Fractal dimension of birds population sizes time series
Transcripción
Fractal dimension of birds population sizes time series
Mathematical Biosciences 206 (2007) 155–171 www.elsevier.com/locate/mbs Fractal dimension of birds population sizes time series Alfonso Garmendia a a,* , Adela Salvador b Department of Agro-Forest Ecosystems, Higher Technical School of Rural Environments and Enology, Polytechnic University of Valencia, Av. Blasco Ibáñez 21, 46010 Valencia, Spain b Higher Technical School of Civil Engineering, Madrid Polytechnic University, Spain Received 31 March 2004; received in revised form 25 January 2005; accepted 3 March 2005 Available online 26 September 2005 Abstract Information about fractal dimension is collected so that it can be applied to time series interpreting Hurst coefficient. The population size of a species is modelled as a dynamic system. The Hurst coefficient is calculated for these times series. A computer programme has been elaborated to compute the Hurst exponent of time series using the algorithms of range increment, second order moment increment and local second order moment increment. It has been applied to time series of birds populations. 2005 Elsevier Inc. All rights reserved. Keywords: Fractal; Time series; Hurst coefficient; Time series of species populations 1. Introduction Fractal geometry is a tool to quantitatively describe objects that are considered as extremely complex and disorderly. The term fractal denotes something irregular, intricate, in which the smallest parts are similar to the overall pattern. The knowledge of self-similar fractals, the quantitative study of the singularities that naturally appear on the iteration theory and the dynamic systems has been quickly developed through the use of the computer, which produces a quick repetition of the process. * Corresponding author. E-mail addresses: [email protected] (A. Garmendia), [email protected] (A. Salvador). 0025-5564/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2005.03.014 156 A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 1.1. Fractal Fractals are mathematic objects that fall within the Geometric Measure Theory. The exact and definitive delimitation of it is yet to be established [4,5,29,26,13]. A self-similar fractal can be defined as the final product of an infinite iteration of a well-specified geometric process. It is the fixed point of a set of contractive applications [22]. It allows the construction and control of extremely complex structures through very simple processes [25]. In literature, we can find different definitions for fractal: an object that has a fractionary dimension, a fixed point of a set of contractive applications. . . The definition we have used is a subset of Rn in which the topological dimension does not coincide with the Hausdorff dimension. 1.2. Dimension In 1919, Hausdorff introduced a key tool to measure those peculiar sets through the introduction of concepts we nowadays call Hausdorff dimension [29,26,57]. Besicovitch, during the 1920s, continued these works and included the geometric measure theory. Hausdorff dimension is a natural generalisation of the topological dimension. Both could are defined by the properties of their minimum open covering. The open ball B(p, r) of radius r and centre p on a metric space is the set Bðp; rÞ ¼ fx : distðx; pÞ < rg; where dist(x, p) is the distance between point x and point p. A set U in a metric space is known as an open set if, given a point p in U there is a distance r > 0 in the way that there is an open ball B(p, r) contained in U. A family of open sets {Ua} is known as an open cover of a set X, if X is contained within the union ¨{a}Ua of sets Ua. The topological dimension of a space is inductively defined as follows: The dimension of the empty set is 1, and the dimension of X is less or equal to n. This is the same as saying that X has a series of open sets with borders less or equal to n 1. Thus, it can be seen that a finite set has topological dimension of 0, a rectifiable curve has a dimension of 1, and a differentiable surface has a dimension of 2. The Hausdorff dimension (or Hausdorff–Besicovitch dimension) can be written as follows: Let X be a subset of Rn covered by N(r) open covers of radius lesser or equal to r. When r tends to 0, N(r) increments according to a power of approximately 1/r, let us say rD. Exponent D is called the Hausdorff dimension or fractal dimension of the set X. That is, for each r > 0, we calculate the lesser number of open covers N(r), of a radius less or equal to r, which is necessary to cover the set X. We can see that the limit log N ðrÞ ð1:1Þ D ¼ lim r!0 log r exists. Value D is called the Hausdorff dimension for X. (Since log r ! 1, the sign is necessary so that D is positive) [10,32,34,7,43,44,1,63]. A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 157 It can be seen that formula (1.1) is equivalent to the approximate exponent expression NðrÞ const rD ; ð1:2Þ D N(r) increments asymptotically with r . Two values x and y are asymptotic when x tends to 0 if the limit log y ð1:3Þ lim x!0 log x exists [41]. An observation that can be drawn is that usual sets have a Hausdorff dimension that coincides with their topological dimension. In a segment, a square or a cube, N(r) increments by 1/r, 1/r2, and 1/r3 respectively. The decade of the 1970 is marked by Mandelbrots intuition, who was the first to notice some of the application possibilities that this field had, and openly proposed them in a widely distributed publication. [26]. Mandelbrot in 1982 [44] says that an object is fractal when there is a discrepancy between these two dimensions. Thus, an object such as the curve of Kochs snowflake is an intermediate object between a line and a plane and its fractal dimension is log 4/log 3 [19,52,53]; the Cantor set has as fractal dimension log 2/log 3; and Sierpinskys triangle, log 3/log 2 [20]. The Hausdorff dimension is difficult to work with, and therefore in self-similar object, it can be calculated using the similarity dimension [36], since these fractals are the fixed point of a finite set of contractive applications. 1.3. Fractals in nature Finding natural fractal elements is easy, since the typical geometry of nature is fractal geometry. We find them in oceanography [50], in the measuring of coasts, mountains [45], islands, coral reefs [61]. Measuring of coasts: The classical example that appears in literature is the measuring of a coastline. It is observed that length L depends on the step size p, and that L is proportional to a power of p LðpÞ ¼ kpd ; therefore lnðLÞ ¼ lnðkÞ þ d lnðpÞ. Therefore, d is the slope of abscissa the logarithm of the step size and of ordinate the logarithm the length of the coast measured [45]. Other examples are the spatial distribution of animals and plants, the steadiness of vegetation patterns [47], the nature of fractures or fracture systems, the porosity of rocks, and the geometry of reefs [9]. Morse et al. [49] studied the fractal character of several branches of trees and river drainage networks. Lovejoy in 1982 [42], proved that the distribution of tropical clouds and rain areas has a fractal structure. The morphology of a flower corolla presents a certain corrugation [21]. The biologist C. Herrera from the Doñana Experimental Station, applied fractal geometry to his characterisation, and classified some flowers through the measuring of the fractal dimension of the corolla perimeter, proving that the fractal dimension was directly related to the absolute production of fruit, and therefore pollination is favoured in flowers that are profusely dissected. Fractal geometry has proved essential to understand chaotic population models [24], species originations and extinctions in the fossil record [56], multi-resource competition models [33], species–area relationships [40] and species diversity indices [8]. 158 A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 Fractal objects or behaviour also emerge in models not explicitly designed as such [51]. The movement of animals, treated as a diffusion problem, was characterised as a fractal object, and treated as a random walk [30]. 2. Dimension of function graphs. Time series dimension The problem of discerning the spectral (or fractal) properties of a model on the basis of a given time series has been addressed by many authors in a number of different contexts [16,14,13,28,65,23]. The graph of Brownian motion (Fig. 1), time continuous random process, is the prototype of random fractals [15,31]. A continuous process {y(t)} is known as a random process or a Brownian process on continuous time, if for a time step Dt, increments Dy(t) = y(t + Dt) y(t) are (i) Normal distribution. (ii) Zero average. (iii) Proportional variance Dt. Or the equivalent to (iii) (instead of (ii)): Successive increments Dy(t) and Dy(t + Dt) are not correlated. The axiom that characterizes random processes can be generalized with the characteristic of a fractal process [43,44] introducing an additional parameter, Hurst exponent H, (0 < H < 1), and replacing (iii) with (iii 0 ) Variance proportional to Dt2H. (Therefore, in the random process H = 1/2.) (iv 0 ) In a fractal process, successive increments have a correlation q, independent of time t, defined by 1 2H ð2:1Þ <q<1 . 2 ¼ 2 þ 2q 2 If {y(t)} is a fractal process with Hurst exponent H, then, "c > 0, the process 1 y c ¼ H yðctÞ c is another fractal process of similar statistical properties. It is renormalization. Fig. 1. The graph of Brownian motion. ð2:2Þ A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 159 In order to introduce the concept of random fractal, let us consider FH as the family of all the graphs of fractal processes with Hurst exponent H. The family FH is closed under renormalization (2.2) and all the elements of FH share the same statistical properties. 2.1. Powers (or power laws) Relationships between the scale measurements of an invariant system take the form of power laws. Functions with self-similar graphs or similar with different scales must be of the form y = f(x) = const xc for some exponent c. Since scale invariance requires f ðaxÞ ¼ bf ðxÞ ð2:3Þ for a constant a and a related constant b dependent of a, scale invariance implies f ðxÞ ¼ const xc ; ð2:4Þ where c = log b/log a. The logarithmic transformation of a power is a linear function log y ¼ logðconstÞ þ c log x. ð2:5Þ This permits the use of linear regressions to determine the fractal dimension of a process. 3. Fourier transform technique This technique can be used to calculate the fractal exponent H for every continuous periodic function of period 1. These functions can be written as a Fourier series through sine and cosine functions [31]: f ðxÞ ¼ 1 X ðan cos 2pnx þ bn sin 2pnxÞ. ð3:1Þ n¼0 The coefficients an and bn indicate the waves amplitudes and the Fourier coefficients set of a function is called spectrum. Therefore, any complex function can be written as a complex Fourier series, using the exponential function f ðzÞ ¼ þ1 X cn e2pinz ; ð3:2Þ 1 where cn is a complex number with real part an and imaginary part bn. The discrete Fourier transform (DFT), requires only a finite sequence of data points to obtain the Fourier coefficients. A function Power spectrum is the succession of the amplitudes squares for the Fourier coefficients and it is explained as the energy associated to each frequency. 160 A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 It is verified that: (1) E(cn) = 0. The mean is 0, both in the real and the imaginary parts. coefficients 2 2 1 (2) Eðjcn j Þ ¼ n2 Eðjc1 j Þ. The coefficients variance satisfies a power law scaling. When n increases, its module decreases. (3) The phases are independent and uniformly distributed on the interval [0, 2p]. Let f be a fractal function on the interval [0, 1] with f(0) = f(1), and with fractal exponent H. [31]. Let {cn; n > 0} be the spectrum. Then each cn is an independent sample from a complex valued normal distribution of expected value 0, and expected variance, the power spectrum, satisfies the power law Eðjcn j2 Þ ¼ const n12H . ð3:3Þ The phases are independent and distribute uniformly on the interval [0, 2p]. The fractal [29] is assembled from a series of sine wave components if different frequencies, the amplitudes of which satisfy a power law relationship Sðf Þ f ð2N þ12DÞ . ð3:4Þ 4. Noise spectrum or colour Noise spectrum or power spectrum in a time series can be split into a spectrum of basic waves each associated with a characteristic wave frequency or colour [29]. A population reddened dynamic imply that power increases with decreasing frequency, what means that successive points in the time series of population densities are more correlated than would be expected purely by chance. In the case that successive entries in a time series are uncorrelated, it is referred to as white noise, in which all frequencies occur with equal power. In the rare case of negative correlation between these terms, the data are said to be blue [59]. A spectrum in which this decrease obeys a power law (y = axb) is called 1/f noise and is a signature of fractal behaviour [58]. Pink noise is a special class of red noise where the log(power) (power is the squared magnitude of the Fourier spectrum) scales linearly with log(1/frequency) [59,54]. Frequency, f, is defined as the number of cycles per N points. To generate pink noise we set the amplitude of the component sine waves as a function of the frequency sffiffiffi rffiffiffiffi 1 i . ð4:1Þ ¼ Sðf Þ ¼ f N 5. Second order moment techniques [31,61] The third axiom offers a way to determine H on the postulate of a fractal model of experimental data. A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 161 For a fractal process and any Dt, the corresponding Dy has an expectation value 0 and equals EðDy 2 Þ ¼ cDt2H ; ð5:1Þ 2 2 where the average of Dy is an unbiased estimator of E(Dy ). Hence, Dx2 = cDt2H, where Dt is a time step and Dx2 is the second order moment of the corresponding spatial increment. 5.1. Determination of H EðDyðDtÞ2 Þ ¼ cDt2H ) ln EðDyðDtÞ2 Þ ¼ ln c þ 2H ln Dt; 2 2H EðDyð2DtÞ Þ ¼ cð2DtÞ 2 ) ln EðDyð2DtÞ Þ ¼ ln c þ 2H ln 2Dt. Subtracting these two equations ln E½yðt þ 2DtÞ yðtÞ2 ln E½yðt þ DtÞ yðtÞ2 ¼ 2H ðln 2Dt ln DtÞ ¼ 2H ðln 2Dt=DtÞ ¼ 2H ln 2. Therefore 2 2 H ¼ 1=ð2 ln 2Þfln E½yðt þ 2DtÞ yðtÞ ln E½yðt þ DtÞ yðtÞÞ g. ð5:2Þ As the Hurst exponent of a fractal process is independent of the time step used, local calculations of the form (5.2) can be made to test whether a process is fractal. Precaution: In the case of long time series, the increment Dy is very small compared to the second order moment, and the terms second order moment and variance are interchangeable. It is not so for short series, in which expectation and second order moment must be tested independently. 6. Local second order moment techniques Hurst exponent can also be determined from the correlation coefficient between successive increments, using formula (2.1). Under the assumption that the Dy has an expected value of 0, we can determine q through the definition Eð½yðt þ 2DtÞ yðt þ DtÞ½yðt þ DtÞ yðtÞÞ q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . Eð½yðt þ 2DtÞ yðt þ DtÞ2 ÞEð½yðt þ DtÞ yðtÞ2 Þ The expectation of [y(t + 2Dt) y(t + Dt)] Æ [y(t + Dt) y(t)], [y(t + Dt) y(t)]2 are unbiased estimators for the expectations. Local exponent H is obtained from formula (2.1) 22H ¼ 2 þ 2q; which is : H ¼ lnð2 þ 2qÞ= ln 4. ð6:1Þ [y(t + 2Dt) y(t + Dt)]2, ð6:2Þ The axiom for a fractal process can be tested by repeating this local computation for different values of Dt. 162 A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 In general, formulae (6.1) and (6.2) compute the local fractal exponent even when the expectation value is not zero and q is not the correlation coefficient. 7. Range increment Range is the difference between the maximum and the minimum values of y(t). R(Dt) denotes the average range of the process {y(t)} on all intervals of deviation Dt. In the equation 1 y c ¼ H yðctÞ c processes {y(t)} and {yc(t)} must have the same expected range. This implies that the range of the process {yc(t)} in an interval of duration Dt is 1/cH times the range of the process {y(t)} in an interval of duration Dt/c. Replacing Dt/c by Dt we obtain RðDtÞ ¼ cDtH ; ð7:1Þ where Dt is the time step and R(Dt) is the average range value in the interval of time of duration Dt. The formula (7.1), for the case of a fractal process in discrete time, is only maintained in long enough intervals of time. In short times, the range cannot be adequately determined, which implies that R(Dt) will increment quicker than DtH. For this reason, in Brownian motion in discrete times, the result is H = 0.63 instead of H = 0.5. 8. Fractal dimension and Hurst coefficient on northern europe passerine species Natural objects are not ideal fractals, but their properties are often sufficiently similar across a wide range of feasible scales that the tools of fractal geometry can be used, providing novel insights where Euclidean tools were found to be insufficient for describing such objects [18,24,3]. The fractal model, like the straight line in linear regression, may be seen as a simplifying frame that help us understand certain features of reality, without necessarily having to be strictly true itself [29,31]. In many ecological cases there is not a stationary state, necessary to understand their dynamics by Euclidean tools. In these cases variance is more informative that mean [6] and stochastic processes defined in term of fractal geometry (1/f noise models) can thus describe much ecological variability [27]. By definition, population levels of a species persistent in time are maintained within limits and therefore the fluctuation range of the population (supposed as random) has an asymptotic value [37]. The variance of population time series increases with observation time, apparently without limit [38,55]. Traditional models of density-dependent growth imply the existence of a basin of attraction, which confines the fluctuation of population abundance to a well-defined range of values about equilibrium [37]. Thus, for tightly regulated populations, the variance should converge to a clear limit in long enough series, but inside these fluctuations there is A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 163 density-independent stochastic growth, the prime example of which is a random walk, for which the variance grows linearly with time [2]. It is therefore a challenge to predict the variation range of a population in the long term, using relatively short time series. The 1/f noise process has been used in simulation models of extinction rates [28] and in laboratory experiments for testing population-dynamic hypotheses [11]. Environmental noise is one of the components affecting the population abundances. This means that changes in extrinsic abiotic factors after the removal of diurnal, lunar and seasonal cycles alter the environment surrounding the population. These changes can be for example variations in the temperature, fires and floods. Environmental forcing has been argued to be one of the main reasons behind the redness of the animal populations [60,2]. Steele [60] examined the records of both terrestrial and marine physical systems and observed that marine environment shows red dynamics and the terrestrial one white up to several decades and after that red spectrum. This gives evident propositions for what kind of autocorrelated noise is appropriate for these two physical systems. Hurst coefficient, measured with the method of range increment, measures the increment of population fluctuations when the time interval increments Dt. This means that a population with a greater Hurst exponent has larger fluctuation range increments. Disregarding the value of constant c of formula (7.1), for the same population sizes, the greater values of Hurst coefficient could be related to a greater danger of extinction [61]. On a time series with a fractal structure, it could be expected that Hurst coefficients, measured by the second order moment method and the range method and the local second order moment method, be similar. Hastings and Sugihara [31] advise us that for short time series, Hurst coefficient calculated by the range increment method offers values that are greater than the real ones. However, there is also a prevailing tendency, across a wide variety of species, for temporal variability to increase with the length of the census [46,17,2,55,62]. It has usually been associated with spectral reddening, (a tendency for low or high abundances to be followed by more of the same) of population dynamics. Dynamics can become reddened in several ways: Redness can be inherited from variation in the environment [60], it may arise through certain types of stochastic density dependence [48,2], or it may be generated through long-range spatial interactions [64]. One would expect a population whose numbers fluctuate more over time to have a greater risk of extinction [37,38,55,39], although this need not always be so [28]. The fractal exponent (Hurst coefficient), i.e. the range increment of population variations, can be due to characteristics intrinsic to the species, or to environmental characteristics. If the range increment rate for the variations of a population depends more on environmental (local) characteristics, the Hurst coefficients of the different populations will not be related. In the case they were, we would be dealing with a characteristic intrinsic to the species (at least in the studied areas). There are three hypotheses to be tested: (1) Check the fractal structure of the time series of passerine populations. In order to carry out this check, we use the three measurements of the fractal exponent by the three aforementioned methods (range increment, second moment and local second moment) and their comparison. (2) Check if Hurst coefficient is maintained for the same species in different places. 164 A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 (3) Check if the range increment of the fluctuations of a population can be related to the danger of extinction or some other population parameter. The data come from real census and therefore are short time series. This means that fractal dimension estimates may be inaccurate due to wandering intercepts [12], but they are still interesting for comparisons (see [29]). 9. Materials and methods To measure Hurst coefficients of different passerine species, the data of the Bird Census News [35] regarding three countries of Northern Europe (Finland (f), Sweden (sw) and Denmark (d)), which form a latitudinal gradient, has been used. In Sweden the time series of both register methods used have been utilized: counts at count points (swp) and parcels registered through the mapping method (swm). The different measurements of Hurst coefficients have been carried out by updating a computer programme. This programme measures Hurst coefficient through range and second order moment increment and through the method of local second order moment. 9.1. Programme The programmes proposed by Hastings and Sugihara [31] to calculate fractal exponents of time series using Hurst exponent with the techniques of range increment, second order moment increment, and local second order increment have been elaborated. In these programmes, in Pascal language, (Turbo Pascal 5.0 de Borland), they have been unified in one, and the correct functioning has been checked with the data used by [31]. It was modified to adapt to the new data that needed to be evaluated. Easy access menus were introduced, graphic proceedings were incorporated, and the screen display and the printing results were improved. The program used to measure Hurst coefficient by the three mentioned methods is freely available on: http://www.bi.upv.es/~algarsal/hurst/hurst.zip. To check if these time series have a fractal structure, despite the fact that the results obtained through the different methods to estimate Hurst coefficient are completely different, we have used linear regressions between the Hurst coefficient obtained by the second order moment increment method and the range increment method. A comparison between both Hurst coefficients obtained from the four sets of data has been carried out though a Spearman correlation analysis and Wilcoxons non-parametric analysis. 9.2. Estimate confidence limits for fractal exponents [31] With P the data set (x2 i, yi) is constructed the regression line y = a + bx, that minimize the squares sum: (yi a bxi) obtaining the distribution functions of a, b and r (correlation coefficient). Let B denote the random variable corresponding to the slope b of a regression line through n points, then the transformed random variable A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 1=2 ðB b0 Þ ðn 2Þr2x T ¼ h i1=2 ð1 q2 Þr2y has a Students t-distribution with n 2 freedom degrees. Solving for B yields h i1=2 ð1 q2 Þr2y B ¼ b0 þ T 1=2 ðn 2Þr2x 165 ð9:1Þ ð9:2Þ a distribution of mean 0 and variance ð1 q2 Þr2y varðBÞ ¼ . ðn 4Þr2x ð9:3Þ Confidence limits are readily obtained from tables of the Students t-distribution. For n larger than 25–30 fitness between the normal approximation and the t-distribution is enough and implies that T is approximately normal with mean 0 and variance 1. Therefore, B is approximately normal with mean b0 and variance varðbÞ ¼ ð1 r2 Þr2y . ðn 4Þr2x ð9:4Þ In this case, confidence limits for b0 are readily obtained using tables for the normal distribution: pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi b 1:96 varðbÞ 6 b0 6 b þ 1:96 varðbÞ ð9:5Þ B statistics can be understood using the residual variance (variance of non explained errors or the differences between predicted and observed values) r2 ¼ ð1 q2 Þr2y ð9:6Þ to rewrite variance of B as varðBÞ ¼ ð1 q2 Þr2y r2 ¼ . ðn 4Þr2x ðn 4Þr2x ð9:7Þ This result can be used to calculate the confidence intervals for the fractal exponent. In the program the standard deviation of the regression line (sdb) is measured as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 r2 Þ ðsumysq k ybar2 Þ sdb ¼ ð9:8Þ ðk 4Þ ðsumxsq k xbar2 Þ being r the correlation coefficient and k the time step used to measure the fractal exponent. Then vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pk 2 ! u u P ln y i k 2 i¼1 uð1 r2 Þ i¼1 ðln y i Þ k k u u ð9:9Þ sdb ¼ u Pk 2 ! u ln xi t ðk 4Þ Pk ðln x Þ2 k i¼1 i i¼1 k 166 A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 when H is calculated by the second order moment technique (mom2), for the confidence interval it has to be multiplied by a correction factor, expfactor = 0.5. For H calculated by the range increment, expfactor = 1. For k 6 4, sdb cannot be calculated. For 4 < k 6 27, the value is on the tables of the Students t-distribution, with k 2 freedom degrees. For k > 27, normal table is used and the confidence interval is H 1:96 sdb expfactor 6 H 6 H þ 1:96 sdb expfactor ð9:10Þ 10. Results The Hurts exponent has been measured by three different methods: Range Increment, second moment growth, and local second moment growth of 20 * 4 real temporal series of passerine populations, two of Sweden (different census methods), one of Denmark and one of Finland (see Table 1). When analysing the data of the different measurements, it can be said that the time series do not seem to have a fractal structure due to the great differences existing between Hurst coefficients measured through the different methods. The correlation coefficients of the regression lines of the range increment method are very high, whereas for the method of second order moment increments are not acceptable in many cases. However, the regression analysis of the Hurst coefficients calculated by the methods of range increment and second order moment is highly significant (p < 0,0001, R2 = 71%), H moment ¼ 60:6 þ 1:4H range. This implies that the great differences observed are due to the shortness of the time series used in this work. In these regressions, it could be clearly observed that the coefficient measured through the method of range increment is larger than the coefficient measured through the method of the second order moment, which was predictable when using time series that were too short (see [31] for similar results). Using Wilcoxons analysis and Spearmans correlation analysis to compare the four time series, only a certain positive correlation (p < 0.1) is observed between the values for Finland and Denmark. Surprisingly, no correlation is observed between the two time series of Sweden. To see the relationship between Hurst coefficient and the different population parameters, the coefficient measured from the range increment has been used, since it is the measure with less variation. No relations are observed with any population parameter (body size, colonisation capability, food type or phylogeny). Fractional Brownian motion can be divided into three quite distinct categories: H < 1/2, H = 1/2 and H > 1/2. The case H = 1/2 is the ordinary Brownian motion, which has independent increments, i.e. y(t + 2Dt) y(t + Dt) and y(t + Dt) y(t) being independent in the sense of probability theory; their correlation is 0 [53]. For H > 1/2 there is a positive correlation between these increments, i.e. if the graph of y(t) increases for same t, then it tends to continue to increase for t 0 > t (Fig. 2). For H < 1/2 the opposite is true. There is a negative correlation between the increments (Fig. 3). Finland Anthus trivialis Motacilla alba alba Prunella modularis Erithacus rubecula Phoenicurus phoenicurus Turdus philomelos Turdus iliacus Sylvia curruca Sylvia borin Phylloscopus sibilatrix Phylloscopus collibita Phylloscopus trochilus Regulus regulus Muscicapa striata Ficedula hypoleuca Parus montanus Parus major Garrulus glandarius Fringilla coelebs Carduelis spinus Emberiza citrinella Sweden mapping method Sweden point counts Denmark Second moment growth Range increment Second moment growth Range increment Second moment growth Range increment Second moment growth Range increment 0.40 ± 0.09 0.41 ± 0.06 0.28 ± 0.03 0.11 ± 0.14 0.17 ± 0.07 0.33 ± 0.07 0.44 ± 0.09 0.13 ± 0.17 0.11 ± 0.08 0.08 ± 0.13 0.49 ± 0.02 0.23 ± 0.07 0.44 ± 0.11 0.14 ± 0.19 0.20 ± 0.08 0.34 ± 0.13 0.02 ± 0.13 0.30 ± 0.13 0.45 ± 0.05 0.01 ± 0.13 0.13 ± 0.10 0.83 ± 0.05 0.71 ± 0.01 0.68 ± 0.04 0.58 ± 0.06 0.38 ± 0.03 0.66 ± 0.05 0.81 ± 0.05 0.47 ± 0.05 0.54 ± 0.03 0.47 ± 0.08 0.79 ± 0.05 0.73 ± 0.05 0.72 ± 0.02 0.39 ± 0.04 0.59 ± 0.03 0.63 ± 0.04 0.45 ± 0.07 0.63 ± 0.02 0.75 ± 0.07 0.48 ± 0.02 0.61 ± 0.04 0.24 ± 0.09 0.63 ± 0.03 0.36 ± 0.09 0.46 ± 0.10 0.22 ± 0.07 0.46 ± 0.10 0.31 ± 0.03 0.62 ± 0.04 0.81 ± 0.04 0.75 ± 0.05 0.71 ± 0.05 0.66 ± 0.04 0.75 ± 0.04 0.60 ± 0.02 0.90 ± 0.03 0.93 ± 0.02 0.73 ± 0.02 0.66 ± 0.02 0.70 ± 0.05 0.73 ± 0.02 0.57 ± 0.06 0.61 ± 0.02 0.71 ± 0.04 0.64 ± 0.02 0.80 ± 0.02 0.60 ± 0.03 0.53 ± 0.01 0.63 ± 0.04 0.76 ± 0.08 0.49 ± 0.04 0.54 ± 0.00 0.66 ± 0.01 0.72 ± 0.02 0.81 ± 0.02 0.50 ± 0.04 0.34 ± 0.06 0.30 ± 0.11 0.22 ± 0.13 0.11 ± 0.19 0.24 ± 0.06 0.22 ± 0.14 0.39 ± 0.07 0.59 ± 0.06 0.31 ± 0.07 0.13 ± 0.07 0.03 ± 0.11 0.43 ± 0.19 0.07 ± 0.21 0.18 ± 0.03 0.42 ± 0.04 0.29 ± 0.07 0.30 ± 0.10 0.07 ± 0.07 0.13 ± 0.12 0.34 ± 0.03 0.15 ± 0.05 0.66 ± 0.03 0.66 ± 0.05 0.42 ± 0.05 0.68 ± 0.05 0.68 ± 0.02 0.50 ± 0.03 0.13 ± 0.10 0.51 ± 0.04 0.60 ± 0.02 0.37 ± 0.05 0.38 ± 0.03 0.85 ± 0.06 0.62 ± 0.02 0.72 ± 0.03 0.90 ± 0.06 0.62 ± 0.04 0.71 ± 0.02 0.68 ± 0.01 0.67 ± 0.02 0.57 ± 0.03 0.79 ± 0.03 0.54 ± 0.03 0.54 ± 0.02 0.46 ± 0.04 0.47 ± 0.07 0.62 ± 0.03 0.72 ± 0.01 0.57 ± 0.02 0.58 ± 0.06 0.71 ± 0.04 0.59 ± 0.04 0.68 ± 0.02 0.63 ± 0.03 0.36 ± 0.06 0.45 ± 0.05 0.39 ± 0.08 0.05 ± 0.23 0.56 ± 0.07 0.19 ± 0.08 0.12 ± 0.06 0.01 ± 0.06 0.29 ± 0.10 0.21 ± 0.11 0.28 ± 0.04 0.14 ± 0.05 0.38 ± 0.08 0.40 ± 0.02 0.24 ± 0.06 0.21 ± 0.09 0.41 ± 0.02 0.28 ± 0.11 0.56 ± 0.02 A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 Table 1 Hurst coefficients calculated from the passerine populations time series 167 168 A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 H2ndM = 0.40; HRange = 0.83 H2ndM = 0.49; HRange = 0.79 Anthus Trivialis Finland 140 120 100 80 60 40 20 0 Phylloscopus collibita Finland 150 100 50 0 1 1 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 Fig. 2. Population time series of different passerine species with Hurst coefficient H > 1/2. H2ndM = 0.14; HRange = 0.39 H2ndM = 0.01; HRange = 0.48 200 Muscicapa striata Finland 150 150 100 100 50 50 Carduelis spinus Finland 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1 2 3 4 5 6 7 8 9 10 11 12 13 Fig. 3. Population temporal series of different passerine species with Hurst coefficient H < 1/2. Theoretically H can be related with the extinction risk [28,61]. If this is true, the species with higher extinction risk would be the species with higher H on the three countries: A. trivialis, M. alba, P. modularis, and on the other side, S. curruca, S. borin, M. striata and P. major are the species with more stable populations. There is not a latitudinal trend for all the species, although some of them are more stable on higher latitudes (P. phoenicurus, P. sibilatrix) or on lower latitudes (E. citrinella). 11. Conclusions The time series of the passerine that we have worked with are too short to permit the confirmation of their fractal structure (if they have it). It can be observed by the linear regressions that the H estimations from different methods are well related in all the cases. The coefficient measured by the range increment method is always bigger than the coefficient measured through the method of the second order moment increase, which was predictable when using time series that were too short (see [31] for similar results). These differences in the H estimations are linearly related. Although the real H value cannot be obtained with these time series (too short), the relation between the values obtained by different methods suggest that still is possible to compare the values from different countries and species. A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171 169 On most cases, for each specie, the Hurst coefficient maintain a high or a low value for all the countries, suggesting that it is an intrinsic variable of the species, independent from environmental factors. But for other species the differences are very conspicuous suggesting precisely the opposite: it is a variable so highly sensitive that it offers different values on the same populations, with the same species but with different registering methods. Moreover, this coefficient does not correlate with any other population parameter studied. 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