Chaotic dynamics in a multispecies

community

M A D H U R A N A N D * and L AÂ SZ L OÂ O R LOÂ CI

Department of Plant Sciences, The University of Western Ontario, London, Ontario, N6A 5B7, Canada

Received April 1996. Revised March 1997

It is shown that community dynamics is neither haphazard nor completely directed. This is quite clearfrom our examination of a concrete example where recovery dynamics in vegetation progressed from anearly phase of strong linear determinism to intense randomness with phase transition defined by density.Is it possible to reconstruct the two phase structure in simple mathematical terms? The results show that itis, and that the model is very simple: a discrete-time Markov chain with white noise. Interestingly, thelong-term behaviour of the model is complex chaotic and explosive, suggesting that progression fromdominant randomness to determinism is a distinctly probable event. And thus a conceptual foundation islaid, through interlinking initial condition, phase structure and explosive chaoticity, for a unifying theory,in which the classical hypotheses of community dynamics appear as special cases.

Keywords: attractor, complexity, ecology, fractal, Markovity, modelling, succession, theory, vegetation

1. Ecological connections

The subject of this paper is dynamics in a natural plant community which has been significantlyperturbed. According to ecological theory, dynamics in such a community should tend towardincreased stability (Odum, 1969). We use the conventional term `succession' (Clements, 1916;Connell and Slatyer, 1977; McIntosh, 1980; Finegan, 1984; Fekete, 1985) to describe this type ofdynamics, but avoid replacing `stability' with the attendant term `climax' because of diversetechnical connotations which usage has imparted.

Ecological theory also assumes a feedback mechanism, through which population and com-munity level processes change the environment, which in turn change the community itself(Clements, 1916). It is well understood that feedback can amplify small random effects andchange a simple predictable dynamics into a chaotic one (Lorenz, 1963; CË ambel, 1993). Closerto home, biological populations have been known to show chaotic dynamics (May, 1987;Berryman and Millstein, 1989; Tilman and Wedin, 1991; Hastings et al., 1993), but we sharethe view that population variables taken one or two at a time are just part of the communitystory. So we ask: Can a multispecies collection, the community, behave chaotically? The exis-tence of a feedback mechanism in the presence of many small random effects should already besufficient ground to suspect that community level chaotic dynamics exists ± and we are not thefirst to suspect this (Godfray and Blythe, 1990; Vandermeer, 1990; Stone and Ezrati, 1996).

1352-8505 1997 Chapman & Hall

Environmental and Ecological Statistics 4, 337±344

* To whom correspondence should be addressed.

2. Quantification of the succession path

To test and study chaos, we have to quantify the successional path. For this we need tools whichcan handle the multivariate case. Mathematician Esoterica will immediately point to logisticequations and related calculus, but would miss the important experience of ecologists whohave tried this approach, yet did not succeed in carrying the exercise further than the triviallimit of about two or three populations per community. The need for another type of mathe-matics is pressing for yet another reason. To make this point, we have to recall that in a com-munity-level succession study attention shifts from rate of change within populations to the typeand magnitude of transitions in the community by means of replacement among populations(Waggoner and Stephens, 1970; Horn, 1975a,b; van Hulst, 1979; Usher, 1981; Fekete, 1985;Orlo ci et al., 1993). But natural transitions have deterministic and non-deterministic compo-nents, and for this reason the choice of model is one which explicitly incorporates both. Thediscrete-time Markov process with white noise is of this kind and is without limitation on thenumber of populations, other than that dictated by computational power or the lack of it. Weintend to demonstrate that our Markov model (Orlo ci and Orlo ci, 1988; Orlo ci et al., 1993) iscapable of tracing the complex successional path through its phases from striking linear deter-minism to blind randomness. For an explicit definition of (coeno) states and the transitionmatrix we refer the reader to the Appendix. Our first step in showing this is to describe succes-sion in a concrete case (Lippe et al., 1985), and then in subsequent steps we fit model to caseand experiment with the model to explore its long-term behaviour.

Figure 1 displays the multidimensional succession process as an eigenmapping. We refer tothe point configuration in this projection as the 2-D trajectory. The reference space is linear bychoice and for this reason a curved trajectory or trajectory segment is a signature of a nonlinear

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Figure 1. Two-dimensional Eigenmapping of the 19-step recovery trajectory (A) from Atlanticheathland (Lippe et al. 1985) disturbed by fire and heavy grazing. Magnified section of last 11steps (B) involves vertical distortion. The axes are omitted to avoid blurring of the graph. Wehave no other use for them than for parsimonious mapping. The original data represent point-cover estimates of seven species, one species group and bare ground. Step size is one year.Numerals indicate years elapsed. The mapping is 99% accurate, 95% on 1st axis (in A). Notelinear early phase and nonlinear late phase. Break between phases coincides with point cover ofbare ground dropping from 57% to less than 7%.

natural process. The graph is simple enough for visual inspection to reveal the presence of achronologically arranged 2-phase structure. In the early phase (roughly years 1±8), a simplelinear law reigns. In the late phase, the process path has random turns and twists, a phase ofturbulence as it were, under increased random effects. The separation of phases is dramatic.

Ecological studies of succession entail a systematic search for regularity. When a phase struc-ture is discovered, significance is obvious. Not surprisingly, ecologists saw fit to use the phases toclassify plant communities chronologically as seral or climax (Clements, 1916; Nichols, 1923;McIntosh, 1980; Fekete, 1985). We note that the seral vs. climax classification puts emphasis onjust how determinism and randomness trade importance in time. But it should be clear that asimplistic isolation of phases may be deceiving with respect to the relative contributions of deter-minism and randomness to the whole process. This is very clear in the example (Fig. 1) where wemeasure determinism as the level of concordance (C) with the fitted discrete-time stationaryMarkov model (Orlo ci et al., 1993). The term concordance describes the level of matchbetween two structures. To define these structures we first compute the two c� c Euclideandistance matrices DX and DM corresponding to coenostate records X1;X2; . . . ;Xc and theMarkov scores records M1;M2; . . . ;Mc such that Mt�1 �MtP with M1 � X1. The coefficient

s2�DX; DM� �Pc

j<k �dXjk ÿ dMjk�2Pcj<k �dXjk � dMjk�2

is our definition of stress or discordance (Orlo ci et al. 1993), and the difference 1ÿ s2�DX; DM�is proportional to the level of concordance of the observed distance configuration and themodel Markov distance configuration. The tight fit of model and process is observed in Fig. 2for which C is so high that it departs by 7.3 standard deviation units (sd) from what would beexpected if randomness were the sole determinant of change. Intuitively, one would think thatan overall high Markovian determinism is a residual of the early linear phase. But this is notexactly true. In fact when we examine the phases independently we find that sd is 5.6 in theearly phase and a meagre 0.5 in the late phase. Observing that 5:6� 0:5 < 7:3, determinism inthe total process is not simply related to determinism within the phases. This gives further evi-dence to the fascinating observation made by others (Schaffer et al., 1986; Moss and Wiesenfeld,1995) that addition of white noise can actually enhance the appearance of determinism. There isin fact much to be said for the simple-minded wisdom behind the idea that focusing on thewhole reveals more than viewing the sum of its parts (Vandermeer, 1990; van Hulst, 1992).

Chaotic dynamics in multispecies community 339

Figure 2. Joint dispersion of observed (heathland) and simulated (Markov) distance configura-tions (19) before (left) and after (right) random permutation of steps. Distances (171 points) arecomputed between steps (see Fig. 1). The tight elongated cluster on the left indicates consistencyof the observed process with Markov-type determinism.

3. The chaos model

Having considered `reality' in the disguise of a 2D trajectory, we now ask: What are the genericcharacteristics of the trajectory as seen in the Markov model perturbed by arbitrary white noisein the transition matrix? The model, defined by the natural initial state and the natural transi-tion probabilities, passes through a large number of steps under continuous impact by, onaverage, low-level white noise. Long-term behaviour is of interest to discover if we shouldexpect simply more of the same, or if in fact a nested or cyclic phase structure is native tothe process. Notwithstanding well-articulated contrary views (Hastings and Higgins, 1994), webelieve that the discovery of phase is not a hindrance but rather an important revelation in linewith ecological expectations.

The distance profile under allowed 25% maximum white-noise (Fig. 3) and the properties ofthe reconstructed attractor (Fig. 4, Table 1) reveal clear signs of a complex chaotic process:positive Lyapunov exponent (Wolf et al., 1985) and high fractal dimension (Burrough, 1981;Mandelbrot, 1982). In other words, the convergence of the attractor trajectories, starting atdifferent fiducial points, intensifies under increased randomness, and also, the attractor's struc-tural complexity escalates. Thus, notwithstanding the accumulation of random feedback effects,the successional path stays sensitive to initial conditions, while exhibiting explosive behaviour asit tends towards a complex `strange' (Lorenz, 1963) attractor which is fractal.

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Figure 3. Distance profiles of the Markov configuration fitted (Orlo ci, 1988; Orlo ci et al., 1993) toheathland data. Compositional distances between annual states are on the vertical scales andsteps on the horizontal scale. Random perturbation levels: 0% (a), 25% (b). (We note that thepercentages for distortion are upper limits and for this reason the actual random distortion of aspecific transition probability at any step is between 0% and the given upper limit.)

Table 1. Fractal dimension (Burrough, 1981) and Lyapunov exponent (Wolf et al., 1985) ofreconstructed attractors in processes as shown (increased fractal dimension implies increasedcomplexity in the attractor; positive Lyapunov exponent indicates chaoticity)

Process Randomness Fractaldimension

Lyapunovexponent

Pure Markov 0% 1.50 ±1.000Distorted Markov 25% 1.91 0.141

100% 2.00 2.097

4. Unified theory: a first approximation

When we interlink initial condition, linear and nonlinear phase structure and space-time scaledependence, i.e. fractal nature, in terms of our model, the broad outlines of a unifying theory ofcommunity succession emerge. The implications of this theory are intuitively straightforward:the Gleasonian (Gleason, 1926) hypothesis is supported by incorporating the importance ofinitial conditions, which define a stochastic range for the attractor, and by showing the signifi-cant role randomness plays in the process. The succession process is in fact individualistic, i.e.not exactly repeatable, and also site-specific. What may seem strange on first thought is thatClementsian (Clements, 1916) determinism is also supported. This is by recognition of the exis-tence of a clearly two-phase structure dominated by linear determinism. Furthermore, theclimax pattern view (Whittaker, 1953) is also supported in that the attractor is not a fixedpoint, but a ballistic target, as it were, with a defined stochastic range.

These findings should be satisfying to ecologists who face the restrictions of the classicalviewpoints and methodology, in that the window opened in earlier work (Usher, 1981) is nowbroadened. It is clear that the classical theories are not only reconcilable but realistic as specialcases. We dare say in closing that `chaos' is a unifying notion, an `equalizer', in a manner ofspeaking, in the ecologist's world of diverse theories about succession.

Appendix

We refer to Feller (1957) for fundamentals and only give a quick summary here. We will useterms such as coenostate and coenosere to anchor the summary in an ecological terminology.

Chaotic dynamics in multispecies community 341

Figure 4. Phase space mappings of the Markov process (left) and reconstructed attractors (right).Chain length shown is 950 steps. Random perturbation levels: 0% (a,b), 15% (c,d), 25% (e,f).

When we observe a community of organisms we observe what ecologists call a census, empha-sizing that in the community the organisms function in unity using all their `talents' in the art ofcompetition, accommodation, opportunism, and the like, to assure that they survive. Themomentary state of things in such a community is meant when we use the term coenostate.But communities change continuously and the states past and future link up through a chainof events. Based on this we may use the term coenosere to emphasize the serial nature of thestates. Our best model for a coenosere has to be less than `continuous', not by preference butnecessity since the modus operandi can only handle the description of community states at dis-crete intervals. Thus our model coenostates are somewhat isolated but naturally ordered bytheir chronology or our calendar. If we present c coenostates in their chronological orderU1;U2; . . . ;Uc each described by a record set X1;X2; . . . ;Xc then we have quantified the suc-cession path in the most primitive terms. For higher levels of quantification we may record, or,more precisely, define higher order descriptors which have to do with structures such as popu-lation interactions, complexity, stability, etc. Having explained the nature of the record, we canthen try to partition the records into components which are due to the determinism (predict-ability) of the process or to unpredictable random variation. The latter in its crudest definitionhas no simple relationship to time and is what chaotic behaviour generates. Now to quantify thedeterministic component, we have an infinite number of possible mathematical functions linearor nonlinear, most of which can be readily discarded for being too simple and thus biologicallynonsensical. Our choice of Markov mathematics or the Markov chain for our model is anoutcome of past experience of both ours and others. We refer to these in the main text. AMarkov chain is completely defined by the initial state of the community, the first record setX1 and a transition matrix P. The latter is a matrix of proportions which tell us with whatfrequency a given population (taxon) is replaced in one time step by another taxon orremains unchanged. The Markov model is thus a recursive expression Mt�1 �MtP withM1 � X1. In our application, the typical element phi of P expresses the rate at which populationh loses ground (vegetation cover, abundance) to population i when the coenosere moves fromone of its states (t) to the next (t� 1). The transition matrix may be constant or it may changein time. Based on this property, the coenosere (or Markov chain) is said to be stationary ormoving. We cannot imagine a pure stationary process in vegetation succession, yet there maybe lengths of the chain where stationarity is masked only by random variation or `white noise'.This manifests itself in the model through certain levels of random variation in the transitionmatrix. A stationary coenosere keeps itself targeted on the same steady-state, while a movingcoenosere undergoes re-targeting from time to time by significant changes in the transitionmatrix. It is true that in the long run all coenoseres are `moving' in the pure sense of theword, but under the usual study scenario the time span is finite and coenosere stationarity isa valid question. So to discover whether the chain is moving or stationary it is necessary to firstfind out how probable it is that an observed change in transition probabilities exceeds thethreshold beyond which it can no longer be considered as white noise.

The problems of estimating transition probabilities from observed data, and testing for deter-minism in general and Markovity in particular are discussed in Orlo ci et al. (1993). They givedetailed discussions of methods and examples to illustrate methods and concepts. It will besufficient here to say without proofs, that any square matrix of non-negative numbers withunit row totals could qualify as a transition matrix P and when X1 and P are given, a stationaryMarkov chain is completely defined (Feller, 1968, Vol. I, pp. 372 et seq.). Of course we needsuch a square matrix that is consistent with averages in the records. Two things are important inthis: (i) knowing the past state X1 is key to generating a unique model path to succession; and

342 Anand and OrloÂci

(ii) it is sufficient to show that magnitude of variation in P between time steps remains belowset threshold limits to conclude that stationarity exists at that chosen threshold.

Acknowledgements

Our home institution (UWO) granted research travel and other financial support to MA.NSERC of Canada supplied grant support to LO and a generous Graduate Fellowship toMA. We thank two anonymous reviewers for helpful comments.

Biographical sketches

Madhur Anand recently received her PhD degree in the Environmental Science GraduateProgram at UWO. She has received several awards and scholarships from UWO and theNatural Sciences and Engineering Research Council of Canada. She has presented her workin scientific journals and at numerous scientific conferences worldwide. She is a native ofCanada.

LaÂszlo Orlo ci has served as professor at the University of Western Ontario since 1965 withshort interruptions on visiting and research professorships elsewhere. By early training, he isspecialized in forest engineering, biology and quantitative ecology. Peer recognition for hiswork in statistical and evolutionary ecology has earned him rapid promotion through UWOranks and other honours, such as the prestigious memberships in the Hungarian Academy ofSciences and in the Royal Society of Canada, Laurea Honoris Causa from the UniversitaÁ diTrieste, INTECOL's Distinguished Statistical Ecologist award, appointments as CNPgDistinguished Visiting Scientist, and Distinguished Visiting Professor, Honorary Professor andAcademic Advisor at Universities. He also has been honoured with offices in learned societiesand scientific institutions, and editorships at scientific journals. A native of Hungary, LaÂszloÂOrlo ci arrived in Canada in 1957 as a student of the Sopron forestry group at the time of the1956 Hungarian revolution. He is married to MaÂrta Miha ly, a fellow Sopron alumna.

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