NATURE VOL.238 AUGUST 18 1972 413 Received April 20;revised June 19,1972. Table 2 Acute Toxicity of Di-n-butyl Phthalate to Four Species of Fish Stalling,D.L.,Proc.Second Intern.Cong.Pesticide Chem.,Tel Aiv(1971). Temperature TL0*(μg1) US Tariff Commission Plasticisers Rep.(1971). Species (C) 24h 48h 96h Nematollahi,J.,Guess,W.L.,and Autian,J.,J.Pharm.,56,1446 (1967). Fathead minnow 4 Farm Chemicals (Meister,Willoughby,Ohio,1971). (Pimephales promelas) 17 1.490 1,300 s Standard Methods for the Examination of Water and Wastewater Bluegill (Amer.Public Health Assoc..New York.1971). (Lepomis macrochirus） 17 1,230 731 731 Litchfield,jun.,J.T.,and Wilcoxon,T.,Pharmacol.Exp.Ther.,69, 99(1949). Channel catfish 7 Matsuda,K.,and Schnitzer,M.,Bull.Environ.Contam.Toxicol., (Ictalurus punctatus) 17 3,720 2,910 2,910 6,2001971). Rainbow trout Cerbulis,J.,and Ard,J.S..J.Assoc.Offic.Anal.Chem.,50,646 (1967). (Salmo gairdneri) 12 6,470 Nazir,D.J.,Beroza,M.,and Nair,P.P.,Fed.Proc.,26,412(1967). 10 Taborsky.R.G..J.Agric.Food Chem.,15,1073 (1967). Toxicity was measured by standard static bioassay. uMorris,R.J.,Nature,227,1264 (1970). Ogner,G.,and Schnitzer,M.,Science,170,317(1970). *The tolerance limit is the concentration in which 50%of fish survive for a specified time. centration (0.1 ug/1.),respectively,within 14 days.Only 6% of residual di-2-ethylhexyl phthalate remained in the scud GENERAL after 10 days in fresh water. Di-2-ethylhexyl phthalate was examined for reproductive effects in zebra fish (Brachydanio rerio)and guppies (Poecilia Will a Large Complex System reticulata)by dietary exposure.Zebra fish were fed diets con- taining 50 and 100 ug/g of food and guppies were fed 100 ug/g. be Stable? Up to 88.5%of the zebra fish fry died before foraging began as Gardner and Ashbyl have suggested that large complex systems compared with a 50%mortality in control fish.All the dying which are assembled (connected)at random may be expected fry exposed to di-2-ethylhexyl phthalate died in tetany,which to be stable up to a certain critical level of connectance,and suggests that this compound may alter normal calcium metabol- then,as this increases,to suddenly become unstable.Their ism.However,tetany did not occur in the dying controls. conclusions were based on the trend of computer studies of Intraperitoneal injections of di-2-ethylhexyl phthalate (3 gg/kg) systems with 4,7 and 10 variables. increased serum calcium,decreased serum potassium,but did Here I complement Gardner and Ashby's work with an not affect serum sodium and chloride in coho salmon (Oncorhyn- analytical investigation of such systems in the limit when the chus kisutch).All of the adult guppies fed di-2-ethylhexyl number of variables is large.The sharp transition from phthalate became lethargic after 2 months of exposure and an stability to instability which was the essential feature of their 8%incidence of abortions was noted in this group.Continuous paper is confirmed,and I go further to see how this critical exposure of waterfleas (Daphnia magna)to 3 ug/l.of di-2- transition point scales with the number of variables n in the ethylhexyl phthalate significantly (P<0.05)reduced reproduc- system,and with the average connectance C and interaction tion by 60%.Details of these studies will be published magnitude a between the various variables.The object is separately. to clarify the relation between stability and complexity in The incidence of phthalate esters in fish seemed to be greater ecological systems with many interacting species,and some in aquatic areas associated with industrial and heavily popu- conclusions bearing on this question are drawn from the model. lated regions,although hatchery and farmed fish fed diets But,just as in Gardner and Ashby's work,the formal develop- contaminated with the esters also contained residues.Dietary ment of the problem is a general one,and thus applies to the contamination was probably a consequence of the use of wide range of contexts spelled out by these authors. contaminated fish products in feeds.Residues of phthalate Specifically,consider a system with n variables (in an esters previously reported in milke and bovine tissues".10 may ecological application these are the populations of the n have resulted from dietary intake of phthalate esters.Phthalate interacting species)which in general may obey some quite esters have also been found in deep sea jellyfish11 and soil1z. nonlinear set of first-order differential equations.The stability The actual amounts and distribution of these pollutants in of the possible equilibrium or time-independent configurations the environment have not been fully investigated,but a recent of such a system may be studied by Taylor-expanding in the report of 100 mg/l.phthalate esters in a water sample from neighbourhood of the equilibrium point,so that the stability the Ohio River,West Virginia,was not anticipated (R. of the possible equilibrium is characterized by the equation Sandridge,personal communication). The acute toxicity of phthalate esters seems relatively dx/dt=Ax (1) insignificant,but there are indications that these compounds can be detrimental to aquatic organisms at low chronic con- Here in an ecological context x is the nx 1 column vector of centrations.They are produced in large amounts,they are in the disturbed populations x,and the nxn interaction matrix wide use as plasticizers,and,by some means,they are entering A has elements ajs which characterize the effect of species k aquatic ecosystems.Thus,these compounds should be con- on species near equilibrium2.3.A diagram of the trophic sidered as environmental pollutants and a more detailed web immediately determines which ays are zero (no web link), evaluation of toxicological effects of phthalate esters is essential and the type of interaction determines the sign and magnitude to elucidate their impact on these systems. of ajk Following Gardner and Ashby,suppose that each of the n FOSTER L.MAYER,JUN. species would by itself have a density dependent or otherwise DAVID L.STALLING stabilized form,so that if disturbed from equilibrium it would JAMES L.JOHNSON return with some characteristic damping time.To set a time- Fish-Pesticide Research Laboratory, scale,these damping times are all chosen to be unity:a=-1. Bureau of Sport Fisheries and Wildlife, Next the interactions are "switched on",and it is assumed US Department of the Interior, that each such interaction element is equally likely to be Columbia,Missouri 65201 positive or negative,having an absolute magnitude chosen 1972 Nature Publishing Group
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414 NATURE VOL.238 AUGUST 18 1972 from some statistical distribution.That is,each of these It is interesting to compare the analytical results with matrix elements is assigned from a distribution of random Gardner and Ashby's computer results for smallish n.(Their numbers,and this distribution has mean value zero and mean choice of A differs slightly from ours,but in essence they square value a.(For a fuller account of such a formulation, have the fixed value a2=1/3,and diagonal elements intrinsically see refs.2 and 3.)a may be thought of as expressing the -0.55 rather than -1.)Although our methods are based on average interaction"strength",which average is for simplicity the assumption that n is large,and are therefore only approxi- common to all interactions.In short, mations when applied to n=4,7,10,the two approaches in fact agree well when compared,being not more than 30% A=B-I (2) discrepant even for n=4. The central feature of the above results for large systems where B is a random matrix,and I the unit matrix.Thus we is the very sharp transition from stable to unstable behaviour have an unbounded ensemble of models,one for each specific as the complexity (as measured by the connectance and the choice of the interaction matrix elements drawn individually average interaction strength)exceeds a critical value.This from the random number distribution. accords with Gardner and Ashby's conjecture. It is important to note that randomness only enters in the Applied in an ecological context,this ensemble of very initial choice of the coefficients aj,which then define a parti- general mathematical models of multi-species communities,in cular model.Once the dice have been rolled to get a specific which the population of each species would by itself be stable, system,the subsequent analysis is purely deterministic. displays the property that too rich a web connectance (too The system (1)is stable if,and only if,all the eigenvalues of large a C)or too large an average interaction strength (too A have negative real parts.For a specified system size n and large an a)leads to instability.The larger the number of average interaction strength a,it may be asked what is the species,the more pronounced the effect. probability P(n,a)that a particular matrix drawn from the Two corollaries are worth noting,although they should not ensemble will correspond to a stable system.For large n, be taken to have more than qualitative significance. analytic techniques developed for treating large random First,notice that two different systems of this kind,with matrices may be used to show that such a matrix will be average interaction strengths and connectances a,Ci and almost certainly stable (P>1)if az,C2 respectively,have similar stability character if a(n)-12 (4) (small a),and conversely those which interact strongly should do so with but a few species.This is indeed a tendency in The transition from stability to instability as a increases from many natural ecosystems,as noted,for example,by Margalef': the regime (3)into the regime (4)is very sharp for n1; "From empirical evidence it seems that species that interact indeed the relative width of the transition region scales as feebly with others do so with a great number of other species n23 Conversely,species with strong interactions are often part of Such a precise answer for any model in the ensemble in the a system with a small number of species.. limit n1 is a consequence of the familiar statistical fact that, A second feature of the models may be illustrated by using although individual matrix elements are liable to have any Gardner and Ashby's computations (which are for a particular value,by the time one has an nx n matrix with n such statis- a)to see,for example,that 12-species communities with 15% tical elements,the total system has relatively well defined connectance have probability essentially zero of being stable, properties. whereas if the interactions be organized into three separate Next we introduce Gardner and Ashby's connectance,C. 4x 4 blocks of 4-species communities,each with a consequent which expresses the probability that any pair of species will 45%connectance,the "organized"12-species models will be interact.It is measured as the percentage of non-zero elements stable with probability 35%.That is,of the infinite ensemble in the matrix,or as the ratio of actual links to topologically of these particular 12-species models,essentially none of the possible links in the trophic web.The matrix elements in B general ones are stable,whereas 35%of those arranged into now either,with probability C,are drawn from the previous three "blocks"are stable.Such examples suggest that our random number distribution,or,with probability 1-C,are model multi-species communities,for given average interaction zero.Thus each member of the ensemble of matrices A strength and web connectance,will do better if the interactions corresponds to a system of individually stable parts,connected tend to be arranged in "blocks"-again a feature observed in so that each part is affected directly by a fraction C of the many natural ecosystems. other parts.For large n,a2C plays the role previously played This work was sponsored by the US National Science by a2,and we find the system(1)is almost certainly stable Foundation. (P(n,a,C)→1)if ROBERT M.MAY* Q(nC)-12 (6) Present address:School of Physics,University of Sydney, Sydney,NSW. From equation (2)it is obvious that the eigenvalues of A are 7-l,where A are those of B.The "semi-circle law"distribution for 1 Gardner,M.R.,and Ashby,W.R.,Nature,228,784 (1970). the eigenvalues of a particular random matrix ensemble was first 2 Margalef,R. Perspectives in Ecological Theory (University of obtained by Wigner+,and subsequently generalized by him to a Chicago,1968). very wide class of random matrices whose elements all have the 3 May,R.M.,Math.Biosci.,12,59 (1971). same mean square values.Although the matrix B does not in Wigner,E.P.,Proc.Fourth Canad.Math.Cong.,Toronto,174(1959) general possess the hermiticity property required for most of these s Mehta,M.L.,Random Matrices,12 (Academic Press,New York, ieaidteuede 1967. 6 Ginibre,J.J.Math.Phys.,6,44(1965). original style of argument on (B)N(BT)N where N is very large. 7 Margalef,R.,Perspectives in Ecological Theory,7(University of Indirectly relevant is Mehta'and Ginibres Chicago,1968). 1972 Nature Publishing Group
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