Can the fractal dimension of fluctuations in population size be used to estimate extinction risk? The problem with estimating this fractal dimension is that the lengths of the time series are usually too short for conclusive results. This study answered this question with long time series data obtained from an iterative competition model. This model produces competitive extinction at different perturbation intensities for two different germination strategies: germination of all seeds vs. dormancy in half the seeds. This provided long time series of 900 years and different extinction risks. The results support the hypothesis for the effectiveness of the Hurst coefficient for estimating extinction risk

ADELA SALVADOR

ALFONSO GARMENDIA

Barnsley M. F.

Cutler C. D.

Falconer K. J.

García-Ruíz J. M.

Hastings H. M.

LUIS GARMENDIA

Mandelbrot B. B.

Pacheco J. M.

English

Garmendia Salvador, Alfonso

Garmendia Salvador, Luis

Salvador Alcaide, Adela

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

HURST COEFFICIENT IN LONG TIME SERIES 
OF POPULATION SIZE: MODEL FOR TWO 
PLANT POPULATIONS WITH DIFFERENT 
REPRODUCTIVE STRATEGIES 
ALFONSO GARMENDIA 
Instituto Agroforestal Mediterráneo 
Escuela Técnica Superior de Agronomía y Medio Natural 
Universidad Politécnica de Valencia 
Camino de Vera s/n, 4-6022 Valencia, Spain 
algarsal@upvnet. upv. es 
LUIS GARMENDIA 
Department of Programming and Computing Systems 
Complutense University of Madrid, Spain 
lgarmend@fdi. ucm. es 
ADELA SALVADOR 
Technical University of Madrid 
E.T.S.I. Caminos, Spain 
ma09@caminos. upm. es 
Received November 10, 2008 
Revised May 11, 2009 
Accepted March 31, 2010 
Abstrac t 
Can the fractal dimensión of fluctuations in population size be used to estimate extinction risk? 
The problem with estimating this fractal dimensión is that the lengths of the time series are 
usually too short for conclusive results. This study answered this question with long time series 
data obtained from an iterative competition model. This model produces competitive extinction 
at different perturbation intensities for two different germination strategies: germination of all 
seeds vs. dormancy in half the seeds. This provided long time series of 900 years and different 
extinction risks. The results support the hypothesis for the effectiveness of the Hurst coefficient 
for estimating extinction risk. 
Keywords: Homocarpy; Heterocarpy; Practal Dimensión; Extinction; Time Series; Hurst 
Coefficient. 
1. I N T R O D U C T I O N 
This article is the continuation of two previous 
articles.1'2 In "Fractal Dimensión of Birds Popu-
lation Sizes Time Series"1 a fractal was defined 
as an object whose topological dimensión does not 
coincide with the Hausdorff dimensión. The time 
series dimensión was studied, and different methods 
to obtain the Hurst coefficient (H) were offered. H 
was used in time series of Passeriforme populations 
to analyze its usefulness as an indicator of extinc-
tion risk. The main problem in the above study was 
that the 20 years time series used are too short for 
a good H estimate, and therefore it was found that 
longer time series were needed to study H's as an 
extinction indicator. 
"The importance ofthe intensity and frequency of 
perturbations on the germination delay"2 studied a 
model in which two plant populations with differ-
ent reproductive strategies competed for a territory. 
The model evaluated perennial plants, considering 
perturbations as the only cause of death. These per-
turbations were defined by two parameters: occur-
rence probability (OP) and perturbation intensity 
(PI). 
H has been used to estimate fractal dimensión3 7 
to study a variety of problems in ecology8-12 and 
biology.13-18 High H valúes can be interpreted as 
extinction risk.7'18 Hastings and Sugihara19 sug-
gested that increased range of fluctuations with the 
time interval might mean an increased likelihood of 
extinction. The main problem of estimating H for 
natural populations is that time series are always 
very short, and therefore sometimes it is difficult 
even to demónstrate their fractal structure. Iter-
ative population models provide an oportunity to 
check this hypothesis without this problem. 
In this paper, the time series obtained in Ref. 2 
were used to analyze H to test its effectiveness for 
extinction risk estimates. 
This work aims to verify whether H is a good 
indicator of species extinction based on: 
(I) the use of long time series (900 years), suffi-
ciently long for reliable measurement of H; and 
(2) different environments, defined by different per-
turbation frequencies and intensities that pro-
duce differences in extinction risk. 
The study objectives therefore are, firstly, to ana-
lyze variations in H for the two germination strate-
gies under different perturbation frequencies and 
intensities. Secondly, to compare time series prior 
to extinction and series without extinction, and to 
check whether differences in H enable prediction of 
species extinction. 
2. M A T E R I A L A N D M E T H O D S 
2.1. Brief Descript ion of the 
Iterative Model 
See Ref. 2 for a full explanation of the model used 
to obtain the time series. This iterative model was 
used to compare the effect of perturbation on two 
different reproductive strategies, homocarpic plants 
(whose seeds germinate the first year) and hetero-
carpic plants (that form a seed bank, from which 
only half of them germinate each year) competing 
for a territory. The model is used to examine the 
adaptability of these strategies to different pertur-
bations. Seeds that germinate must find and occupy 
an empty place in a particular cell in a grid and thus 
become an adult plant, otherwise they die. 
Perturbations were the only cause of death con-
sidered for adult plants, using random variables 
with a chosen probability, OP that indicates pertur-
bation occurrence probability, and PI that is defined 
as the likelihood of death for each plant when the 
perturbation occurs. 
In Ref. 2, it was concluded that heterocarpy is a 
competitive strategy in perturbed environments, as 
heterocarpic plants were dominant at high PI, and 
that the extinction of homocarpic plants was more 
dependent on PI than OP. In the present study, 
therefore, H was analyzed at different PI valúes, 
keeping OP at 0,5. 
The study área consisted of 100 cells distributed 
in a 10 x 10 toroidal grid, i.e. avoiding the presence 
of borders, and the independent variables were fixed 
for all the time series: the number of seeds produced 
by each adult plant was assumed to be three per 
year, randomly dispersed between the plant's own 
cell and the eight neighbouring cells. Five was the 
máximum number of adult plants in each cell. 
2.2. Programs 
Two computer programs were used for this paper. 
but both has already been used before: To obtain 
the H valúes, the program called "Hurst Coeffi-
cient" (HC) from Ref. 1 was used, and to obtain 
the long time series (1000 years) at different pertur-
bation intensities (and therefore, different extinc-
tion risks), the program "Time series" (TS) was 
used.2 
For each P /and each OP valué from 0.1-0.9, in 
steps of 0.1, the TS program was run iteratively to 
obtain 10 x 1000 years time series. The first 100 
years of each time series was eliminated so that the 
series only represent the phase in which populations 
have achieved a certain independence from initial 
conditions. This gave time series of 900 years if the 
species did not become extinct or less (up to the 
year of extinction) for the cases where the species 
became extinct. 
These time series were entered in the HC pro-
gram to calcúlate H. This program calculates H 
from several methods, but for clarity's sake, only 
one was chosen: the range increment method. Dif-
ferent methods can produce different results, but 
there is a linear relationship between the results 
of all of them1'12 and consequently no variation of 
the conclusions would emerge from which method 
is used. Nevertheless, results from second order and 
local methods were also compared but they are not 
included because trends were very similar. 
The range increment method was chosen because 
it is more closely related to the theoretical basis for 
using H to estimate extinction risk.12 The range is 
the difference between the máximum and minimum 
valúes in the time series in a given time interval. 
H variability was calculated by running the pro-
gram ten times for each OP and PI valué and 
obtaining the average and standard deviation from 
the ten H estimates. To observe the changes caused 
by PI (the factor which increases the extinction 
risk) in H, measurements were taken from PI of 
0.1-0.9 in steps of 0.1. 
The HC program1 is implemented in Pascal 
and can be downloaded free of charge from: 
http://personales.upv.es/~algarsal/hurst.zip 
The TS program2 that simulated compe-
tition for space between plants with homo-
carpic and heterocarpic reproductive strategies 
was implemented in C++ and can be seen at: 
http://personales.upv.es/~algarsal/plantas.zip 
2.3. Hurst Coefflcient Range 
Increment Method 
The range increment method obtains H by the 
expression R(At) = cAtH, where Ai is a given 
time step, and R(At) denotes average valúes in 
the process range {y(i)}, for all time steps of dur-
ation Ai. 
To test the hypothesis that the scaling is applica-
ble, tests of linearity has been made for several time 
increments (an example can be seen at Fig. 1). For 
comparison between different perturbation intensi-
ties and plant strategies, the time increment Ai was 
set at 10 and 100. 
H measured by the range increment method pro-
vides a measure of increase in the range of fluc-
tuations in a time series as the time interval is 
increased, which could be an indicator of extinction 
risk.12 Hastings and Sugihara19 suggested that as 
the range grows, fluctuations increase and so does 
the extinction risk. They also observed that H cal-
culated by this method in short time series could 
be higher than the real valué. 
As species sometimes become extinct in a period 
of 200 years (i.e. within the time series used) H 
could not be calculated by the range increment 
method using Ai = 100, and therefore the results 
are analysed using the range increment method with 
Ai = 10. 
0.6 n 1 
0.1 
0 ^ 1 1 1 1 
0 50 100 150 200 250 
At 
Fig. 1 Linearity test for the Hurst coefflcient (H) measured 
for the heterocarpic population with PI = 0.8, OP = 0.8 
{PI, perturbation intensity; OP, occurrence probability for 
perturbation). 
ANOVA was used to analyze H variability in the 
ten time series for each given valué of OP and PI, 
to evalúate the differences, and measurements were 
compared using Student's í-test. 
3. RESULTS 
3.1. Range Fluctuations Analysis 
In Ref. 2 it was found that range valúes were 
affected by PI with little influence by the pertur-
bation frequency. 
Fluctuation range behaviour in homocarpic and 
heterocarpic plants in 1000 years time series with 
OP = 0.5 and for different PI valúes is shown in 
Fig. 2. 
For PI = 0.2 the fluctuation range was 
around 100 plants; for homocarpic plants PI was 
approximately 200-300 plants, and heterocarpic 
150-250 plants. For PI = 0.5 the fluctuation range 
was around 200 plants; for homocarpic plants PI 
was 100-250 plants and for heterocarpic 100-300 
plants. For PI = 0.8 homocarpic plants became 
extinct within 200 years, and due to lack of compe-
tition, the number of heterocarpic plants increased, 
with wide fluctuations of 100-500 plants. The seed 
bank strategy favoured plants in environments with 
very intense perturbations. 
3.2. Hurst Coefflcients Measured by 
the Range Increment Method 
for Different OP and PI Valúes 
Table 1 shows the averages and standard devia-
tions for H in ten time series of homocarpic and 
PI = 0.2 Homocarpic PI = 0.2 Heterocarpic 
0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 
PI = 0.5 Homocarpic PI = 0.5 Heterocarpic 
500 
400 
300 
200 H 
100 
0 
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¡IfflfP l'IFIi'ipfflF'IIIPIiPllllipil' 
iiiiiii uní ii iiiiiiiiiiiii'iiiiiiiiiiiiiiii ni ...ni 
l||l|l|P|||l|Pf|¡|PI|Pl|l 
0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 
PI = 0.8 Homocarpic PI = 0.8 Heterocarpic 
500 
400 
300 
200 
100 I' .I , 
0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 900 
Fig. 2 A 1000 years temporal series for different PI with OP = 0.5, and different reproductive strategies, homocarpic and 
heterocarpic. Each pair of graphs represents one particular case. Increasing fluctuations in amplitude are evident with higher 
PI valúes (PI, perturbation intensity; OP, occurrence probability for perturbation). 
Table 1 The Range Increment Method Provided the Hurst Coefficient (H) for the 
Equation: fl(At) = cAt , where A i is the t ime-step, fl(At) denotes the average 
range valué of the process { Í / ( Í ) } , at all t ime intervals of duration A i . Range is 
the difference between the máximum and the mínimum valúes of y(t). This table 
shows the average H in ten different t ime series for homocarpic and heterocarpic 
strategies with different PI and OP valúes. The measures were made by the range 
method with A i = 100 years and A i = 10 years. Error was measured as the 
standard deviation for ten different t ime series. Different letters on H measures 
represent signiflcant mean differences ( P < 0.05) (PI, perturbation intensity; OP, 
occurrence probability for perturbation). 
OP = 0.2 
Homocarpic plants (Ai = 100) 
PI= 0.2 0.27 ± 0.01 e 
PI= 0.5 0.23 ± 0.01 c 
PI= 0.8 0.25 ± 0.01 d 
Heterocarpic plants (Ai = 100) 
PI= 0.2 0.29 ± 0.01 g 
P / = 0 . 5 0.23 ± 0.01 f 
PI= 0.8 0.23 ± 0.01 f 
Homocarpic plants (Ai = 10) 
PI= 0.2 0.52 ± 0.02 e 
PI= 0.5 0.49 ± 0.02 d 
PI= 0.8 0.51 ± 0.02 e 
Heterocarpic plants (Ai = 10) 
PI= 0.2 0.52 ± 0.01 e 
PI= 0.5 0.49 ± 0.02 d 
P / = 0 . 8 0.48 ± 0.02 cd 
OP = 0.5 
0.20 ± 0.00 b 
0.15 ± 0.01 a 
Extinct 
0.21 ± 0.00 d 
0.15 ± 0.01 b 
0.08 ± 0.01 a 
0.37 ± 0.03 ab 
0.33 ± 0.04 a 
0.40 ± 0.03 Extinct b 
0.37 ± 0.03 b 
0.33 ± 0.04 ab 
0.30 ± 0.04 a 
OP = 0.8 
0.27 ± 0.01 e 
0.20 ± 0.01 b 
Extinct 
0.28 ± 0.01 g 
0.19 ± 0.01 c 
0.22 ± 0.01 e 
0.46 ± 0.02 c 
0.45 ± 0.02 c 
0.52 ± 0.02 Extinct e 
0.46 ± 0.02 c 
0.46 ± 0.02 c 
0.48 ± 0.02 cd 
heterocarpic plants for 900 years (1000 years minus 
the first 100 years) of competition for each different 
P/and OP. For example, for PI= 0.8 and OP = 0.2 
from the range increment method with Ai = 10, in 
homocarpic plants, the mean valué of H = 0.51 was 
calculated from ten time series of 900 years each. 
The standard deviation was 0.01. 
H had uniform valúes for each of the ten time 
series used in each case, as the typical deviations 
were all very low (< 0.05). These typical deviations 
were lower (and the differences clearer) for Ai = 100 
than for Ai = 10, although the general behaviour 
was very similar. 
For PI = 0.8 and OP = 0.5 homocarpic plants 
became extinct in about 200 years and H = 0.4 for 
the time series before extinction. For OP = 0.8 and 
PI= 0.8, H = 0.52, and therefore, H increased with 
increased PI At the same time, H for heterocarpic 
populations was significantly lower (P < 0.05) when 
the homocarpic population became extinct. 
The highest H valúes (Ai = 10) for homocarpic 
plants were obtained from OP = 0.5, PI= 0.8 (H = 
0.40) and OP = 0.8, PI = 0.8 (H = 0.52), which 
suggests that H is a good indicator of extinction 
risk. A similar observation applied to Ai = 100. 
3.3. Study of H for the PI Variable 
For the results in Table 1, the average valúes for ten 
time series of H obtained by the range increment 
method for Ai = 10 and Ai = 100, for different 
valúes of P /and OP = 0.5, are found in Fig. 3. 
When PI increased, H decreased, fitting a 
straight line in heterocarpic plants. There was a 
very different shape for homocarpic plants, which 
decreased initially to a minimum of PI = 0.5 
and then increased. H increased when homocarpic 
plants showed an extinction risk. 
4. CONCLUSIONS 
The Hurst coefficient is a good indicator of extinc-
tion risk for the long 900 years time series used in 
this study. There was a strong relation between the 
H and PI as the fractal dimensión of the time series 
and fluctuations decreased (lower H) with P/in het-
erocarpic plants, which are the most resilient, best-
adapted to perturbation species. In homocarpic 
plants, which are more sensitive to perturbations, 
the coefficient and fractal dimensión reached high 
valúes for high PI on situations when there was a 
high extinction risk. 
0,4 
U ~~1 
1 1 
Homocarpic pañis 
H Helerocarpic pañis 
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 
PI 
(a) 
PI 
(b) 
• H_„ » , 
0,8 0,9 
Fig. 3 The mean Hurst coefñcient (H) valúes for homocarpic and heterocarpic plants during 1000 years of competition 
with different PI OP = 0.5 for all cases. The first 100 years of the temporal series were eliminated to include only the 
"stable" dynamic. (a) Measured with Ai = 10 years. (b) Measured with Ai = 100 years, H for PI = 0.8 and PI = 0.9 could 
not be measured, because extinction occurred in about 200 years (PI, perturbation intensity; OP, occurrence probability for 
perturbation). 
In general, H increased only when the extinc-
tion risk increased. No such increase occurred when 
there were no extinctions, even when perturbation 
conditions were intermedíate or even intense. The 
lowest H valúes occurred in heterocarpic plants 
when they were favoured by perturbations which 
eliminated competition (despite the fact that in 
such cases PI was very high), and in homocarpic 
plants when there were modérate perturbations 
with no extinction risk. 
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germination delay, Math. Bíoscí. 211 (2008) 153-
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Hurst Coefficient in long time series of population size: Model for two plant populations with different reproductive strategies

Archivo Digital UPM

A multiplicative multifractal model for originations and extinctions,

A review of the theory and estimation of fractal dimensión,

Applications of fractal in ecology,

El uso de la geometría fractal en las ciencias naturales,

Fractal dimensión of birds population sizes time series,

Fractal dimensión of vegetation and the distribution on anthropod body lengths,

Fractal species distributions do not produce power-law species-area relationships,

Fractal: Punto ñjo de aplicaciones contractivas,

Fractales y oceanografía,

Fractals Everywhere

Fractals in ecology: Methods and interpretation.

Fractals: A User's Guide for the Natural Sciences

Fractals: Form, Chance and Dimensión (Freeman,

Fundamental unpredictability in multispecies competition,

Scale invariante in biology: Coincidence or footprint of a universal mechanism?

Species-area curves, diversity Índices and species abundance distributions: A multifractal analysis,

The Fractal Geometry of Nature.

The Geometry of Fractal Sets (C.U.P.,

The importance of the intensity and frequency of perturbations on the germination delay,

 Can the fractal dimension of fluctuations in population size be used to estimate extinction risk? The problem with estimating this fractal dimension is that the lengths of the time series are usually too short for conclusive results. This study answered this question with long time series data obtained from an iterative competition model. This model produces competitive extinction at different perturbation intensities for two different germination strategies: germination of all seeds vs. dormancy in half the seeds. This provided long time series of 900 years and different extinction risks. The results support the hypothesis for the effectiveness of the Hurst coefficient for estimating extinction risk. </jats:p

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Fractals

HURST COEFFICIENT IN LONG TIME SERIES OF POPULATION SIZE: MODEL FOR TWO PLANT POPULATIONS WITH DIFFERENT REPRODUCTIVE STRATEGIES

Hurst Coefficient in long time series of population size: Model for two plant populations with different reproductive strategies

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