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Temporal properties of some biological systems and their fractal attractors
Bulletin of Mathematical Biology Vol. 51, No. 6, pp. 785-800, 1989. Printed in Great Britain.
0092-8240/8953.00 + 0.00 Pergamon Press plc © 1989 Society for Mathematical Biology
TEMPORAL PROPERTIES OF SOME BIOLOGICAL SYSTEMS A N D T H E I R F R A C T A L A T T R A C T O R S n
EMILIA GUTIERREZ
University of Barcelona, Avgda Diagonal, 645, 08028 Barcelona, Spain •
HELENA ALMIRALL
Dept of Psychiatry and Clinical Psychobiology, University of Barcelona, Adolf Florensa s/n, 08028 Barcelona, Spain In this paper we analyse time series data as the growth of organisms using markers such as treerings and otolith deposits (fish). The series studied belong to two tree species (Pinus uncinata, Fagus sylvatica) and one fish species (Dicentrarchus labrax). Spectral analyses of the time series growth show that the main frequencies of fluctuation may be due to variations of the energy input. However, any causal explanation must consider the internal continuous readjustment in the system as reported by the corresponding chaotic properties of the asymptotic decay of the spectra time structure. Since the output of noisy and chaotic systems tend to show similar spectral densities, an attempt to differentiate them has been carried out. The chaotic behaviour has been characterized by the study of the attractors. The dimensions of these multiple topologies were 3.2 and 3.4 for the tree species and 2.3 for the fish species, Therefore, we are dealing with fractal attractors and the minimum number of variables that can be used to describe the systems are 4 and 3 respectively. It is suggested that some of the variables that most influence growth are those obtained by the response functions in the case of trees.
It is well known that information is a non-random interaction with the environment that allows, at least in our planet, highly organized structures (Christensen et al., 1980). Biological systems are considered not to observe classical thermodynamics in dealing exclusively with equilibrium states (Prigogine, 1962; 1978; Johnson, 1981; Ulanowicz, 1986). On the interchange of matter and energy, biological systems acquire information, i.e. structures, differences or local complexities, which influence later responses (memory). According to Margalef (1980) and Odum (1983), the information that is present in contemporary structures can be used to reconstruct the past and can be considered to reflect the energy used and degraded as well. Somewhat more realistically, the information or the form always appears as being associated with historical development. Moreover, the utility of this information lies in the fact that accumulated structures increase the efficiency of the future degradation of energy. Any interaction between matter and energy, which implies an increase in entropy, modifies the structure and condition; future changes to be somewhat more predictable. On the other hand, owing to discontinuities in the input of energy, biological systems are submitted to a fluctuating energetic regime of different intensity, 1. Introduction.
785
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E. GUTIERREZ AND H. ALMIRALL
duration and frequency. Among these variations, organisms have internalized those that present regular time constants, for example daily and seasonally changes, coupling their activities to these fluctuations. (In the sense of Allen and Starr (1982), we call perturbations those variations characterized by their unpredictability which are not internalized by the considered biological system.) Series of growth data will be studied and analysed in this work. The annual growth increment in secondary growth in mountain pine (Pinus uncinata Ram) and beech (Fagus sylvatica L.), and daily growth increment in the otoliths (ear stones) of sea bass (Dicentrarchus labrax L.). We shall characterize the behavior of the systems, their time scales, and the frequencies of fluctuation, by means of time series spectral analysis. Our interest is focused more on discussing the characteristics of spectral density or variance as a function of frequency than on testing the statistical significance of rhythms, (for a discussion on this point, see Pittock, 1983). The spectral decomposition of a series, based on the periodogram, shows its "energy" or variance at each of the Fourier frequencies (Jenkins and Watts, 1968; Koopmans, 1974; Bracewell, 1986). But the discussion of the shape of the spectra must keep in mind, quoting Kloeden and Mees (1985) that: " . . . both noisy and chaotic systems tend to have similar looking spectra; that is, both are typically characterized by a broadband spectral background, perhaps with spikes or peaks corresponding to periodic components."
As Grossmann and Sonneborn-Schmick (1982) have fully argued, that the chaotic properties correspond to the asymptotic decay of the time correlations. Among others, Grassberger and Procaccia (1983a and b) have proposed some measures of the chaotic behaviour of the systems by determining the attractor dimension. In this way, Nicolis and Nicolis (1984; 1986) have analysed the paleoclimatic proxy records previously studied by Berger (1979), proposing a reinterpretation of the spectrum. According to the results, another kind of question that comes out is whether the spectral properties are the result of random fluctuations over given dominant frequencies or whether there exists some underlying determinism in the process we observe in the system. If the answer is affirmative, then we need to ask the minimum number of variables that could define this behaviour. For this we shall use the theory of non-linear dynamic systems in order to carry out the characterization of the series by studying the attractors. A few studies in this direction have been done in ecology (e.g. Schaffer, 1984). In other scientific fields these kinds of studies are more frequent (Kadanoff, 1983; Cvitanovic, 1984; Holden, 1986), and also analysing small data sets (Abraham et al., 1986), although as Blythe and Stokes (1988) sharply pointed out the problem of how to subdivide the time series and the overlap of time-lagged variable has yet to be resolved.
TEMPORAL PROPERTIES OF BIOLOGICAL SYSTEMS
787
2. Data. Tree-ring width series have been extensive and exhaustively studied in multiple fields including: ecology (Hustich, 1948; Borel and Serre, 1969; Fritts, 1974; LaMarche and Stockon, 1974; Tessier, 1986; Jacoby and Hornbeck, 1986; Guti6rrez, 1988; inter alia); paleoclimatology (Douglass, 1919, 1936; Lamb, 1964; 1972; LaMarche, 1974; Fritts, 1976; Hughes et al., 1982; LaMarche and Hirschboeck, 1984); forestry and demography (Prodan, 1968; Harper, 1977); and icthyology (Panella, 1980; Campana, 1984; MoralesNin, 1984; Guti6rrez and Morales-Nin, 1986; Thresher, 1988; inter alia). The measurement of the thickness of a growth ring sequence gives us a time series and, as we can obtain series from another specimen (sample replication), this, when averaged out, provides us with information which can be considered representative of a specific situation. A total of 10 and 13 trees, respectively, were sampled for Fagus sylvatica and for Pinus uncinata in the Spanish Prepyrenees mountains (Table I). Tables were all growing on rocky soils, undamaged, and where there were no obvious signs of disturbance (Guti6rrez, 1987). In order to relate tree-ring growth series with climatic variables, a series of 42 years of monthly precipitation and temperature was obtained from a weather station (42 20'09"N and 01 47'06"E; 1712 m a.s.1, and 80 km away from the sampled sites).
TABLE I Growth Records Time Series of the Species Studied. N, Number of Specimens Averaged Out. Length, Span of the Series..~, Mean of the Ring Width Series. SD, Standard Deviation of the Series Species
Fagus sylvatica Pinus uncinata Dicencharchus labrax
N
Length
10 13 40
271 years 183 years 114 days
Width growth ring .((mm) SD 1.05 0.99 1 × 10-2/ira
0.30 0.41 0.38
Due to the changes that organisms undergo with age, we should not compare averaged out single series belonging to trees of different ages with different mean levels of growth tendency. To keep the series stationary in the mean, the trend was removed by fitting the growth function [ Y t = a t**b exp(-ct)] (Prodan, 1968). New stationary series, series of indices, were obtained by dividing observed values by those estimated by the function. In dendrochronological studies this process is called standardization (Fritts, 1976; Warren, 1980); and, in this way, series belonging to organisms of different ages can be compared and averaged out to obtain a chronology with those well crossdated tree-ring width series (Fritts, 1976). A chronology is an averaged series which is
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E. GUTIERREZ AND H. ALMIRALL
considered representative of a given site and situation. The characteristics for the two chronologies used in this work are shown in Table I. The otoliths used (n = 40) were obtained from sea bass up to 116 days old. Sea bass were kept indoors from January to March (1980) in two 500 dm 3 tanks filled with freely circulating seawater. Specimens were killed at regular intervals and otoliths were removed and treated for later measurements. The formation of growth increments was found to take place daily, starting 2 days after hatching (Morales-Nin, 1984). Otoliths are thin oval bodies, translucent enough to enable differentiation and measurement of daily growth increments. The width of the increments observed on the same day in the otoliths was averaged. The resulting series of consecutive otolith growth was used in the analysis after a first order differentiation to achieve stationarity, so that the mean and variance were constant over the course of time. Descriptive statistics for the series are given (Table I). The major environmental variable influencing growth and not manipulated in the experimental procedure, was water temperature, which was measured daily at n o o n (Gutirrrez and Morales-Nin, 1986). 3. Data Analyses and Results 3.1. Time scale characterization. A series of exact observations on time are worthy of consideration when described in terms of the frequency at which determined events are produced and the variance associated with them. We are referring to the power spectrum of the observations. The power spectrum has some interesting advantages. Among them we can mention the distribution of variance as a function of frequency, which typifies fluctuations in relation to energy sources (Odum, 1983). The characterization of the series by means of spectral analysis is determined by the observation interval and by the length of the series. Thus, in the case of trees we can say nothing about events on time scales of less than a year; in the case of the fishes we can make no inference about the hours. The events occurring within these time scales are integrated in the sample interview of the variable under observation. At low frequency extremes, fluctuations may appear as constants if the number of observations is insufficient. In Figs 1 and 2, power spectra corresponding to the systems under study are shown. Of the spectra presented, we can underline certain c o m m o n characteristics. (1) Frequency (cycles per year and cycles per day for the trees and fish series respectively) is inversely related to intensity: as frequencies increase the intensity of the phenomenon decreases. (2) Parcelling out of variance is not homogeneous, there is more accumulated variance in determined frequencies. This discreteness can be interpreted as greater concentrations or fractions of energy at these frequencies. And, in
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Figure 1. Power spectra of: (A) Fagus sylvatica with a spectral bandwidth (BW) of 0.0184; (B) Pinus uncinata, BW = 0.0267; (C) spring air temperature, BW = 0.1190; (D), autumn precipitations, BW = 0.1190. principle, we can flag them as variations in energy input to the system. F o r example, the cycle of 11 years of the solar spots (that correspond to a frequency of 0.09 in the trees spectra). We shall return to this point in Section 3.2.
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Figure 2. Power spectra of: (A) Dicencharchuslabrax; (B) water temperature.The spectral bandwidth used in both cases was 0.0439. In the first case (1), the relationship between spectral intensity and frequency is usually expressed by the power function: S(f)=af**b, where, S(f) is the spectral intensity andfis the frequency. Logarithmically (log-log) this function is a straight line. The exponent b is normally called spectral roll-off and has a value of - 5/3 when a turbulence spectrum is represented at certain determined spatial scales (Kolmogorov, 1962). In this case the regularity of the spectrum, that is the distribution of the disturbances depends on physical causes. It represents a process of slow energy degradation, or energy transfer in swirls growing ever smaller until they dissipate and whose limit is viscosity. Empirical data on temperature and chlorophyll in the oceans adjust themselves to this law (Platt and Denman, 1975). (Insofar as the structure of phytoplankton is
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considered to depend on the structure of turbulence.) Anyhow, fish do not depend on this structure and the slope is not as great (b = - 1. 603) (Fig. 3A). For water temperature the slope value was (b = - 1.58) (Fig. 3B). Nevertheless as in our experimental data the water was closed in a 500 dm 3 tank, the structure of turbulence, if any, would be broken. =E 3.0*
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A similar fact may be observed on analysis of the climatic data from the observatory near where we took our tree samples. Certain climatic variables also adjust themselves to the value of - 5/3 (Figs 4C and D). The fit of power spectra by models of the type S(f)=af**(b), is a standard technique in the study of the atmospheric turbulence (Lumley and Panofsky, 1964), the discussion about the validity of b = - 5/3 in one or another scale is classical (Gage, 1979).
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But trees (at least in their secondary growth) do not reflect this physical structure of the atmosphere (Figs 4A and B). The possibility of accumulating reserves allows them to overcome changes and differences between the seasons of the year, or of night and day, which at the same time implies a longer life. It is natural to find a longer life together with greater corpulence, both characteristics being, very frequently, forms of selection. As an initial conclusion we may say that historical development is conducive to making innocuous impacts more energetic. In relation to this it is obvious that a tree is not affected in the same way by a wind, a fire, a drought when it is a seedling, a sapling or a pole. 3.2 Phase space description of the biolooical time series. With respect to the spectrum discreteness (second point in Section 3.1 ), in a first approximation we can attribute it to variations in energy input in the system. However, in the spectra we observe given, regular fluctuations, as well as others which are not so regular. Once recorded, the oscillations can serve to make inferences on the system. But, let us suppose that a simple harmonic oscillator can be used to realistically model a biological unit, where oscillations are to be measured. If the model takes into account the energy dissipation then the asymptotic trajectory of a dissipative harmonic oscillator is a stable point, fixed in the space. This signifies that the amplitude of the oscillator is increasingly smaller
TEMPORAL PROPERTIES OF BIOLOGICAL SYSTEMS
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up to eventual repose. In order for the oscillator to maintain periodicity, the system must be provided with energy in such a form that the continual loss associated with the dissipation is compensated for. If this relation is kept stable, then the orbit in the phase space is converted into a stable limit cycle, that is all the orbits around this orbit fuse with it asymptotically, and the oscillator maintains a stationary oscillation. From this point of view, we have the same problem as when we use models which are normally employed in ecology--models which are based on the hypothesis of a stationary state or in sequences of regular fluctuations. The typical example of this sort of model is that normally used to describe predator-prey interactions. This simple model oscillator allows us to study more meaningfully the relation between the distinct scales of complex oscillation which characterize the system, in the sense in which we have been explaining it; but it is lacking in some characteristics which are very apparent in biological systems. What happens when a system, as often occurs, is subject to disturbances which are infrequent or of a greater intensity than that which characterizes the cohesive forces of the system? One answer is that the unusual disturbances in one scale can be recurrent in another (Allen and Starr, 1982). Let us go back to the spectra (Figs 1 and 2). By way of these we obtain information about the forms of variation of the system as well as a certain degree of variability which is non-systematic or irregular fluctuations in the series, any of which prevents us from arriving at an adjusted prediction of the system although such a red-type spectra implies some predictability. Looking at the spectra one question which occurs to us is whether the shape of the spectra is a result of random fluctuation at dominant frequencies or whether there exists some underlying determinism in the process we observe in the system. Dynamic systems of non-linear characteristics do not have, over the course of time, a stationary or periodic state. The appearance of irregular fluctuations, their strange, unpredictable behaviour over the course of time, and the system's sensitivity to initial conditions causes us to think of chaos. (The chaotic properties corresponding to the asymptotic decay of the time structure of the spectra, Fig. 1.) The question is whether this type of behaviour can be characterized. It would seem so, by way of the study of the behaviour of the series in a multi-dimensional phase space. A time series can be employed by itself to put into relief the multivariate internal dynamic of the system shifting its values a fixed lag of time. An instantaneous state of the system will be defined by a point P and a sequence of states of the system will be defined by successive trajectories (Nicolis and Nicolis, 1984). If these converge to and define a subset in the phase space, where the system seems to be trapped, we term this invariant subset as attractor, where d is the dimension of the topology formed by this subset and n the minimum number of variables that should be involved in the description of the
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system. Evidently the trajectories are conditioned by: the number of variables used to define the phase space and the existence or nonexistence of an attractor. The topology might be a point (then: d = 0 , n = 1), a line (d= 1, n = 2 ) , a surface (d = 2, n = 3) or a multiple topology. Nevertheless, the value o f d can be a noninteger and then the attractor i s termed a fractal attractor. If d reaches a saturation limit in relation to n, that is d becomes asymptotic at a certain number of n this limit will be called, d~, the saturation value of the topology dimension. To compute the attractors dimension we have used the algorithm given by Grassberger and Procaccia (1983a): N
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C(r)ocr", where/~ is closely related to d, the dimension of the attractor. To calculate the phase space from the series N variables where obtained by shifting its values a fixed lag of time, Z = 5 years, for the trees series and Z = 5 days for the fish series, this lag was chosen by observing power spectra. At this period the signal power is not significant. Regarding the length to compute a phase space is only possible for the longest series (F. sylvatica, P. uncinata, D. labrax, and water temperature). In graphic form (Fig. 5) we have represented d in relation to n and notice that there exist a d s for these series. For the fish series ds has the value of 2.3; for the trees it is 3.4 for Pinus uncinata and 3.2 for Fagus sylvatica. This means that we need 3 and 4 variables to define the behaviour of the attractor. More data points were computed for the fish series attractor dimension taking lags of z = 2, z = 3, and z = 4. The results confirm that the dimension oscillates between 2 and 3. Also computations were made taking lags of z = 6, z = 7, z = 8 and z = 9, the fractal dimension then died out probably due to the small number of points included in the analysis. According to Blythe and Stokes (1988) if d is not a fixed value this means that we are dealing with variables and not parameters. According to the results in Fig. 5, all the attractors are fractal, such as those found in non-linear dissipative dynamic systems. On the other hand the attractor calculated for water temperature is similar to that found for the otolith series, and this suggests that fishes do not simplify external conditions. The same comment can be pointed out in relation to tree growth considering the dimension of the climatic attractor (d = 3.1) computed by Nicolis and Nicolis (1984).
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3.3. Response and transfer functions. Further to what we have discussed, we m a y ask ourselves: what environmental variables have greater influence on growth and how are these variables internalized? F o r reasons of length and data availability, we have answered the first question by analysing the growth
796
E. GUTIERREZ AND H. ALMIRALL
series of trees as a function of monthly mean temperatures for an observation period of N = 4 2 years. The method used to examine climate-ring width relationships is termed response function (Fritts, 1976). The statistical analysis performed was multivariate regression after extracting principal components, a process fully described elsewhere (Fritts, 1976; Draper and Smith, 1981). An example of this approach is that of Hughes et al. (1982). This method has proven to be one of the most accurate as it takes into account the multicollinearity of the predictors (Cropper, 1984) and does not violate the statistical constraints oforthogonality between independent variables (Draper and Smith, 1981). The unit record of temperature and precipitation was a 14month period starting in June of the year prior to growth, and ending in July of the year of current growth over the period 1941-82. The confidence intervals for each element of the response function were calculated at a significant level of p < 0.05, and the variance accounted for by the climatic variables is high, as can be deduced by the R2 value (Fig. 6). The horizontal axis represents the 14 month period and the vertical axis the value of the standardized response function coefficients. Vertical bars delimiting the 95% confidence interval for each element are also drawn. If the confidence intervals of any response functions element does not include zero, the element is said to have a significative effect on growth. The climatic variables which most affect their growth are the high temperatures as the growing season advances and the lack of precipitation, among others (Gutirrrez, 1987; 1988). Although the obtained response functions are quite complex for both P. uncinata and F. sylvatica (Fig. 6), some mean monthly temperature and precipitation have a great influence on growth and could be considered when modelling tree growth. In order to respond to the second question of how the environmental variables are internalized, we used the growth series of the fish as a function of temperature records of the water for a total of N = l 1 4 observations. Temperature plays an important role in determining the variability in thickness and type of increment observed. Time series analyses (Box and Jenkins, 1976; Liu and Hanssens, 1982) are used to analyse and to identify the dynamic transfer function of the bivariate process formed by temperature and otolith growth. Temperature exhibits auto-correlative structure. This feature underlines the dynamic nature of the growth process in relation to environment, which should be taken into account in the statistical analysis of the data. The dynamic relationship between the input and the output was well described by a third-order transfer function model. The impulse response function describes how the output is related to the input of the linear system. In this case it occurred at regular intervals of three time units (days) and decreased exponentially (Gutirrrez and Morales-Nin, 1986). As stated by the time series model, otolith growth is a conservative process
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Figure 6. Response functions obtained for: (A) Pinus uncinata; (B) Fagus sylvatica taking into account monthly mean precipitations and temperatures from 1941 to 1982, unit period record from June of the year prior to growth (t-I) to July of the year of current growth (t). involving memory of the previous growth and also of the environmental conditions. Growth of a new increment integrates only a proportion of the disturbances which are absorbed and have a lasting influence. It would seem that the growth process includes a mechanism to adapt to changes in the environment based on a short-term memory. Such a situation would have to be explained in terms of fish's physiological mechanisms and hormonal and metabolite concentration levels. Daily changes in water temperature (perturbations at organism scale, i.e. molecular) cause an increase or in general a change, in the metabolic rate of the fish which returns "slowly" to the initial
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E. GUTIERREZ AND H. ALMIRALL
ground state. But this state has been modified previously and so on. The final steady-state is achieved at a new level in which entropy production is least (Prigogine and Wiame, 1946; Johnson, 1981; Ulanowicz, 1986). The organism is very sensitive to each set of initial conditions being the adult state, the end product of its historical process (see references above mentioned for further discussion). 4. Conclusions. We have studied some time series belonging to biological systems. Their time scales have been analysed through spectral analysis, and the structure of the variance by means of the spectral roll-off. According to the results the power spectra of these organisms is due to an evolutionary process of assimilation and internalization of impacts and not exclusively dependent on external physical causes. For the tree-rings g r o w t h series we have determined, from a given set of climatic variables, those which have a greater influence on growth; they are mainly low precipitations and high temperatures during the growing season. And in the case of the sea bass D. labrax, the internalization of fluctuations of temperature occurred at regular intervals and decreased exponentially. As the attractor dimensions are fractals any discussion of causal mechanisms, which tries to explain the behaviour of the systems studied, must consider the possibility of a continued internal readjustment and also the possibility that the current state of the system is not its unique state, given any particular set of external variables. As a clear and immediate co0sequence, it can be said that spectral analysis is incapable of distinguishing the attractor output of what we generally call random noise processes. Moreover, the studied biological systems do not simplify the complexity of external conditions. Finally, a realistic model of the growth activity of the species dealt with ought to consider the relevant characteristics described in relation to these systems. The authors gratefully acknowledge the technical assistance and comments of Jordi Flos. They thank Beatriz Morales-Nin for providing the temperature and Dicencharchus labrax series.
LITERATURE Abraham, N. B., A. M. Albano, B. Das, G. de Guzm/m, S. Yong, R. S. Gioggia, G. P. Puccioni and R. C. Tredicce. 1986. "Calculatingthe Dimension of Attractors from Small Data Sets." Phys. Lett. I14A, 217--221. Allen, T. F. H. and T. B. Start. 1982. Hierarchy: Perspectivesfor Ecological Complexity. The University of Chicago Press. Berger, A. 1979. "Spectrum of Climatic Variations and their Causal Mechanisms." Geophys. Surveys 3, 351-402. Blythe, S. P. and T. K. Stokes. 1988. "Biological Attractors, Transients and Evolution." In:
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Ecodynamics: Contributions to Theoretical Ecology, W. Wolf, C.-J. Soeder and F. R. Drepper (Eds). Berlin: Springer-Verlag. Borel, L. and F. Serre. 1969. "Phytosociologie et Analyse des Cernes Ligneux: l'exemple de Trois Forets du Haut Var (France)." Oecol. Plant. IV, 155-176. Box, G. E. P. and G. M. Jenkins. 1976. "Time Series Analysis: Forecasting and Control. San Francisco; Holden-Day. Bracewell, R. N. 1986. The Fourier Transform and its Applications (2ndedn). New York: McGraw Hill. Campana, S. E. 1984. "Lunar Cycles of Otolith Growth in the Juvenile Starry Flounder Platichthys stellatus." Mar. Biol. 80, 239-246. Christensen, B., H. Krausz, R. Perez-Polo, and W. D. Willis Jr. 1980. "Communication among Neurons and Neuroscientists." In Information Processing in the Nervous System, H. M. Pinsker and W. D. Willis Jr (Eds), pp. 339-359. New York: Raven Press. Cvitanovic, P. (Ed.) 1984. Universality in Chaos. Bristol: Adam Hilger. Cropper, J. P. 1984. "Multicollinearity within Selected Western North American Temperature and Precipitation Data Sets." Tree-Ring Bull. 44, 29-37. Douglass, A. E. 1919. "Climatic Cycles and Tree Growth--I?' Carnegie Inst. Wash. Publ., 289. --. 1936. "Climatic Cycles and Tree Growth--III. A Study of Cycles." Carnegie Inst. Wash. Publ., 289. Draper, N. R. and H. Smith. 1981. Applied Regression Analysis. New York: Willey. Fritts, H. C. 1974. "Relationship of Ring Widths in Arid-Site Conifers to Variations in Monthly Temperature and Precipitation." Ecol. Mort. 44, 411-440. --. 1976. Tree-Rings and Climate. New York: Academic Press. Gage, K. S. 1979. "Evidence for a k**-5/3 Law Inertial Range in Mesoscale Two-Dimensional Turbulence." J. Atmosph. Sci. 36, 1950-1953. Grassberger, P. and I. Procaccia. 1983a. "Characterization of Strange Attractors." Phys. Rev. Lett. 50, 346--349. --. 1983b. "Measuring the Strangeness of Strange Attractors." Physica 9D, 198-208. Grossmann, S. and B. Sonneborn-Schmick. 1982. "Correlation Decay in the Lorenz Model as a Statistical Physics Problem. Phys. Rev. A25, 2371-2384. Guti6rrez, E. 1987. Dendrocronologia de Pinus uncinata, Pinus sylvestrys y Fagus sylvatica en CataIunya. Ph.D. Dissertation, Universitat de Barcelona. ----. 1988. "Dendroecological Study ofFagus silvatica in the Montseny Mountains (Spain)." Oecol. Plant. 9, 1-9. and B. Morales-Nin. 1986. "Time Series Analysis of Daily Growth in Dicentrarchus Labrax L. Otoliths." J. Exp. Mar. Biol. Ecol. 103, 163-179. Harper, J. L. 1977. Population Biology of Plants. London: Academic Press. Holden, A. V. (Ed.). 1986. Chaos. Manchester University Press. Hughes, M. K., P. M. Kelly, J. R. Pilcher, and V. C. LaMarche Jr. (Eds). Climate From Tree Rings. Cambridge University Press. Hustiche, I. 1948. "The Scotch Pine in Northernmost Finland and its Dependence on Climate in the Last Decades. Acta Bot. Fenn. 42, 1-75. Jacoby, G. C. and J. W. Hornbeck (Eds). 1986. Proceedings of the International Symposium on Ecological Aspects of Tree-Ring Analysis. Oak Ridge: U.S. Dept of Energy. Jenkins, G. M. and D. G. Watts. 1968. Spectral Analysis and its Applications. San Francisco: Holden-Day. Johnson, L. 1981. "The Thermodynamic Origin of Ecosystems." Can. J. Fish Aquat. Sci. 38, 571-590. Kadanoff, L. P. 1983. "Roads to Chaos." Phys. Today 36, 46-53. Kloeden, P. E. and A. I. Mees. 1985. "Chaotic Phenomena." Bull. math. Biol. 44, 29-37. Kolmogorov, A. N. 1962. "A Refinement of Previous Hypothesis Concerning the Local Structure of Turbulence in a Viscous Incomprehensible Fluid at High Reynolds Number." J. Fluid Mech. 13, 82-85. Koopmans, L. H. 1974. The Spectral Analysis of Time Series. New York: Academic Press. -
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Lamb, H. H. 1964. "Trees and Climatic History in Scotland." Q. Jl R. met. Soc. 90, 382-394. --. 1972. Climate: Present, Past and Future, Vols I and 11. London: Methuen. LaMarche, Jr, V. C. 1974. "Paleoclimatic Inferences from Long Tree-Rings Records." Science 183, 1043-1048. and C. W. Stockon. 1974. "Chronologies from Temperature-Sensitive Bristlecone Pines at Upper Treeline in Western U.S.A." Tree-Ring Bull. 34, 21-45. and K. K. Hirschboek. 1984. "Frost Rings in Trees as Records of Major Volcanic eruptions." Nature 307, 121-126. Liu, L-M. and D. M. Hanssens. 1982. "Identification of Multiple-Input Transfer Function Models." Commun. Stat. All, 297-314. Lumley, J. L. and H. A. Panofsky. 1964. The Structure of Atmospheric Turbulence. New York: Wiley. Margalef, R. 1980. La Biosfera: Entre la termodingtmica y el Juego. Barcelona: Omega. Morales-Nin, B. 1984. Caracteristicas, Composici6n y Microestructura de los Otolitos de los Tele6steos. Ph.D. Dissertation, Universitat de Barcelona. Nicolis, C. and G. Nicolis. 1984. "Is there a Climatic Attractor?" Nature 311,529-532. and .1986. "Reconstruction of the Dynamics of the Climatic System from Time Series Data." Proc. natl Acad. Sci. U.S.A. 83, 536-540. Odum, H. T. 1983. Systems Ecology: An Introduction. New York: Wiley. Panella, G. 1980. "Growth Patterns in Fish Sagittae." In Skeletal Growth of Aquatic Organisms: Biological Records of Environmental Changes, D. C. Rhoads and R. A. Lutz (Eds), pp. 519-560. New York: Plenum Press. Pittock, A. B. 1983. 'Solar Variability, Weather and Climate: An Update." Q. Jl R. met. Soc. 109, 23-55. Platt, T. and L. D. Denman. 1975. "Spectral Analysis in Ecology." Ann. Rev. Ecol. Syst. 6, 189-211. Prigogine, I. 1962. Introduction to Thermodynamics of Irreversible Processes. New York, Interscience. --. 1978. "Time, Structure, and Fluctuations." Science 201,777-785. and J. M. Wiame. 1946. "Biologic et Thermodynamique des Ph6nomenes Irr6versibles." Experimentia 2, 451-453. Prodan, M. 1968. Forest Biometrics. London: Pergamon Press. Schaffer, W. M. 1984. "Stretching and Folding in Lynx Fur Returns: Evidence for a strange attractor in nature? Am. Natur. 124, 798-820. Tessier, L. 1986. "Approche Dendroclimatologique de l'Ecologie de Pinus sylvestris L. et Quercus pubescens Willd. dans le Sud-Est de la France." Oecol. Plant. 7, 339-355. Thresher, R. E. 1988. "Otolith Microstructure and the Demography of Coral Reef Fishes." Trends Ecol. Evol. 3, 78-80. Ulanowicz, R. E. 1986. Growth and Development: Ecosystems Phenology. New York: Springer. Warren, W. G. 1980. "On Removing the Growth Trend from Dendrochronological Data." TreeRing Bull. 40, 35-44. -
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Received 14 S e p t e m b e r 1988 Revised 27 J u n e 1989
0092-8240/8953.00 + 0.00 Pergamon Press plc © 1989 Society for Mathematical Biology
TEMPORAL PROPERTIES OF SOME BIOLOGICAL SYSTEMS A N D T H E I R F R A C T A L A T T R A C T O R S n
EMILIA GUTIERREZ
University of Barcelona, Avgda Diagonal, 645, 08028 Barcelona, Spain •
HELENA ALMIRALL
Dept of Psychiatry and Clinical Psychobiology, University of Barcelona, Adolf Florensa s/n, 08028 Barcelona, Spain In this paper we analyse time series data as the growth of organisms using markers such as treerings and otolith deposits (fish). The series studied belong to two tree species (Pinus uncinata, Fagus sylvatica) and one fish species (Dicentrarchus labrax). Spectral analyses of the time series growth show that the main frequencies of fluctuation may be due to variations of the energy input. However, any causal explanation must consider the internal continuous readjustment in the system as reported by the corresponding chaotic properties of the asymptotic decay of the spectra time structure. Since the output of noisy and chaotic systems tend to show similar spectral densities, an attempt to differentiate them has been carried out. The chaotic behaviour has been characterized by the study of the attractors. The dimensions of these multiple topologies were 3.2 and 3.4 for the tree species and 2.3 for the fish species, Therefore, we are dealing with fractal attractors and the minimum number of variables that can be used to describe the systems are 4 and 3 respectively. It is suggested that some of the variables that most influence growth are those obtained by the response functions in the case of trees.
It is well known that information is a non-random interaction with the environment that allows, at least in our planet, highly organized structures (Christensen et al., 1980). Biological systems are considered not to observe classical thermodynamics in dealing exclusively with equilibrium states (Prigogine, 1962; 1978; Johnson, 1981; Ulanowicz, 1986). On the interchange of matter and energy, biological systems acquire information, i.e. structures, differences or local complexities, which influence later responses (memory). According to Margalef (1980) and Odum (1983), the information that is present in contemporary structures can be used to reconstruct the past and can be considered to reflect the energy used and degraded as well. Somewhat more realistically, the information or the form always appears as being associated with historical development. Moreover, the utility of this information lies in the fact that accumulated structures increase the efficiency of the future degradation of energy. Any interaction between matter and energy, which implies an increase in entropy, modifies the structure and condition; future changes to be somewhat more predictable. On the other hand, owing to discontinuities in the input of energy, biological systems are submitted to a fluctuating energetic regime of different intensity, 1. Introduction.
785
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E. GUTIERREZ AND H. ALMIRALL
duration and frequency. Among these variations, organisms have internalized those that present regular time constants, for example daily and seasonally changes, coupling their activities to these fluctuations. (In the sense of Allen and Starr (1982), we call perturbations those variations characterized by their unpredictability which are not internalized by the considered biological system.) Series of growth data will be studied and analysed in this work. The annual growth increment in secondary growth in mountain pine (Pinus uncinata Ram) and beech (Fagus sylvatica L.), and daily growth increment in the otoliths (ear stones) of sea bass (Dicentrarchus labrax L.). We shall characterize the behavior of the systems, their time scales, and the frequencies of fluctuation, by means of time series spectral analysis. Our interest is focused more on discussing the characteristics of spectral density or variance as a function of frequency than on testing the statistical significance of rhythms, (for a discussion on this point, see Pittock, 1983). The spectral decomposition of a series, based on the periodogram, shows its "energy" or variance at each of the Fourier frequencies (Jenkins and Watts, 1968; Koopmans, 1974; Bracewell, 1986). But the discussion of the shape of the spectra must keep in mind, quoting Kloeden and Mees (1985) that: " . . . both noisy and chaotic systems tend to have similar looking spectra; that is, both are typically characterized by a broadband spectral background, perhaps with spikes or peaks corresponding to periodic components."
As Grossmann and Sonneborn-Schmick (1982) have fully argued, that the chaotic properties correspond to the asymptotic decay of the time correlations. Among others, Grassberger and Procaccia (1983a and b) have proposed some measures of the chaotic behaviour of the systems by determining the attractor dimension. In this way, Nicolis and Nicolis (1984; 1986) have analysed the paleoclimatic proxy records previously studied by Berger (1979), proposing a reinterpretation of the spectrum. According to the results, another kind of question that comes out is whether the spectral properties are the result of random fluctuations over given dominant frequencies or whether there exists some underlying determinism in the process we observe in the system. If the answer is affirmative, then we need to ask the minimum number of variables that could define this behaviour. For this we shall use the theory of non-linear dynamic systems in order to carry out the characterization of the series by studying the attractors. A few studies in this direction have been done in ecology (e.g. Schaffer, 1984). In other scientific fields these kinds of studies are more frequent (Kadanoff, 1983; Cvitanovic, 1984; Holden, 1986), and also analysing small data sets (Abraham et al., 1986), although as Blythe and Stokes (1988) sharply pointed out the problem of how to subdivide the time series and the overlap of time-lagged variable has yet to be resolved.
TEMPORAL PROPERTIES OF BIOLOGICAL SYSTEMS
787
2. Data. Tree-ring width series have been extensive and exhaustively studied in multiple fields including: ecology (Hustich, 1948; Borel and Serre, 1969; Fritts, 1974; LaMarche and Stockon, 1974; Tessier, 1986; Jacoby and Hornbeck, 1986; Guti6rrez, 1988; inter alia); paleoclimatology (Douglass, 1919, 1936; Lamb, 1964; 1972; LaMarche, 1974; Fritts, 1976; Hughes et al., 1982; LaMarche and Hirschboeck, 1984); forestry and demography (Prodan, 1968; Harper, 1977); and icthyology (Panella, 1980; Campana, 1984; MoralesNin, 1984; Guti6rrez and Morales-Nin, 1986; Thresher, 1988; inter alia). The measurement of the thickness of a growth ring sequence gives us a time series and, as we can obtain series from another specimen (sample replication), this, when averaged out, provides us with information which can be considered representative of a specific situation. A total of 10 and 13 trees, respectively, were sampled for Fagus sylvatica and for Pinus uncinata in the Spanish Prepyrenees mountains (Table I). Tables were all growing on rocky soils, undamaged, and where there were no obvious signs of disturbance (Guti6rrez, 1987). In order to relate tree-ring growth series with climatic variables, a series of 42 years of monthly precipitation and temperature was obtained from a weather station (42 20'09"N and 01 47'06"E; 1712 m a.s.1, and 80 km away from the sampled sites).
TABLE I Growth Records Time Series of the Species Studied. N, Number of Specimens Averaged Out. Length, Span of the Series..~, Mean of the Ring Width Series. SD, Standard Deviation of the Series Species
Fagus sylvatica Pinus uncinata Dicencharchus labrax
N
Length
10 13 40
271 years 183 years 114 days
Width growth ring .((mm) SD 1.05 0.99 1 × 10-2/ira
0.30 0.41 0.38
Due to the changes that organisms undergo with age, we should not compare averaged out single series belonging to trees of different ages with different mean levels of growth tendency. To keep the series stationary in the mean, the trend was removed by fitting the growth function [ Y t = a t**b exp(-ct)] (Prodan, 1968). New stationary series, series of indices, were obtained by dividing observed values by those estimated by the function. In dendrochronological studies this process is called standardization (Fritts, 1976; Warren, 1980); and, in this way, series belonging to organisms of different ages can be compared and averaged out to obtain a chronology with those well crossdated tree-ring width series (Fritts, 1976). A chronology is an averaged series which is
788
E. GUTIERREZ AND H. ALMIRALL
considered representative of a given site and situation. The characteristics for the two chronologies used in this work are shown in Table I. The otoliths used (n = 40) were obtained from sea bass up to 116 days old. Sea bass were kept indoors from January to March (1980) in two 500 dm 3 tanks filled with freely circulating seawater. Specimens were killed at regular intervals and otoliths were removed and treated for later measurements. The formation of growth increments was found to take place daily, starting 2 days after hatching (Morales-Nin, 1984). Otoliths are thin oval bodies, translucent enough to enable differentiation and measurement of daily growth increments. The width of the increments observed on the same day in the otoliths was averaged. The resulting series of consecutive otolith growth was used in the analysis after a first order differentiation to achieve stationarity, so that the mean and variance were constant over the course of time. Descriptive statistics for the series are given (Table I). The major environmental variable influencing growth and not manipulated in the experimental procedure, was water temperature, which was measured daily at n o o n (Gutirrrez and Morales-Nin, 1986). 3. Data Analyses and Results 3.1. Time scale characterization. A series of exact observations on time are worthy of consideration when described in terms of the frequency at which determined events are produced and the variance associated with them. We are referring to the power spectrum of the observations. The power spectrum has some interesting advantages. Among them we can mention the distribution of variance as a function of frequency, which typifies fluctuations in relation to energy sources (Odum, 1983). The characterization of the series by means of spectral analysis is determined by the observation interval and by the length of the series. Thus, in the case of trees we can say nothing about events on time scales of less than a year; in the case of the fishes we can make no inference about the hours. The events occurring within these time scales are integrated in the sample interview of the variable under observation. At low frequency extremes, fluctuations may appear as constants if the number of observations is insufficient. In Figs 1 and 2, power spectra corresponding to the systems under study are shown. Of the spectra presented, we can underline certain c o m m o n characteristics. (1) Frequency (cycles per year and cycles per day for the trees and fish series respectively) is inversely related to intensity: as frequencies increase the intensity of the phenomenon decreases. (2) Parcelling out of variance is not homogeneous, there is more accumulated variance in determined frequencies. This discreteness can be interpreted as greater concentrations or fractions of energy at these frequencies. And, in
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Figure 1. Power spectra of: (A) Fagus sylvatica with a spectral bandwidth (BW) of 0.0184; (B) Pinus uncinata, BW = 0.0267; (C) spring air temperature, BW = 0.1190; (D), autumn precipitations, BW = 0.1190. principle, we can flag them as variations in energy input to the system. F o r example, the cycle of 11 years of the solar spots (that correspond to a frequency of 0.09 in the trees spectra). We shall return to this point in Section 3.2.
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Figure 2. Power spectra of: (A) Dicencharchuslabrax; (B) water temperature.The spectral bandwidth used in both cases was 0.0439. In the first case (1), the relationship between spectral intensity and frequency is usually expressed by the power function: S(f)=af**b, where, S(f) is the spectral intensity andfis the frequency. Logarithmically (log-log) this function is a straight line. The exponent b is normally called spectral roll-off and has a value of - 5/3 when a turbulence spectrum is represented at certain determined spatial scales (Kolmogorov, 1962). In this case the regularity of the spectrum, that is the distribution of the disturbances depends on physical causes. It represents a process of slow energy degradation, or energy transfer in swirls growing ever smaller until they dissipate and whose limit is viscosity. Empirical data on temperature and chlorophyll in the oceans adjust themselves to this law (Platt and Denman, 1975). (Insofar as the structure of phytoplankton is
TEMPORAL PROPERTIES OF BIOLOGICAL SYSTEMS
791
considered to depend on the structure of turbulence.) Anyhow, fish do not depend on this structure and the slope is not as great (b = - 1. 603) (Fig. 3A). For water temperature the slope value was (b = - 1.58) (Fig. 3B). Nevertheless as in our experimental data the water was closed in a 500 dm 3 tank, the structure of turbulence, if any, would be broken. =E 3.0*
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A similar fact may be observed on analysis of the climatic data from the observatory near where we took our tree samples. Certain climatic variables also adjust themselves to the value of - 5/3 (Figs 4C and D). The fit of power spectra by models of the type S(f)=af**(b), is a standard technique in the study of the atmospheric turbulence (Lumley and Panofsky, 1964), the discussion about the validity of b = - 5/3 in one or another scale is classical (Gage, 1979).
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But trees (at least in their secondary growth) do not reflect this physical structure of the atmosphere (Figs 4A and B). The possibility of accumulating reserves allows them to overcome changes and differences between the seasons of the year, or of night and day, which at the same time implies a longer life. It is natural to find a longer life together with greater corpulence, both characteristics being, very frequently, forms of selection. As an initial conclusion we may say that historical development is conducive to making innocuous impacts more energetic. In relation to this it is obvious that a tree is not affected in the same way by a wind, a fire, a drought when it is a seedling, a sapling or a pole. 3.2 Phase space description of the biolooical time series. With respect to the spectrum discreteness (second point in Section 3.1 ), in a first approximation we can attribute it to variations in energy input in the system. However, in the spectra we observe given, regular fluctuations, as well as others which are not so regular. Once recorded, the oscillations can serve to make inferences on the system. But, let us suppose that a simple harmonic oscillator can be used to realistically model a biological unit, where oscillations are to be measured. If the model takes into account the energy dissipation then the asymptotic trajectory of a dissipative harmonic oscillator is a stable point, fixed in the space. This signifies that the amplitude of the oscillator is increasingly smaller
TEMPORAL PROPERTIES OF BIOLOGICAL SYSTEMS
793
up to eventual repose. In order for the oscillator to maintain periodicity, the system must be provided with energy in such a form that the continual loss associated with the dissipation is compensated for. If this relation is kept stable, then the orbit in the phase space is converted into a stable limit cycle, that is all the orbits around this orbit fuse with it asymptotically, and the oscillator maintains a stationary oscillation. From this point of view, we have the same problem as when we use models which are normally employed in ecology--models which are based on the hypothesis of a stationary state or in sequences of regular fluctuations. The typical example of this sort of model is that normally used to describe predator-prey interactions. This simple model oscillator allows us to study more meaningfully the relation between the distinct scales of complex oscillation which characterize the system, in the sense in which we have been explaining it; but it is lacking in some characteristics which are very apparent in biological systems. What happens when a system, as often occurs, is subject to disturbances which are infrequent or of a greater intensity than that which characterizes the cohesive forces of the system? One answer is that the unusual disturbances in one scale can be recurrent in another (Allen and Starr, 1982). Let us go back to the spectra (Figs 1 and 2). By way of these we obtain information about the forms of variation of the system as well as a certain degree of variability which is non-systematic or irregular fluctuations in the series, any of which prevents us from arriving at an adjusted prediction of the system although such a red-type spectra implies some predictability. Looking at the spectra one question which occurs to us is whether the shape of the spectra is a result of random fluctuation at dominant frequencies or whether there exists some underlying determinism in the process we observe in the system. Dynamic systems of non-linear characteristics do not have, over the course of time, a stationary or periodic state. The appearance of irregular fluctuations, their strange, unpredictable behaviour over the course of time, and the system's sensitivity to initial conditions causes us to think of chaos. (The chaotic properties corresponding to the asymptotic decay of the time structure of the spectra, Fig. 1.) The question is whether this type of behaviour can be characterized. It would seem so, by way of the study of the behaviour of the series in a multi-dimensional phase space. A time series can be employed by itself to put into relief the multivariate internal dynamic of the system shifting its values a fixed lag of time. An instantaneous state of the system will be defined by a point P and a sequence of states of the system will be defined by successive trajectories (Nicolis and Nicolis, 1984). If these converge to and define a subset in the phase space, where the system seems to be trapped, we term this invariant subset as attractor, where d is the dimension of the topology formed by this subset and n the minimum number of variables that should be involved in the description of the
794
E. G U T I E R R E Z A N D H. A L M I R A L L
system. Evidently the trajectories are conditioned by: the number of variables used to define the phase space and the existence or nonexistence of an attractor. The topology might be a point (then: d = 0 , n = 1), a line (d= 1, n = 2 ) , a surface (d = 2, n = 3) or a multiple topology. Nevertheless, the value o f d can be a noninteger and then the attractor i s termed a fractal attractor. If d reaches a saturation limit in relation to n, that is d becomes asymptotic at a certain number of n this limit will be called, d~, the saturation value of the topology dimension. To compute the attractors dimension we have used the algorithm given by Grassberger and Procaccia (1983a): N
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C(r)ocr", where/~ is closely related to d, the dimension of the attractor. To calculate the phase space from the series N variables where obtained by shifting its values a fixed lag of time, Z = 5 years, for the trees series and Z = 5 days for the fish series, this lag was chosen by observing power spectra. At this period the signal power is not significant. Regarding the length to compute a phase space is only possible for the longest series (F. sylvatica, P. uncinata, D. labrax, and water temperature). In graphic form (Fig. 5) we have represented d in relation to n and notice that there exist a d s for these series. For the fish series ds has the value of 2.3; for the trees it is 3.4 for Pinus uncinata and 3.2 for Fagus sylvatica. This means that we need 3 and 4 variables to define the behaviour of the attractor. More data points were computed for the fish series attractor dimension taking lags of z = 2, z = 3, and z = 4. The results confirm that the dimension oscillates between 2 and 3. Also computations were made taking lags of z = 6, z = 7, z = 8 and z = 9, the fractal dimension then died out probably due to the small number of points included in the analysis. According to Blythe and Stokes (1988) if d is not a fixed value this means that we are dealing with variables and not parameters. According to the results in Fig. 5, all the attractors are fractal, such as those found in non-linear dissipative dynamic systems. On the other hand the attractor calculated for water temperature is similar to that found for the otolith series, and this suggests that fishes do not simplify external conditions. The same comment can be pointed out in relation to tree growth considering the dimension of the climatic attractor (d = 3.1) computed by Nicolis and Nicolis (1984).
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Figure 5. Relationship between the dimension d of the topologies formed in the phase space using n variables from n = 2 to n = 8. The horizontal line represents the saturation limit of the attractor dimension, ds. The diagonal represents random noise with no saturation limit. The question according to Schaffer (1984) is now: does this signify that the cycles are governed by the attractor? And to this we do not know how to answer, but because the attractor's dimension is not an integer (i.e. a fractal attractor), 3 and 4 variables respectively are sufficient to describe the systems we are studying. One conclusion that comes out easily is that it is not necessary to appeal to external mechanisms in order to explain the cycles. Any discussion of causal mechanisms must also consider the possibility of a continual readjustment in the System, and its possible current state as non-unique, given any particular set of external variables. Stated somewhat differently, the effect of a given initial configuration leads to considerable differences in the final state the system reaches. This property is generally known as sensitivity to initial conditions.
3.3. Response and transfer functions. Further to what we have discussed, we m a y ask ourselves: what environmental variables have greater influence on growth and how are these variables internalized? F o r reasons of length and data availability, we have answered the first question by analysing the growth
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series of trees as a function of monthly mean temperatures for an observation period of N = 4 2 years. The method used to examine climate-ring width relationships is termed response function (Fritts, 1976). The statistical analysis performed was multivariate regression after extracting principal components, a process fully described elsewhere (Fritts, 1976; Draper and Smith, 1981). An example of this approach is that of Hughes et al. (1982). This method has proven to be one of the most accurate as it takes into account the multicollinearity of the predictors (Cropper, 1984) and does not violate the statistical constraints oforthogonality between independent variables (Draper and Smith, 1981). The unit record of temperature and precipitation was a 14month period starting in June of the year prior to growth, and ending in July of the year of current growth over the period 1941-82. The confidence intervals for each element of the response function were calculated at a significant level of p < 0.05, and the variance accounted for by the climatic variables is high, as can be deduced by the R2 value (Fig. 6). The horizontal axis represents the 14 month period and the vertical axis the value of the standardized response function coefficients. Vertical bars delimiting the 95% confidence interval for each element are also drawn. If the confidence intervals of any response functions element does not include zero, the element is said to have a significative effect on growth. The climatic variables which most affect their growth are the high temperatures as the growing season advances and the lack of precipitation, among others (Gutirrrez, 1987; 1988). Although the obtained response functions are quite complex for both P. uncinata and F. sylvatica (Fig. 6), some mean monthly temperature and precipitation have a great influence on growth and could be considered when modelling tree growth. In order to respond to the second question of how the environmental variables are internalized, we used the growth series of the fish as a function of temperature records of the water for a total of N = l 1 4 observations. Temperature plays an important role in determining the variability in thickness and type of increment observed. Time series analyses (Box and Jenkins, 1976; Liu and Hanssens, 1982) are used to analyse and to identify the dynamic transfer function of the bivariate process formed by temperature and otolith growth. Temperature exhibits auto-correlative structure. This feature underlines the dynamic nature of the growth process in relation to environment, which should be taken into account in the statistical analysis of the data. The dynamic relationship between the input and the output was well described by a third-order transfer function model. The impulse response function describes how the output is related to the input of the linear system. In this case it occurred at regular intervals of three time units (days) and decreased exponentially (Gutirrrez and Morales-Nin, 1986). As stated by the time series model, otolith growth is a conservative process
TEMPORAL PROPERTIES OF BIOLOGICAL SYSTEMS
797
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Figure 6. Response functions obtained for: (A) Pinus uncinata; (B) Fagus sylvatica taking into account monthly mean precipitations and temperatures from 1941 to 1982, unit period record from June of the year prior to growth (t-I) to July of the year of current growth (t). involving memory of the previous growth and also of the environmental conditions. Growth of a new increment integrates only a proportion of the disturbances which are absorbed and have a lasting influence. It would seem that the growth process includes a mechanism to adapt to changes in the environment based on a short-term memory. Such a situation would have to be explained in terms of fish's physiological mechanisms and hormonal and metabolite concentration levels. Daily changes in water temperature (perturbations at organism scale, i.e. molecular) cause an increase or in general a change, in the metabolic rate of the fish which returns "slowly" to the initial
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ground state. But this state has been modified previously and so on. The final steady-state is achieved at a new level in which entropy production is least (Prigogine and Wiame, 1946; Johnson, 1981; Ulanowicz, 1986). The organism is very sensitive to each set of initial conditions being the adult state, the end product of its historical process (see references above mentioned for further discussion). 4. Conclusions. We have studied some time series belonging to biological systems. Their time scales have been analysed through spectral analysis, and the structure of the variance by means of the spectral roll-off. According to the results the power spectra of these organisms is due to an evolutionary process of assimilation and internalization of impacts and not exclusively dependent on external physical causes. For the tree-rings g r o w t h series we have determined, from a given set of climatic variables, those which have a greater influence on growth; they are mainly low precipitations and high temperatures during the growing season. And in the case of the sea bass D. labrax, the internalization of fluctuations of temperature occurred at regular intervals and decreased exponentially. As the attractor dimensions are fractals any discussion of causal mechanisms, which tries to explain the behaviour of the systems studied, must consider the possibility of a continued internal readjustment and also the possibility that the current state of the system is not its unique state, given any particular set of external variables. As a clear and immediate co0sequence, it can be said that spectral analysis is incapable of distinguishing the attractor output of what we generally call random noise processes. Moreover, the studied biological systems do not simplify the complexity of external conditions. Finally, a realistic model of the growth activity of the species dealt with ought to consider the relevant characteristics described in relation to these systems. The authors gratefully acknowledge the technical assistance and comments of Jordi Flos. They thank Beatriz Morales-Nin for providing the temperature and Dicencharchus labrax series.
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Received 14 S e p t e m b e r 1988 Revised 27 J u n e 1989