j. Assuming that

a positive equilibrium vector Q"(qi)ni/1

exists,the Jacobian matrix of the LVCM is Df"diag (Q) )P. For simplicity, we chose q

i"1, for

i"1,2,n, so that Df"P. This approach isequivalent to normalizing the system by repla-cing x

iby y

i"x

i/q

ifor all i. This makes the equi-

librium of yibe 1 for all i and replaces the matrix

of interaction coe$cients P by P@"Pdiag (Q).The Jacobian matrix at the equilibrium 1 for thenormalized system is then P@. We just generate P@within the interval of (0, 1). Both endpoints 0and 1 of the uniform distribution are arbitrarychoices in this case.

For each con"guration of r, s and t, 10,000such stochastic systems are generated, and therelative frequency with which these systems arequalitatively globally asymptotically stable andlocally asymptotically stable is used to approxim-ate the probability of having QGAS and LAS.Similarly, we also "x the connectance (Ct"0.4)and vary the number of species (n) to determinethe probability of stability in relation to the num-ber of species in model food webs. The upperbound of the standard deviation of each relativefrequency p is

S.D."Jpq/n)J0.5 ) 0.5/10000"0.005.

The identi"cation of permanence is computation-ally expensive, so we simulate only 1000 samplesystems to estimate the probability of perma-nence (except for Table 1).

MEASURES OF STABILITY

Several conditions determine the QGAS ofthe Lotka}Volterra model (Quirk & Ruppert,1965; Bone et al., 1988; Redhe!er & Zhou, 1989;Logofet, 1993). Logofet listed "ve conditions (1993)for determining the QGAS of the Lotka}Volterramodel:

(i) pit)0 for all i and at least one p

ii(0;

(ii) pijpji)0 for any iOj;

(iii) the digraph D (P) has no k-cycles for k*3;

TABLE 1Frequency of randomly assembled ¸otka }<olterracascade model systems having local asymptoticstability (¸AS) and satisfying the su.cient condi-tion for permanence. ¹he model food webs all haveten species. <alues outside the parentheses are thefrequency of webs with t/10"0.5. <alues insidethe parentheses are the frequency of webs with

r/10#s/10"0.5 (and r/10"s/10)

¸AS

Yes No Total

Permanence Yes 7758 (2522) 122 (534) 7980 (3056)No 515 (183) 1605 (6761) 2020 (6944)

Total 8273 (2705) 1727 (7295) 10000


(iv) there exists a non-zero term in the stan-dard expansion of the determinant

det (P)"+ sgn (p)n

<i/1

pp (i)i , (2)

where p is a permutation of the numbersM1, 2,2,nN and sgn(p)"$1 is the sign of p;

(v) the trophic graph fails the color test.A trophic graph is an undirected graph witha node corresponding to each species and anundirected edge between any two nodes, if andonly if, at least one of the two correspondingspecies eats the other species. A trophic graph issaid to pass the color test if each of its vertices canbe colored black or white in such a way that: (1)all self-limited vertices are black; (2) there existwhite vertices, each of which is linked to at leastanother white vertex; (3) if a black vertex is linkedto a white one, then it is also linked to at leastanother white vertex. If not so, the graph fails thetest.

Conditions (i), (ii), (iv) and (v) are immediatelysatis"ed in the LVCM since p

ii(0 for all i and

pijpji)0 for any iOj. The determination of

QGAS in the LVCM reduces to checking that thedigraph of a LVCM food web has no k-cycles fork*3.

The LAS is determined by the eigenvalues ofthe Jacobian matrix of the LVCM. A system issaid to have LAS, if and only if, all the eigen-

values of its Jacobian matrix are negative or havea negative real part, i.e. Re(j

i)(0 for all i.

Permanence measures the ability of a systemto stay bounded inside the positive orthant ofthe state space. A system is said to be perma-nent if the boundary (include the in"nity) is arepeller (Hofbauer & Sigmund, 1998), that is,if there exists constants 0(d

l(d

usuch that

xi(0)'0 for all i"1,2,n implies d

l(

limt?`=

infxi(t)( lim

t?`=supx

i(t)(d

ufor

i"1,2,n (Hofbauer & Sigmund, 1998). Asu.cient condition for permanence is the exist-ence of an average Lyapunov function (Hofbauer,1981; Hutson, 1984; Jansen, 1987). For aLotka}Volterra system with a unique positiveequilibrium (x*

i'0 for all i ) and no trajectory

tending to in"nity, the test of an averageLyapunov function is reduced to solving the lin-ear programming problem (Jansen, 1987):

Minimize zsubject to

n+i/1

hiAe

i#

n+j/1

pijyB(k)j B#z*0,

and hi'0 for all i"1,2,n.

Here hi, i"1,2,n, and z are the variables in

the linear programming problem and yB(k)j

( j"1,

2,n and k"1,2,m) is the equilibrium densityof species j at the k-th boundary equilibrium. Ask"1,2,m, the m boundary equilibria yieldm linear constraints. The boundary is a repellorand, hence, the system is permanent if the opti-mum z

minis negative. For more information on

the su.cient condition for permanence of theLotka}Volterra systems, see Jansen (1987), Law& Blackford (1992) and Law & Morton (1993).We solved this linear programming problemusing software of the MOSEK optimizationtoolbox for Matlab (by EKA Consulting APS).

The second way to check whether the bound-ary is a repellor is numerical invasibility analysis,done by deleting species one at a time and seeingif the deleted species has a positive per-capita rateof change at the boundary equilibrium. The sec-ond approach could sometimes be computation-ally less expensive, but with e$cient linearprogramming software such as MOSEK, the "rstapproach is faster.

FIG. 1. Perspective view of the probability of stabilityin the Lotka}Volterra cascade model (LVCM), as measuredby di!erent stability criteria, as a function of r#t and s#t.(a) Asymptotic (nPR) probability of qualitative globalasymptotic stability (Pr(QGAS)) (redrawn from Cohen et al.,1990). (b) Pr (QGAS) in the LVCM with n"10 species.(c) Probability of local asymptotic stability (Pr (¸AS)) in theLVCM with n"10. Graphs (b) and (c) are plotted witht"0, each representing the lower bound of a family ofqualitatively similar surfaces.


A way to determine the persistence (not perma-nence) of a dynamical food web model is tosimulate numerically its trajectories from a largenumber of initial conditions. Unfortunately, forthis approach, some non-permanent systems mayhave a non-zero probability of survival for thenecessarily "nite duration of the numerical simu-lation while some permanent systems may notpersist when started from some starting points(Law & Morton, 1993). This numerical approachis computationally very expensive.

Currently, conditions that are both necessaryand su$cient for permanence in Lotka}Volterrasystems with more than three species arenot known. Jansen's (1987) condition is only asu.cient condition for permanence in Lotka}Volterra systems with more than three species. Itmay miss an unknown proportion of permanentsystems. For large Lotka}Volterra systems, it isso far not clear how large a proportion of perma-nent systems may go undetected (Law & Morton,1992). Below, when we say the probability ofpermanence for brevity, we refer to the probabil-ity that systems will satisfy the su$cient condi-tion of Jansen (1987).

Results

COMPLEXITY-STABILITY IN LVCM

We plot the probability surfaces of QGASand LAS in the LVCM having ten species withrespect to r#t and s#t [Fig. 1(b and c)]. Likethe asymptotic (nPR) probability surface(Cohen et al., 1990b) [Fig. 1(a) here], the twoprobabilities of stability in the LVCM withn"10 decrease with increasing r#t and s#t(Fig. 1). The asymptotic probability of QGAS asnPR shows a sharp transition from a positivevalue to zero as t increases (Fig. 2). In contrast,the probabilities of QGAS, LAS and permanencefor the LVCM food webs with ten species showa gradual transition from a positive value towardzero with increasing t (Fig. 2). When all links areconsumer}victim (t) links (r"s"0), the prob-ability of QGAS is smaller than the probability ofpermanence which is, in turn, smaller than theprobability of LAS [Fig. 2(a)]. However, when allthe links are an equal number of r and s links(r"s and t"0), the probability of permanence

becomes greater than the probability of LAS[Fig. 2(b)]. Table 1 also shows that with tenspecies and the same directed connectance, thesubset of webs that have permanence but not

FIG. 2. The probabilities of stability of the Lotka}Volterra cascade model (LVCM), as measured by di!erent criteria: (a)with respect to t, assuming s"r"0. (b) with respect to r and s, assuming r"s and t"0. The four curves are the asymptotic(nPR) probability of qualitative global asymptotic stability ( ), the probability of the qualitative global asymptoticstability ( ), the probability of local asymptotic stability ( ) and the probability of permanence ( ) in LVCM withten species.

FIG. 3. The probabilities of stability (Pr(stability)) of theLotka}Volterra cascade model, as measured by di!erentcriteria, with respect to the number of species (n). The threecurves represent the probability of qualitative global asymp-totic stability ( ), the probability of local asymptoticstability ( ) and the probability of permanence ( ),respectively. In each case, the LVCM food webs are assem-bled with t links only with an undirected connectance of 0.4.


LAS is smaller than the subset of webs that haveLAS but not permanence for model webs witht links, while the opposite is true for model webswith the same probability of r and s links.

When the number of species n is increasedwhile the undirected connectance is "xed atC

u"t/n"Ct"0.4, with r"s"0, the probabil-

ities of QGAS, LAS and permanence of theLVCM all decrease (Fig. 3). With increasing n,the probability of QGAS drops very sharply. Theprobabilities of LAS and permanence decreasemore slowly (Fig. 3).

QUALITATIVE GLOBAL ASYMPTOTIC STABILITY

AND FOOD WEB CONFIGURATION

For the LVCM with nPR, the limiting prob-ability of QGAS is uniquely determined by the

values of r#t and s#t irrespective of the pro-portion of t in those values (Cohen et al., 1990b).A t link has the same e!ect on the probability ofQGAS as do an r link plus an s link. Since a t linkrepresents a pair of directed links, a coupledpositive}negative pair of unidirectional links hasexactly the same e!ect on the asymptotic prob-ability of QGAS as does an uncoupled posit-ive}negative pair of unidirectional links.

In contrast, in LVCM food webs with tenspecies, with equal directed connectances, i.e.,t/n"(r#s)/(2n), the LVCM food webs witht links (r"s"0) have greater probabilities ofQGAS than do the webs with paired r and s links(r"s and t"0) [Fig. 4(a)].

On the other hand, if a t link is counted as anundirected link, given equal undirected connec-tances, i.e., t/n"(r#s)/n, the LVCM food webswith t links (r"s"0) have smaller probabilitiesof QGAS than do food webs with paired r ands links (r"s and t"0) [Fig. 4(b)].

This result leads to our prediction 1: when foodwebs are looked upon as directed graphs, thenconsumer}victim links, equivalent to coupledpairs of directed links, lead to a higher probabil-ity of QGAS than do the same number of pairs ofdonor-controlled and recipient-controlled links;when food webs are regarded as undirectedgraphs, then consumer}victim links lead to alower probability of QGAS than do the samenumber of donor-controlled and recipient-controlled links, assuming there are equal num-bers of donor-controlled and recipient-controlledlinks.

FIG. 4. The probabilities of qualitative global asymptotic stability (Pr(QGAS)) of the Lotka}Volterra cascade model infood webs connected by t links ( ) and by paired r and s links ( ). (a) Given equal directed connectances, i.e.,t/n"(r#s)/(2n), Pr(QGAS) is greater in webs with t links than Pr (QGAS) in webs with paired r and s links. (b) Given equalundirected connectance, i.e., t/n"(r#s)/n, Pr(QGAS) is smaller in webs with t links than Pr(QGAS) in webs with pairedr and s links.

FIG. 5. The probability of qualitative global asymptoticstability (Pr (QGAS)) in the Lotka-Volterra cascade modelwith n "10. For the same values of r#s, the probabilitybecomes larger as the absolute value of di!erence betweenr and s increases. Here t"0 for all three cases. The threecurves represent the Pr (QGAS) with Dr!s D"0 ( ),Dr!sD"0.4 ( ) and Dr!s D"0.8 ( ), respectively.


Like the asymptotic probability surface[Fig. 1(a)], the probability surface of QGAS withten species is symmetric with respect to r#t ands#t [Fig. 1(b)]. If we transect the probabilitysurface, given by a speci"c value of t, by a planes"c and a plane r"c (where c is a positiveconstant), respectively, the two resulting curvescoincide.

Like the asymptotic probability, with the samevalue of (r#t)#(s#t), the probability ofQGAS in the LVCM with n"10 decreases as thedi!erence between r and s becomes smaller, andreaches the minimum when r"s (Fig. 5).

This result gives our prediction 2: a donor-controlled link and a recipient-controlled linkhave an identical e!ect on the probability ofQGAS in the LVCM; with "xed values of t andr#s#t, food webs with more even proportions

of r (recipient-controlled) links and s (donor-controlled) links are less likely to be QGASthan webs with less even proportions of these twotypes of unidirectional links. Results omitted forbrevity also show that the second part of predic-tion 2, that webs with more even proportions ofdonor-controlled and recipient-controlled linksare less likely to be stable than those with lesseven proportions, applies equally to LAS and tothe lower bound of, or su$cient condition for,permanence.

LOCAL ASYMPTOTIC STABILITY AND

FOOD WEB CONFIGURATION

As with the probability of QGAS for LVCMfood webs with ten species, the probability ofLAS is not uniquely determined by the values ofr#t and s#t. The probability increases withthe proportion of t in the combination of r#tand s#t. In Fig. 6, we compare the probabilityof LAS in food webs with t links (r"s"0) withthe probability of LAS in food webs with r ands links (r"s and t"0). The comparison is basedon equal directed connectances [Fig. 6(a)], i.e.,t/n"(r#s)/(2n), and equal undirected connec-tances [Fig. 6(b)], i.e., t/n"(r#s)/n, respec-tively. In both cases, the LVCM food webswith t links (r"s"0) have greater probabilitiesof having LAS than do the webs with paired rand s links (r"s and t"0) (Fig. 6).

This result yields our prediction 3: when re-garded as coupled pairs of directed links, con-sumer}victim links produce a higher probabilityof LAS than the same number of pairs of

FIG. 6. The probability of local asymptotic stability (Pr (¸AS)) of the Lotka}Volterra cascade model in food websconnected by t links ( ) and by paired r and s links ( ). (a) Given equal directed connectances, i.e., t/n"(r#s)/(2n),Pr(¸AS) is greater in webs with t links than Pr(¸AS) in webs with paired r and s links. (b) Given equal undirectedconnectances, i.e., t/n"(r#s)/n, Pr(¸AS) is greater in webs with t links than Pr (¸AS) in webs with paired r and s links.

FIG. 7. Di!erences between e!ects of coupled positive}negative pairs of links (t) and e!ects of uncoupled positive}negative pairs of links (r"s) on the probability of qualitat-ive global asymptotic stability (Pr(QGAS), ), the prob-ability of local asymptotic stability (Pr(¸AS), ) and theprobability of permanence ( ) in Lotka}Volterra cas-cade model with ten species. Pr (.)(t) represents probability ofstability as a function of t with r"s"0; Pr (.)(r/s) representsprobability of stability as a function of r and s (r"s) witht"0. Undirected connectance is measured by t/n or(r#s)/n, respectively.


recipient-controlled and donor-controlled links;when regarded as undirected links, consumer}victim links produce a higher probability of LASthan the same number of recipient-controlledand donor-controlled links, given that the num-ber of recipient-controlled links equals the num-ber of donor-controlled links.

For n"10 species, the di!erence betweenthe e!ects of t links (i.e., coupled positive}nega-tive pairs of directed links) and the e!ects of thesame number of r and s links (i.e., uncoupledpositive}negative pairs of directed links) on theprobability of LAS is much greater than the dif-ference of e!ects on the probability of QGASwhen C

u'0.12, and is slightly smaller when

Cu)0.12 (Fig. 7).

Unlike the probability surface of QGAS, wherethe recipient-controlled and donor-controlledlinks have identical e!ects on the probability ofglobal asymptotic stability, the probability ofLAS is asymmetric with respect to r links and slinks. To facilitate the comparison, we de"nethe probability of LAS as partial functions ofr (with s and t "xed) and s (with r and t "xed),respectively,

Pr (¸AS)(r)"PrMmax(Reji(P) D i"1,2,n)(0:

r3[0, n!sc!t

c], s"s

c, t"t

cN ,

(2a)

Pr (¸AS)(s)"PrMmax(Reji(P) D i"1,2, n)(0:

s3[0, n!rc!t

c], r"r

c, t"t

cN,

(2b)

where rc, s

cand t

care positive constants. We

transect the probability surface of LAS [Fig. 1(c)]for t

c"0 along planes parallel to the

r!Pr (¸AS) plane at s"c (c is a positiveconstant) and along planes parallel to thes!Pr(¸AS) plane at r"c, respectively, for sev-eral di!erent values of c. The transects in eachpair are superimposed and the horizontal axis istransformed from the value of r or s into partialconnectance (by dividing by 10) to compare theprobability of LAS as a function of the partialconnectance of r links (Cr) with the probability ofLAS as a function of the partial connectance ofs links (Cs) (Fig. 8). The curve Pr (¸AS)(r) crosses

FIG. 8. The probability of local asymptotic stability(Pr(¸AS)) with respect to the partial undirected connectanceof r links (Cr, ) and s links (Cs, ), respectively, withno t links. (a) When varying Cr with Cs"0.1 and varyingCs with Cr"0.1, the two curves cross at 0.1; (b) Whenvarying Cr with Cs"0.2 and varying Cs with Cr"0.2, thetwo curves cross at 0.2; (c) When varying Cr with Cs"0.3and varying Cs with Cr"0.3, the two probabilities cross at 0.3.


the curve Pr (¸AS)(s) where Cr"Cs. Where Cr(

Cs for Pr (¸AS)(r) and Cr'Cs for Pr (¸AS)(s), we"nd Pr (¸AS)(r)'Pr (¸AS)(s). Where Cr'Cs forPr (¸AS)(r) and C r(Cs for Pr (¸AS)(s), we "ndPr (¸AS)(r)(Pr (¸AS)(s) (Fig. 8).

FIG. 9. The probability of permanence of the Lotka}Volterrby paired r and s links ( ). (a) Given equal directed connecgreater in webs with t links than the probability of permanenceconnectances, i.e., t/n"(r#s)/n, the probability of permanencwith paired r and s links.

This result leads to our prediction 4: with thesame number of t links and the same number ofunidirectional links r#s, systems with s'r aremore likely to have LAS than systems with r's.The biological implication of this property is thatif two sets of food webs have the same number ofspecies, the same total number of trophic links(r#s#t), and the same number of con-sumer}victim links (t), the set of webs with moredonor-controlled (s) than recipient-controlled (r)links is more likely to be locally asymptoticallystable than the set with the reverse relationship ofthe two types of links.

THE PROBABILITY OF PERMANENCE AND

FOOD WEB CONFIGURATION

The probability of permanence in LVCM foodwebs with t links (r"s"0) is compared with theprobability in webs with paired r and s links(r"s and t"0), based on equal directed connec-tance [t/n"(r#s)/(2n), Fig. 9(a)] and equal un-directed connectance [t/n"(r#s)/n, Fig. 9(b)],respectively, for ten species. The probability ofpermanence with respect to both directed andundirected connectance of t links is greater thanthe probability with respect to the equivalentconnectance of r and s links (Fig. 9).

This result leads to prediction 5: when re-garded as coupled pairs of directed links, or asundirected links, consumer}victim links yield alarger probability of permanence than the samenumber of pairs of recipient-controlled anddonor-controlled links, given equal number ofrecipient-controlled and donor-controlled links.

The di!erence between the e!ects of t links (i.e.,coupled positive}negative pairs of directed links)

a cascade model in food webs connected by t links ( ) andtances, i.e., t/n"(r#s)/(2n), the probability of permanence isin webs with paired r and s links. (b) Given equal undirectede is greater in webs with t links than the probability in webs

FIG. 10. The probability of permanence in the LVCMwith ten species and no t links as a function of the partialundirected connectance of r links (Cr, ) and s links (Cs,

), respectively. (a) When varying Cr with Cs"0.1 andvarying Cs with Cr"0.1, the two curves cross at 0.1; (b)When varying Cr with Cs"0.2 and varying Cs withCr"0.2, the two curves cross at 0.2; (c) When varyingCr with Cs"0.3 and varying Cs with Cr"0.3, the twoprobabilities cross at 0.3.


and the e!ects of the same number of pairs ofr and s links (i.e., uncoupled positive}negativepairs of directed links) on the probability of per-manence is smaller than that on the probabilityof LAS but generally greater than that on theprobability of QGAS (Fig. 7).

Like the probability of LAS, the probability ofpermanence shows asymmetric e!ects of r linksand s links. We denote the probability of perma-nence with respect to r by PP(r) and the probabil-ity of permanence with respect to s by PP(s). Thevalues of r and s are transformed into partialconnectance C r and C s. The probability curvescross where Cr equals Cs. Where Cr is smallerthan Cs for PP(r) and C s is smaller than Cr forPP (s), PP (r) is greater than PP(s) (Fig. 10). WhereCr is greater than Cs for PP(r) and Cs is greater thanCr for PP(s), PP(r) is smaller than PP(s) (Fig. 10).

This result leads to our prediction 6: with thesame number of bidirectional links t and uni-directional links r#s, systems with s'r aremore likely to be permanent than systems withr's. In biological terms, if two sets of food webshave the same number of species, the same totalnumber of trophic links (r#s#t), and the samenumber of consumer}victim links (t), the webswith more donor-controlled (s) than recipient-controlled (r) links are more likely to be perma-nent than those with more recipient-controlledthan donor-controlled links.

Since currently no conditions that are bothnecessary and su$cient for permanence areknown, the proportion of permanent webs unde-tected by the su$cient condition cannot be accu-rately determined. Here, we use the probability ofsystems satisfying the su$cient condition for per-manence as a lower bound and the probability ofsystems satisfying the necessary conditions forpermanence as an upper bound to enclose theprobability of permanence. The probability thatsystems persist in numerical simulations is alsocompared with these two bounds.

For a Lotka}Volterra system of n species witha community matrix P and Jacobian matrix A,necessary conditions for permanence are that(!1)n det A'0, trA(0, and (!1)ndet P'0(Hofbauer & Sigmund 1998).

We determined the persistence of a systemnumerically by tracing the trajectory startingfrom random values for 5000 time steps. If the

populations of all species remain greater than1]10~8 during the 5000 time steps, we regard thesystem as being persistent. For each food webcon"guration, 1000 sample systems are simulated.

We compared the three probabilities alongfour lines: (a) the partial connectance of t linkswas increased with no other links; (b) the partialconnectance of r and s links (r"s) is varied with

FIG. 11. The probabilities of LVCM systems satisfying the su$cient conditions ( ) for permanence and the necessaryconditions ( ) for permanence and the probability of persistence ( ) in the LVCM with ten species. (a) Webs withvarying connectance of t links only. (b) Webs with varying connectance of r and s links only. (c) Webs with varying partialconnectance of r links, "xed partial connectance of t links (C t"0.5) and no s links. (d) Webs with varying partial connectanceof s links and "xed partial connectance of t links (C t"0.5) and no r links.


no t links; (c) the partial connectance of r links isvaried with a "xed partial connectance oft (t/n"0.5) and no s links; (d) the partial connec-tance of s links is varied with a "xed partialconnectance of t (t/n"0.5) and no r links.

The probability of persistence falls between thelower and upper bounds set by the su$cientcondition and the necessary conditions for webswith only t links [Fig. 11(a)], webs with t andr links [Fig. 11(c)] and webs with t and s links[Fig. 11(d)], while the probability of persistencealmost coincides with the lower bound for sys-tems with only r and s links (r"s) [Fig. 11(b)].

5. Discussion and Conclusion

This study appears to be the "rst to exploreexplicitly the general relationship between com-plexity and stability of food webs when perma-nence is used to measure stability. Our resultshows that three measures of stability predictqualitatively consistent relationships betweencomplexity and stability of food webs. The prob-abilities that LVCM food webs have QGAS, LASand permanence all decrease monotonically asfood web complexity increases, when complexityis measured by the number of species or by

connectance. However, there are importantquantitative di!erences among the probabilitiesof the three measures of stability.

Law & Blackford (1992) and Law and Morton(1993) reported that communities with moreomnivory links have more prevalent permanentpaths and suggested that communities of highconnectance are more ready to reassemble them-selves. Based on that, they argued that complexcommunities may be less vulnerable to distur-bance than simple ones. Their results do notnecessarily have to be considered as being incontradiction with our results as well as resultsfrom previous asymptotic stability analysis. Theresults of Law and colleagues (1992, 1993) em-phasize that, within self-assembled communities,complex communities have more alternative per-manent states than simple ones. Our results em-phasize that within stochastically assembled foodwebs, complex food webs are less likely to bepermanent than simple webs.

This study generates six predictions about thee!ects of di!erent types of trophic links on the sta-bility measured by the three criteria. Predictions 1,3 and 5 concern the e!ects of coupled positive}negative pairs of directed links (t or consumer}victim links) and uncoupled positive}negative


pairs of unidirectional links (r and s, or recipient-controlled and donor-controlled links) on theprobability of stability measured by QGAS, LASand permanence, respectively. Prediction 1 statesthat, if food webs are considered as directedgraphs, coupled positive}negative pairs (t or con-sumer}victim links) yield a larger probability ofQGAS than the same number of uncoupled pos-itive}negative pairs of directed links (r and s, orrecipient-controlled and donor-controlled links).On the other hand, if food webs are considered asundirected graphs, consumer}victim links yielda smaller probability of QGAS than the samenumber of r and s links, assuming t"r"s. Incontrast, predictions 3 and 5 state that, whetherfood webs are viewed as directed or undirectedgraphs, t links yield larger probabilities of LASand permanence than the equivalent number ofpaired r and s links.

Predictions 2, 4 and 6 are concerned with thee!ects of uncoupled negative links (r or recipient-controlled links) and uncoupled positive links(s or donor-controlled links) on the probabilityof QGAS, LAS and permanence, respectively.Prediction 2 says that with the same numberof consumer}victim links and the same numberof recipient-controlled plus donor-controlledlinks, webs with more even proportions of posit-ive (donor-controlled) and negative (recipient-controlled) unidirectional links are less stablethan those with less even proportions, regardlessof the sign of the imbalance. This prediction im-plies that food webs with even proportions ofdonor-controlled and recipient-controlled linksare expected to be rare compared with webs withless even proportions of these two types of links.In contrast, predictions 4 and 6 state that foodwebs with more donor-controlled links havea greater probability of LAS and permanencethan do the webs with more recipient-controlledlinks. This prediction agrees with the observationthat donor-controlled interaction is rather com-mon in natural communities (Hawkins, 1992;Hall & Ra!aelli, 1993; Polis & Strong, 1996). It isalso consistent with the evolutionary trend thatmany species that serve as food resources of otherspecies have evolved traits to secure reproduc-tion, such as protection of the part of organismsor populations that are important for reproduc-tion, while allowing predators to feed on

the parts that have a trivial contribution toreproduction. There has been hardly any theoret-ical proof or empirical evidence for the ecologicalor evolutionary bene"ts of recipient-controlledinteractions. Stability constraints may be a mech-anism favoring the donor-controlled interaction,in addition to energetic and demographic mecha-nisms. Previous theoretical studies showed thatfood webs rich in donor-controlled interactionsrequire shorter times of return to the equilibriumfollowing a small perturbation (Pimm, 1982;Chen & Cohen, 2001), but have greater transientgrowth of perturbation (Chen & Cohen, 2001)than webs rich in recipient-controlled interactions.

Combining the above predictions, the foodweb con"gurations favored by stability criteria inLVCMs emerge: consumer}victim and donor-controlled links are expected to be the majortrophic interactions and recipient-controlledlinks are expected to be rare.

If food webs are constrained by stability, thenwhich stability measure represents the operativeconstraint in natural systems? Perhaps the threestability measures operate in di!erent situations.In #uctuating environments, where food websundergo frequent perturbations of both thepopulation sizes and the coe$cients of interac-tions and reproduction, QGAS is more likely thestability constraint in operation, since LAS andthe probability of permanence are structurallyunstable and LAS is only locally stable. Here,being structurally unstable means that a smallchange in the parameters (interaction coe$-cients) may change qualitatively the dynamicbehavior of the system (Hofbauer & Sigmund,1998). In constant environments, LAS or perma-nence may be su$cient to maintain food webpersistence. Donor-controlled links seem morelikely to become established in constant environ-ments than in #uctuating environments, sincesuch links are more likely to appear in web con-"gurations constrained by LAS and permanence.These predictions should be tested empirically.

This work was supported in part by U.S. Natio-nal Science Foundation grants BSR9207293 andDEB9981552. J.E.C. acknowledges with thanks thehospitality of Mr. and Mrs. William T. Golden duringthis work. We thank four anonymous referees forhelpful comments


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