The complexity-stability problem in food web theory

In Advances in Environmental and Ecological Modelling, F. Bias('() and A. Weill, cds. pp. 91-119. Elsevier, Paris ( 1999)

The complexity-stability problem in food web theory. What can we learn from exploratory models? Jerzy Michalski, Roger Arditi

Résumé l_es modèles ècolog1queo peuvent avo1r tro1s ràleo

une description des observat1or1o,

une representation des relatior1s b1olog1ques CiHJSnles, ou un outil pour explorer comnlent formuler les queotions attribuables à la nature. Les nwdeles exploratcmes ne pretemJent pas que les valeurs des paramètres ut1l1ses do1vent ètre eot1més par les observaliOil'>, ou qu'1ls dé>mvent quant1tat1vement des écosystemes reels, ou b1en qu'ils demvent des rnècan1smes b1olog1ques spec1f1ques Ils a1dent plutàt a trouver les cornportements qualitatif'> possibles de systèmes reels, a1dent à comprendre comment certalrles c,lr,Kterlotlques peuvent affecter les propriétes gènerales des oystemes r·éels et a1der1t à orienter observations et experrmentat1ons. ür1 décrit ICI deo modeles exploratoires qur sor1t utrlrsés pour etud1er le problème anCien des relat1ons entre complexité et stabi11tè dans les modeles de communautés et cela permet dc rn1eux comprendre le l1en entre la complex1te des reseaux trophiques et le fonctionnement des ecosystemes

1. Introduction The -.tability of an ccological community depends on its structure. ln 1955. VI ac Arthur 1111 argucd thal populatinn dcn-,itic-, of spccic-, -.hou Id be more >table in communitic.-. with more complcx food wcbs. His argument was thal aftcr the lo" of a -,inglc prey specics. spccics thal fccd on fcw 'pccie' wou Id he more likely to hccomc very rare than \Vould spccics thal arc more polyphagous. Similarly. an outhrcak of a spccics population would he dampcu hettcr if the spccics had more prcuator-,. For a long ti mc the ·convcntional wisdom ·of ccologists was th at complC\ity hcgch .\lahility

171. The mcaning of stability was not Jcfincd prcci-

>L'Iy but. intuitivcly. it was unucrstood as an opposition to variahility: a sy-;tcm wa-, Uln-,idcrcd \table if ih cmnponcnb diu not vary too much in timc. or in respon'c to cxtcrnal perturbations.

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This convcntional wisdom was scriously undcrmincd hy the rcsults of Gardner and Ashhy 181. They considcrcd a very simple dynamic mode! (sec 1/flfli'lldix A for the rationale hchind this modcl ):

dxi dt=

"L

Ai 1 x 1 ,

i.j

= 1.2 .... N

( 1)

J

whcrc _ri is a dynamic variahlc, and A ii dcscrihcs the inllucncc of _ri on the rate of change of xi. Evcry variahlc docs not ncccssarily have to he inllucnccd hy ali the othcrs. Somc clements of the matrix (A ii) cou id he zero- A 11111 = 0 mcans that the clement x 11 has no direct cflcct on the rate of change of the clement \n· Gardner and Ashhy dcfïncd conncctancc as the pcrccntagc of non-zero extra-diagonal clements of the matrix (A..). The values of the diagonal clements A..Il wcrc choscn l.f randomly from a unifonn distribution hctwccn -1 and -0.1: the non-zero extradiagonal clements A .. (i ..~-,·) wcrc distrihutcd cvcnly hctwccn -1 and L rcsulting ~. in a randomly conncctcd system (/ïgure 1). Gardner and Ashhy cxamincd the asymptotic stahility of such constructcd system ( 1 ). A system is considcrcd to he asymptoticaliy stahlc iL and only iL the variahlcs ali rcturn to thcir cquilihrium values aftcr displaccmcnt from them. Using computer simulations, Gardner and Ashhy showcd thatthc prohahility of a randomly intcracting system to he asymptotically stahlc dcCI-cases whcn the system sizc or the conncctancc incrcasc. The asymptotic stahility. studicd hy Gardner and Ashhy, is not dircctly conncctcd to the intuitive mcaning of stahility as opposcd to variahility. Indccd, whcn an asymptotically stahle system is pcrturhcd. its variahlcs may, u priori, vary widcly hcforc rcturning to the ir cquilihrium values. Convcrscly, variahles of an unstahlc system may tluctuatc without approaching very smali values. i.e. without loo much risk of spccics extinction. Ncvcrthclcss, cxisting ccosystcms wc re considcrcd as 'stahlc' ( indcpcndcntly of wh at this word cou id cxactly mean), and the conclusion from the study of Gardner and Ashhy, hrought to ecoL

(/

L

0 .______ 0 0

0

92

Figure 1. A randomly connected system. Each arrow represents an influence (positive of negative) of one species on another

The complexity-stability problem in food web theory

log y hy May Il :n wa~ thal complcx cclogical systems mc not likcly to cxist in Jl totally Ulhtructured. The values of the parametcrs A ii were chosen randomly from a prohahility distribution. ln particular. a parameter Au was complctely independent of ,\ 11,. The conclusion about the relationship hctwccn the complexity and the asymptotic stahility of the syslL'm ( 1) may change. if one impose'> some structure on interspecilïc interactions. Lawlor 1101 noticcd that "if May·s conclusions arc to he relevant to the complcxity-stahility question in ccological sy\lems. his randomly constructed matrices must. of cour-,e. be representative of real ccosystcJm. Lvcnthough thcre may he billions or even googols (one googol = 10 1110 ) of hiologically reasonablc systems. they may still fonn only a minisculc fraction of the univcrsc of randomly constructcd systems." Lawlor proposed thrce criteria for a food web to be biologically rcasonahlc: 1) thcrc arc no more than lïvc to scvcn trophic lcvcls: 2) thcrL' arc no thrce-species food loups (of the kind ;\cats B cats C cal'> A):

3 l thcrc is at lcast one plant in the system. He cstimatcd thal. even using only the '>ccond rcquiremcnt. the dcnsity of biologically rcasonablc system'> in the univcrse of random sy-,tcms dccrcascs very rapidly with the numbcr of spccics. ln particular. he fou nd that for the numbcr of spccic-, \1 = 1O. 20. lO and 40. the probabilitics of lïnding a random sy~tcm \\itlwut loops of type A-B-C-A arc rc-,pectively 10 2 . 10 u,_ 10 'ill_ and J0- 1' 7 La\\ lor concludcs: "\Vith Jess than one chance in a googol of constructing an ecosystem with a random numher gcncrator. any analysis of the complexity-stahility question in ccological system' clearly must begin hy L'Xamining systems known a priori to he hiologically acceptable. The question is not whcther randomly coJl\tructcd sy~tems hecome more (or lcss) stable as thcir complcxity i' increasccl. but rather what the specifie structural patterns di-,tingui.-,hing real ccmystems from randomly Cllll'>tructed syslL'ms arc and how thc-,c ohservcd structural propertic~ of real ccosystems contrihutc to their .',\ability (or instahilityl."

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Following Lawlor·s suggestions. Pi111m [ 1~] constructcd food wch modcls \\ ith somc hiologically rca-,onahlc con-,trainh. He considcrcd a general Lotka~ Volterra mode! (sec OfJf!Cndix /J for the rationalc hehind this mode!): dX, ( h, ~-=X, dt

s + Lu,

1

X 1)

1.2 ..... N.

•

(2)

;=1

\\hcrc xi is the density of specics i. /Ji the intrinsic growth (or decline) of spccics i in the ahsencL' of othcr specics. and oi1 dcscrihcs the inllucnce of specie-,j on the growth rate of speeics i. The constraints wcrc the following ([ 14]. pp. 76 and 132): 1 J Only trophic interactions wcrc considcred. i.e. ui 1 = o 1i = 0 unless i was a prcdator ofj. ln the case where i did prey on j. the paramctcrs were choscn ran~ domly with a uniform prohahility from the following intcrvals: O. 1

10

{/fi

()

2) Only the ha-,al spccics (plants) wcrc sclf~limiting: o 11 wcrc selcctcd ovcr the intnval (-1.0): ai,= 0 for hctcrotrophic spccics. J) The structures of the trophic interactions wc re not random. Only food wc os of particular and ·rcasonahlc' structures weJT considnL·d: chains with (or \\ ithout) omnivory. systems with two. thrcc or four trophic levels. etc. ((ïgurc 2 ). Pimm intcrprctcd the variables X, as numher-; of individuals. His argument in fa vour of the choicc (J) wa-, th at prcdators arc usually larger then the ir prey and a large predator must consume many smaller prey to sun ive and rcproducc. This lcads to an asymmctry in the pcr capita cffccts. and thus in the \;ducs of the inter~ action parameters ( 3 ). The sign -,tructurc in ( 3) rcllects the l'act th at an cllcct of a

0

0

0

t' t"'t 0

94

0

0

t/ Xt 0

0

0

0

0

0

t t t

Figure 2. Structured food web. The arrows indicate trophic interactions, i.e., they show who eats whom

The complexity-stability problem in food web theory

prey on its prcdator is al ways pmitivc; thal or the prcdator on its prey is al ways negative. In (2). the variable., Xi can also he intcrprctcd in tcnns of hiomass rather than as numhcrs of individuals (sec ilfifN'ndi.r C for the rationalc hchind this interpretation). Th en the absolu tc value of the cfkct of a prcdator on its prey is al ways highcr (and ncvcr lmvcr) than thal of the prey on its prcdator. hccau;,c caten prey hiomass is only partially transformcd into prcdator hiomass. This hiological (or. rathL'L phy-,ical) rcality i-. rdlectcd in (3) only statistically- whilc the average cllect of a predator on its prey ( = -5 J is a hundrcd times higher than thal of a prey un il.'> prcdator (= 0.05J. therc is still 0.5 'X prohahility thal a prcdator would grow more than it has eatcn. (This looks paradoxical only whcn the variables Xi arc interprctcd as biomas-.cs. With Pimm's original interpretation [ 1-J.[ (i.e. in tenns of numbers of individuab) thi.'> would simply mean thal therc is 0.5 ';( probability thal one predator individual can consume lcss than one prey individual and -.till be able to sun ive and reproduœ- a biologieally quite reasonahle po-.sibility. cspecially in in.-,eet-plant system-.. Hcre. we diseuss Pimm's mode! in tenw., of biomasses in order to makc possible a eomparison with the mudcl of DeAngelis [6[). To complete the description or Pinun·s mode!. we mention thal he tïxed each hi= 1 for basal species and h, = -0.02 for specic-. not at the base of the web. L:sing the mode! (2) with the constraints 1 to 3. Pimm [ 14[ .showed thal biologic~tlly rca-.onable modch are much more likely to be stable than random oncs for a given comhination of species. interaction strengths and connectance. 1\evcrthcless. he found thal. for a given rcasonahle constraint. the probahility or stahility decrcases when complexity increases. contïrming thus the general L'Otll'lusion drawn by May [ 12[ from Gardner and Ashhy's work [X[. DL'i\ngeli-, [6 [ studied asymptotic stability of food web sy-.tcms with Cllll'>lrainh imposed in a slightly different manncr than Pimm did. He also ex:tmincd a sy-.tem with a structure imposed by the assumption thal interspecitïc inter:tctions are purely trophic. i.e. of lhL' predator-prey type. but he considered more general interaction !enns than those of (2). His general food weh mode! is:

=v

i.j = 1.2 ..... N 1

./1

l j 1 prcdatm ol ,

(4)

\\ hcrc the tïrst lerm reprcsents the contribution to hiomass from specic-. of lower trophic levcls. the second lerm rcpresenh los-. caused by predation from higher trophic levcJs. and .I)X) denotes the intrin.sic tendency or a speciL'S to increasc or decrease in the absence of interactions with other species . .f,/X1• X1) is a function dcscribing the trophic interaction between species i and specie.'> j. and y the assimilation dficiency. i.e. the fraction of the ki lied prey biomass thal is transformed into predatnr biomass. The system (4) only contains interactions through which biomass ami energy pass - thcre are no interference competition. no mutualism. no amcnsalism and no commen-.alism.

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The -.ystcm 14) is. in general. non-li1war. If the food web syslL'nl hasan equilihrium. thcn one Clructurc imposcd on the food web system 14) is thus tran-.,latcd into the -.,tructurc ol the Jacobi nwtrix l'il. ln particular. tilL' diagonal clements ol the Jacobi mat rix ('i) depend rJn lincar combinations of thL· extra-diagonal clements (in Gardner ami /\shhy'-., work ali matrix clements werc totally indcpcndent from one imilation elïicicncy ol ali heterotrophs to the same value. y. that could easily he varicd. For smali values of this parameter. his conclusion was opposcd to Pimm·-.,. 2) Pimm lixed once and for ali h1 = 0.02. DeAngelis cuuld \ary d1 1 /dX, ( which corresponds closl'iy to Pinnn·-. parametcr h 1 ). For strong negative values ol d.11 /dXi. hi-., conclusion was upposed tu Pimm's.

The complexity-stability problem in food web theory

.~) DcAngl'lis could vary the proportion of interactions of the donor control 1\- pc hv.., varvinl.', the fraction of (J( f())(.1 (reaction of prey j 12rowth rate to the chan.., '-· IJ - · '\.!l'of prcdator i densitv near cquilihrium) that arc smaller than If fe))(.1 (reaction or prcdator i growth rate to the change or prey j dcnsity ncar cquilihrium). ln l'i mm· s wmk. thL·sc de ri vat iv es correspond to u 1iX1 and 11 11 X 1 • respect ivc 1y. whL·rc i is the prcdator and j is the prey, Pi mm could not vary the proportion of interactions hia-;cd toward donor depL·ndencc hccause the cquilihium dcnsitics. x·t . WerL' functions of al] the parallll'll'rS (U 1/ and /1 1 ), Which WerC always chosen from thL· samc intcrvals. This comparison -,uggcsts that the range from which paramctcr values arc L'imscn and the way in which this is donc is \cry important for the OU(COml' of l'\ en very simple modcls. Would Pi mm ha\'C arrivcd at different conclusions. had hL· chosen his parametl'r values in a different way'' Prcsumahly yes. but this rcnwins to he \crilïcd. More importantly. Pimm's work eontrihutl'd decisivcly to the \Cicnce of L'CO]ogy hy \(ating ~\ \cl of p!n]ictions thal COU]d he VL'rificd hy oh'L'J'\ation' or c·xpcrimcnts. More importantly than the precise rclationship hetwc·cn complcxity ~!nd asymptotic stahility prcdictL·d hy givcn mode!-,. what rcally nwttcr-, is thal the-,e modcls -,]w\\ clcarly tlwt this rclationship cxists. and thal thn indic~Ill' how it mav be ohsL·rvcd a11d wherc to look for it in natural systems. L

"

ar •

3. Community assembly and succession The co1nnwn biological rationalc hchiml till' studiL·s of Gardner and Ashhy

lXI. l\Liy 1121. De;\ngclis lhl ami Pinm1 1I...J.I is the following: takc

S r~u1domly

ChOSL'n \peciL'S, put them togct!Jer and look at the ]oca] asymptotic Sl arL' cxamplc-, of such growth for -,ume\ ariahlc-, ahove the ir equilihrium \;ducs. \Vith the Lotb- Volterra sy (sec hclow), the initial numhn of populations (from 20 to 100), the initial si;e for ali populations (0.1) ami the extinl'lion threshold tO.OO 1). 2. Generale the initial system of S spccics in interaction . .\. Simulate the ti mc evolution of the system hy numcrical integration. 4. Uiminate any population thal falls he lm\ the extinction thresholcL resulting in a new srnallcr system.

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.'ï. Continue the simulation until a stcady ~tate i~ achicvcd or ~omc populations ·explode'. i.e. incrcase without limit. 6. R.epeat '-leps 2-.'ï l'or a ~ample of initial systems suhjcct to the spccilïc Vl/c have proplhcd lo mode! them as rhcugogin: modifications of truphic interaction' (./ÏgHrc 3 ). The equations thal dcscribc our mode! arc thL' gcncr;tliscd LotkaVoltnra equations ( 2). moditïcd to includc trophic structure. nutricnt cycling. nutrient conscn·ation and rhcagogics: Soi 1:

t7a)

Plants:

t7b)

102

Animal~,:

L /(l'(lll'-(i)

Îl, 1

x~). '

(7c)

The complexity-stability problem in food web theory

8 ... ~1----'--+-

c

A Figure 3. A rheagogy is a modification of a trophic interaction. Species Beats species A, i.e., nutrients flow from A to B. Species C (that must belong to the same food web as species A and B) influences this flow either by reinforcing it or by weakening it.

with

(7d)

whcrc Xi is the amount of nutricnts in the hiomass of spccics i. hi the internai growth rate (positive for autotrophs and negative for hctcrotrophs). liii the influence of spccics j on the growth rate of spccics i in the absence of ali othcr spccics. cii" dcscrihcs the rhcagogical influence of spccics k on the trophic interaction hctwccn spccics i and spccic~ j. c the conversion crtïcicncy of consumcrs. ·cons(i) · stands for the set of ali consumcrs of spccics i. and 'rcs(i)' stands for the re sources of spccics i. ln (7 ). in ordcr to make cl car the mcaning of cach tcrm. wc have assumcd thal ali paramctcrs hi and uii have positive values- thus. the !osses of spccics i arc dcscrihcd hy tcrms with a ;_. hcforc it and the gains hy tcnns with a '+'.Th us. for ali spccics (7h. c). the paramctcrs hi dcscrihc the natural mortality rates. As indicatcd in (7d). the modification of a trophic interaction is houndcd from hclow. This is ncccssary in ordcr to forhid the reversai of the direction of the nu trient flow hctwccn spccics i and spccics j - whatcvcr the influences of othcr spccics. a barc will ncvcr cal a lynx. Note thal only plants can use nutricnts from the soi! X0 :u 0 i = 0 if spccics j is not a plant. Animais can consume plants or othcr animais from the adjacent infcrior trophic lcvcl. The sum of ali right-hand sidcs of ( 2) is zero. i.e. the total nutricnt content of the system docs not vary. This rcflccts our assumption thal the system is closed. Using the equations in (7). wc cxamincd communitics constructcd via invasions from a spccics pool. Our simulations uscd the following schcmc: 1) The reservoir of spccics (the spccics pool) is crcatcd. It consists of a food wch with four trophic lcvcls with cqualnumhcrs of spccics on cach leve!. without omnivory and charactcriscd hy the paramctcrs hi. c1ii and cijk' choscn randomly from a uniform distribution ovcr appropriatc intcrvals (sec hclow). The connectance of trophic interactions. the numhcr of spccies and the dcnsity of rhcagogies can he modi fied. 2) A specics, chosen randomly from the spccics pool, invadcs.

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Advances in Environmental and Ecological Modelling

_l l Calculatc the temporal evolution ol' the assemhlcd 'Y'll'm hy intcgrating the· equations in ( 7 ). ...f) Eliminatc nery spccil's whose ahundanl'c drop-. hl'low the cxtinl'tion threshold (().00)). )) During the lïrst 200 lime steps. thl're is a prohahility /' = 0.1 thal a new spl'c·ics l'mill the spL·cics pool L·oloni,cs thL' system at the end ol' cacl1 stcp. 6) /\l'ter 200 timl' units. intcrrupl colonisations ami wail until it arrives at an c·quilihrium or the li mit ol') 000 timc units is rcacill'd. 7 J Repcat slL'ps 2-6 l'or dillcrcnl colonisation scqul'nccs. The paraillctcr 1 a lucs arc takL·n lrom unil'onn distrihutillll'> ovcr till' l'ollowing inter\ ais: ;,,

1

H.:'i 1l'or planh. thcn diminishcd hy a l'actor

1/11 l'or cach highcr trophic

le• \ L' 1: (/.=0: Il uli

= u 1, = 0 with prohahility

( '1 :

111i

r:

i/ 11

= cu, 1 1\hcrL' i is rL'sourL·c tprq:J amlj i.s ils consumer tprcdator) and

,,

= 0 .. \ the coll\ l'rsion cllicicncy ol' consu1ncrs.

10.001.0.011 1vith pmh~1hility C 1:

Till' prohahility ol' trophic intcractiun-.. ('!' is hc·rcal'tcr c~dlcd the trophic L·onncctpecics pool. Thi' mean' thal the non-trophic interaction' (mutualism. amensalisn1. commcnc,alism. competition. cllccts of engineering species. etc., mmlelled hy the rhc) arc very important for cco'Y'lem functioning. Without them. availahlc nutricnh cannot he uscd cllicicntly. i.e. without them. nutricnts cannot he the limiting factor. \1odilïcations of the trophic interactions arc nL'CL''''li"Y lor the construction of -,pccics-rich ccological comn1unitic' thatlully exploit a\ailahlc rc,ourcc-,.

6. Summing up The fir't mmkl' ol randomly conncctcd '>)!->lem' [ ~- 12 [ !->howcd th at the prolxthility ol· a ramlomly intcracting \)Stem hcing a'ymptotic:tlly !->lahk dccrca'>L'S \\"hL'n thL' ')'lem -.,ilL' ami conncctancc increasc. The widcsprL·ad cmlL"Iu-,ion ol thi' rL·-,ult \\as that complcx ccological 'Y'll'tll'- . ThL'!->L' models lac"ed a very important katurL' of natural communitic': thcsc L·ommunitiL'' have hccn huilt \ ia lL'm. lïltcrs par;unctcr \;duc comhinations down to a suhsct ol ali po'>Sihlc random sy,tclll!->. Modcl 'Y'lclll!-> from thi' suhset ctn tell us more about natural !->)stems than L"Lltnplctcly ramlom mode! !->)Sll'lll!->. Trcgonning and Roberts [20[ noticcd thal the procec,s of succc-,.siVL' elimination olthc lese, adaptcd spccies elahk !->)Stem of arbitrat")· compkxity. C\Cn if the initial randomly connlTlcd 'Y'lcm is unstahlc. Taylor [IL)[ rcpcated the wor" of Trcgonning ami Roberts [20 [ using a dillcrcnt computation;tl technique. He confirmcd thcir main rc,ult. He also found thal '>Ollll' unc,tahlc c,yc,tcms could 'L'\plode·- the variable value' grcw toward-, inlïnity. He di-,cardcd thesc explosions a' 'mathcmatical pathologies·.

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ln contras!. wc consider these explosions as a mathematical expressiOn of what really happens in nature. Many ecological communities are very effective in using their ahiotic resources. ln such systems. the total hiomass is determined most! y hy the total amount of availahle resources and not hy the details of interspecifie interactions. The exploding systems are. in our opinion. a mathematical analogy to such effective natural communities. ln order to explore this analogy. we have huilt a food weh mode! in which nutrient limitation is explicitly laken into account. We have considered mode! communities assemhled from a species pool. in which species were arranged in hiologically reasonahle structures: four trophic leve! s. no loops. nutrients flowing only Yia trophic links. Our study demonstrated the great importance of rheagogies (modifications of trophic interactions). which are necessary to descrihe nont roph ic interactions (amen sai ism. commensal ism. interference competition. mutualism. etlects of ecosystem engineers. etc.). Systems with kw (or weak) rheagogies are less likely to use their ahiotic resource ellectively. ln particular. systems with only purely trophic interactions (such as those considered hy lkAngelis and Pimm) are not effective and thus. they may not represent many real ecosystems. such as tropical rain forests. Rheagogies play a crucial role in ecosystem assemhly and they constitute a link hetween ecosystem functioning and the structure of ecological communities.

References [1] Begon M., Harper J.L., Townsend C.R., Ecology. lndividuals, Populations and Communities, 3rd ed., Blackwell, Oxford, 1997. [2] Cohen J.E., Ecologists' Co-Operative Web Bank. Machine-readable data base of food webs, version 1.0, The Rockefeller University, New York, 1989. [3] Cohen J.E., Newman C.M., A stochastic theory of community food webs. 1. Models and aggregated data, Proc. R. Soc. Land. B 224 (1985) 421-448. [4] Cohen J.E., Briand F., Newman C.M., A stochastic theory of community food webs. Il. lndividual webs, Proc. R. Soc. Land. B 224 (1985) 448-461. [5] Cohen J.E., Briand F., Newman C.M., A stochastic theory of community food webs. Ill. Predicted and observed lengths of food chains, Proc. R. Soc. Land. B 228 (1986) 317-353. [6] DeAngelis D.L., Stability and connectance in food web models, Ecology 56 (1975) 238-243. [7] Elton C.S., The Ecology of Invasions and Animais, Chapman and Hall, London, 1958. [8] Gardner M.R., Ashby W.R., Connectance of large dynamic (cybernetic) systems: critical values for stability, Nature 228 (1970) 784. [9] Hall S.J., Raffaelli D., Food web patterns: lessons from a species-rich web, J. Anim. Ecol. 60 (1991) 823-842.

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[1 0] Lawlor L.R., A comment on randomly constructed madel ecosystems, Am. Nat. 112 (1978) 445-447. [11] MacArthur R., Fluctuations of animal populations, and a measure of community stability, Ecology 36 (1955) 533-536. [12] May R.M., Will a large complex system be stable7 Nature 238 (1972) 413-414. [13] Paine R.T., Food webs: raad maps of interactions or grist for theoretical development? Ecology 69 (1988) 1648-1654. [14] Pimm S.L., Food Webs, Chapman and Hall, London, England, 1982.

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[15] Pimm S.L., The Balance of Nature' Ecological Issues in the Conservation of Species and Communit1es, University of Chicago Press, Chicago, 1991. [161 Pol1s G.A, Complex trophic interactions in deserts: an empirical critique of food web theory, Am. Nat. 138 (1991) 123-155. [171 Tavares-Cromar A.F, Williams D.D., The importance of temporal resolution in food web analysis: evidence from a detritus-based stream, Ecol. Monogr. 66 (1996) 91-113. [18] Taylor P.J., Consistent scaling and parameter choice for linear and generalized LotkaVolterra models used in community ecology, J. Theor. Biol. 135 (1988) 543-568. [191 Taylor P.J., The construction and turnover of cornplex community models having generalized Lotka-Volterra dynarnics, J. Theor. Biol. 135 (1988) 569-588.

120] Tregonning K., Roberts A, Complex systems which evolve towards homeostasis, Nature 281 (1979) 563-564. [21] Winemiller K.O., Spatial and temporal variation in tropical fish trophic networks, Ecol. Monogr. 60 (1990) 331-367 .

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Appendix A. The linear model \Vhy is the linear modl'l i.j

=

1.:2 .... . N

( !\ 1)

relevant to ecology 1 Considera general system of many interacting species. The dynanlic-, of specil'S densitics. X, (i = 1. .... :V). i> givcn hy the follcming -,y,tcm: li.Y1 dt

=

~~

( )( X' ..... X" l .

i

1 •

= 1. :?. ..... N

( ;\:?_)

\\hcre l·,tX 1.X:' ..... X.yi is a function (dcpl'llding. in gcnl·ral. on ;ill specic-, den>itiL's) dcscrihing inlcr>pccilïc interactions. Equilihrium solution-, of(;\:21 arc givcn h\ 0

=

~~

(.Y

1 •

X~ ..... X 1 1 .

i

=

1. :?. ..... ;\'

(i\.î 1

ln ordn to >tudy the hehaviour of lill· dynamic system tA:?.). one Gill lineariSL' tilL' r.h.-,. of(;\:?_) and ohtain the lincar >)Stem(;\ 1). 1\hl'!'c 1 1 =~\-X, is the dc1 itemiA:?.) is asymplotically -,tahk hy stllllying the sy>tcm (;\ 1). WhL·n \\l'do not 1--now l'\actly \\hat intcrspccilïc interaction> i-, frcquently the Ctahility propLTtiL'' of a general multi>pecie> -,y stem ( ;\:?_ ). \\'L' may -,tudy syslL'm ( ;\ 1) in-,tL·ad.

.. 108

The complexity-stability problem in food web theory

Appendix B. The generalised Lotka-Volterra model The )!CncJ mulliplicd hy one of thL' \ariahiL·s. The J'unction/,(X X ,1 can he intcrprctcd