霊長類研究

総説

Primate Res. doi：10.2354 / psj.30.011（早期公開）〈高島賞受賞記念〉

The Fractal Primate: Interdisciplinary Science and the Math behind theMonkeyAndrew James Jonathan MacIntosh1）,2）
1）Kyoto University Wildlife Research Center, Kyoto, Japan
2）Center for International Collaboration and Advanced Studies in Primatology(CICASP), Kyoto University Primate Research Institute, Inuyama, Japan“Nature exhibits not simply a higher degree but analtogether different level of complexity”

generally the emerging ﬁeld of complexity science,has provided us with the tools necessary to assess
such a theory and its inﬂuence has now penetrated
all ﬁelds of science, including behavioral ecology.
In this essay, based on the Takashima Prize
lecture I was invited to give at the 29th Congress of
the Primate Society of Japan in Okayama, 2013, I
modestly attempt to exemplify how theinterdisciplinary study of complex phenomena — inthis case fractal patterns in individual-level
behavioral organization — can provide novel
insights into important ecological processes. I begin
with a brief (and admittedly simpliﬁed) introductionof key concepts in fractal geometry (Section 2: A
Geometry for Nature), and then provide examples
of how and why its analysis has been employed inbiological/ecological studies (Section 3: Fractals in
Ecology and Evolution). I then provide evidence- Benoit B. Mandelbrot, 1924-2010
1. Introduction
In the 4th century BCE, the eminent Greek
mathematician Euclid laid the foundations formathematics in the Elements, a work that persisted
as the main text for teaching mathematics andespecially geometry for as many as two millennia
and still features prominently in the curriculum
today. However, as Benoit Mandelbrot - the“father
of fractals”- notes in his seminal work The Fractal
Geometry of Nature (1977a, p. 1),“Clouds are notspheres, mountains are not cones, coastlines are
not circles, and bark is not smooth, nor does
lightning travel in a straight line”. Therefore, whilemathematics has always sought to formalize thenatural world, we remained until recently unable tocharacterize even the most common of naturalobjects and other natural phenomena. What theworld needed, Mandelbrot concluded, was a‘theory of roughness’. His fractal geometry, alongwith the closely related chaos theory and morefrom my own and other work that animal behavior
sequences occur in fractal time, before illustrating
how fractal analysis can be used as a sensitive
indicator of behavioral‘quality’(Section 4: Toward
the Diagnostic Fractal). I argue that complex
behavior sequences both result from and arenecessary
to
cope
with
environmental

2014 年 4 月 23 日受付，2014 年 4 月 30 日受理，2014 年 6 月 17 日早期公開（J-STAGE）
e-mail: andrew.j.jmacintosh@gmail.com

1

2

Andrew James Jonathan MacIntosh

heterogeneity. This body of work supports the
hypotheses that complexity is biologically adaptive
and that complexity loss, i.e. greater periodicity or
stereotypy, is not only indicative of altered or
impaired condition but may in addition carry with it
significant fitness consequences.
Despite the title, my aim throughout this review
is to facilitate conceptual understanding only; very
few equations will appear in these pages. Interested
readers seeking more rigorous explanations ofmathematical concepts, however, are directed toany of a number of excellent texts on this andvarious related topics (Mandelbrot 1977a; b; Beran
1994; Vicsek et al. 1994; Vicsek 2001; Peitgen etal. 2004; Seuront & Strutton 2004; Seuront 2010;
Viswanathan et al. 2011). Furthermore, this essay
does not present an extensive or critical review ofthe evidence supporting or rejecting the use of
fractal geometry in behavioral ecological studies,but instead highlights a number of ways in which
the approach has been used and with hope, thereby
presents an array of avenues for future studies. Fora critique of the application of fractal tools in
ecological studies, readers are directed to Halley etal. (2004).
My own work in this context focuses on
exploring complex and variable qualities insequences of behavior at the individual level,
particularly in relation to the various and often
increasing ecological challenges that individual
animals are likely to encounter in their changing
environments. In The Foundations of Ethology(1981, p. 50), Konrad Lorenz notes that,“so much
instinctive behavior is directly dependent on thephysical condition of the animal”. Yet, directly
linking altered condition and behavior under natural
conditions remains exceptionally challenging. The
work presented here therefore grew out of a great
desire, if not need, to search for answers outside ofour reductive and intra-disciplinary traditions, andinstead incorporate ideas and methodologies fromthe non-reductive and multidisciplinary science ofcomplexity. But more generally, I must echo thesentiments of a recent essay aimed at behavioral
ecologists, in which Bradbury & Vehrencamp(2014) note, likely prophetically, that“complexity
is one of the major remaining frontiers in our field”.
In one of his many classics of non-fiction,
Chaos: making a new science, James Gleick (1987,
p. 38) notes that “Ideas that require people toreorganize their picture of the world provoke
hostility.”He later goes on to quote Mandelbrot
himself, however, in stating that“Science would beruined if … it [withdrew] entirely into narrowly
defined specialties”(Gleick 1987, p. 90). I sincerely
hope that the ideas presented in this review
encourage the receptive reader to continue in the
tradition invoked by such ideals.
2. A Geometry for Nature
2.1 Mathematical Monsters
Before fractal geometry was consolidated into a
coherent field of study by Mandelbrot in the secondthhalf of the 20 century, various complex curves that
defiedmathematicalexplanation
appearedsporadically beginning in the late 19th century.
These ‘mathematical monsters’included among
other things the famous Cantor set, Koch snowflake
and Sierpinski Gasket (Fig. 1). It was Mandelbrot
who held that “The existence of these patterns
challenges us to study these forms that Euclid
leaves aside as being ‘formless’, to investigate themorphology of the ‘amorphous’” (Mandelbrot
1977a, p. 1). Through such investigations,
Mandelbrot’s work led to the creation of an entirely
new field: fractal geometry. Ultimately, this branchof mathematics took hold because some curves are
so intricate that they elude simple measurement
properties such as length and surface area. These
fractal shapes are self-similar, meaning that as you
zoom in, the same patterns appear over and overThe Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

3

th
th
Figure 1: These three‘mathematical monsters’appeared in the late 19 and early 20 centuries. These
complex shapes are generated using simple iterative processes. For example, the Cantor set (top)is formed by simply removing the middle third of each line recursively. Similarly, construction of
the Koch snowflake (bottom left) also begins by dividing each line segment into three equal parts,
but instead of removing the middle third, an equilateral triangle is placed with the removed middle
segment acting as the base. The Sierpinski triangle (bottom right) is a little more complicated.
Beginning with a filled in equilateral triangle, place a point at the center of each of its three sides.
By connecting these three points with new vertices, and keeping the old vertices intact, four
smaller triangles are produced, among which the center triangle is removed. The result is threefilled in equilateral triangles each with vertices half the length of the original and each touchingboth other triangles at a corner such that a triangle-shaped‘hole’is left in the center. Iterationof this process reveals the shape illustrated in this figure, but in fact, any shape subjected to thesame procedure will magically produce more or less similar results. What is perhaps most striking
about these curves is that their lengths are infinite, which can be understood clearly by looking attheir generative processes. The fractal dimensions of these shapes are 0.63, 1.26 and 1.59 for the
Cantor set, Koch snowflake and Sierpinski triangle respectively. For more on these curves see forexample (Mandelbrot 1977a; b; Peitgen et al. 2004).

again as reduced-scale images of the whole, andtheir properties can be related to those at otherscales through a scaling exponent (see below). For
mathematical fractals of course, these patterns are
reproduced with perfect fidelity, but nature tends
insteadtoproducestatistical
fractals,
generalizations of their ideal counterparts but often
following the same, though imperfect, generative
rules. It is interesting to note that the natural
structures we see all around us more closely

resemble those early monsters than they do
Euclidean shapes. As a result, fractal geometry hastaken over as the geometry of nature (Mandelbrot
1977a).
2.2 The Fractal Dimension
The exact definition of fractal has remained
elusive, as was intended by Mandelbrot himself,
but mathematically this class of shapes can becharacterized as having Hausdorff-Besicovitch (H-4Andrew James Jonathan MacIntoshB) dimensions which exceed their topological(Euclidian) dimensions (Mandelbrot 1977a). For
non-fractal, Euclidian shapes like points, lines,
planes, and spaces, the H-B dimensions are equal totheir topological dimensions of 0, 1, 2, and 3
respectively (Peitgen et al. 2004). For lines, squares
and cubes, for example, a strict power law with an
exponent equal to the topological dimension
governs their scaling behavior. In other words, you
can reduce a line by 1/3 and find that making two
copies of the reduced version and aligning them
together reproduces the original. Therefore, theself-similarity dimension - a type of fractal
dimension estimate based on the relationshipbetween the reduction factor and the number ofpieces into which the object can be cut - is 1, thesame as a line’s topological dimension. If you take1/3 of a square and want to reproduce the original,
however, you will need nine pieces, or 32. Here, theself-similarity dimension is two, as indicated by thescaling exponent. As was the case for the line, thisis equal to the topological dimension of a square.
Similarly, twenty-seven pieces (33) that are 1/3 thesize of a given cube are necessary to reproduce theoriginal, and so on. There are two key differences,then, between these shapes and fractal shapes. The
first is that the self-similarity dimension is an
integer, i.e. not fractional. The second is that the
reduction factor can be absolutely anything and thesame power law holds, they are completely
differentiable; for fractal shapes that even havethem, the reduction factors themselves arecharacteristic (Peitgen et al. 2004). For example,
the Koch snowflake shown in Figure 1 has a selfsimilarity dimension of 1.26, a non-integer, and can
only be reproduced using reduction factors
following 1/3k .
The above relationships hold for all strictly selfsimilar objects, but the statistical fractals observedin nature are far more complicated. Nonetheless,the above description should illustrate somefundamental properties of fractal shapes, i.e. selfsimilarity and scaling. Ultimately, though, what thefractal dimension gives us is a quantitative measureof inherent structural complexity, representing therelationship between patterns within a structure andthe scale at which they are measured (Mandelbrot
1967; 1977a). It is thus an indispensable tool in themensuration of roughness. The fractal dimension
can likewise describe the geometry of physicalstructures and the correlational properties of
temporal processes, but the methods involved in its
calculation tend to vary across systems. A classic
example that can serve to illustrate the use of thisparadigm can be found in Mandelbrot’s (1967)
Science article “How long is the coast of Britain?”.
Using a divider method, in which measuring sticksof variable length were used to measure the lengthof coastlines, Mandelbrot showed that, as predicted,
the length of the coast varied as an inverse functionof the measuring stick length, but there was acharacteristic scaling relationship between the two,
which differed across coastlines (D=1.25 for thecoast of Britain, which is remarkably similar to thatof the Koch snowflake shown in Fig. 1!).
Over the past three decades there has been an
explosion of interest in the development and use of
fractal geometry in diverse fields of study, frompure mathematics and statistical physics tocomputer science and networking, economics,
atmospheric sciences and even medical physiology.
It is rather telling that an ISI Web of Science searchwith“fractal”as a search term revealed a total of
49,330 indexed publications between 1980 & 2013,
doubling between 1990 and 1991 and increasing
steadily thereafter (Figure 2). Subject filters
restricting the same search to publications in thefields of ecology and evolution produce 2,058
results, just 4% of that original figure but stillsuggestive of genuine interest in the topic. There
have been some areas in particular in which fractal
geometry has provided insight into ecological andThe Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

5

Figure 2: A search of the ISI Web of Science using the key word“fractal”returns a total of 49,330
articles published since 1980, but only 2,058 (4%) remain when filtering search results
by subject fields related to ecology and evolution (Environmental Sciences; Ecology;
Evolutionary Biology; Zoology; Behavioral Sciences; Anthropology; Paleontology).

evolutionary processes, and it is to these which I
now turn my attention.
3. Fractals in Ecology and Evolution
I begin this section with a rather poignant
example of how complexity science and fractal
geometry can radically transform our ideas about
natural processes. In the 1970’s, Eldridge and
Gould introduced and developed the theory ofpunctuated equilibria to explain temporal
distributions in speciation events (Eldridge &Gould 1974; Eldridge & Gould 1977). Interestingly,
the“burstiness”inherent in this model of evolution
can be reproduced quite readily by invoking selforganized criticality (SOC) (Bak et al. 1987). SOC
is rather hard to define, but typifies dynamical selforganized systems operating at disequilibrium (at,and continually attracted to, a critical point), withintermittency, spatio-temporal scaling ‘( avalanchesof all sizes’), and non-linear interactions being
characteristic of their components (Bak et al. 1987).
This is perhaps best illustrated using the canonicalexample of SOC, that of the sand pile on top ofwhich individual grains of sand are perpetually
added; once at its critical point, the sand pile
releases its energy through cascades of avalanchesof all sizes, but with smaller avalanches being
much more frequent than larger avalanches, i.e. a
power-law distribution of events.
The relationship between SOC and evolution is
as follows: ecological communities evolve into a
critical state at which the fitness landscapes of eachspecies are maximized given the landscapes of allother interacting species, as if in a state of Nashequilibrium (Kauffman & Johnsen 1991; Bak &Sneppen 1993). In this state, small changes canhave cascading effects because of the vastinterconnectedness existing at all ecological scales.
At least for the Bak-Sneppen model, the key is thatthese critical states are reached naturally with noexternal stimuli or‘fine-tuning’necessary, i.e. they
emerge as the result of simple interaction rules
followed by each species (Bak & Sneppen 1993). A
common feature of systems exhibiting SOC is thatthey produce fractal dynamics, like the power-law
distribution of avalanches in the sand pile model.
Both speciation and extinction events appear to6

Andrew James Jonathan MacIntosh

exhibit such fractal characteristics (Raup 1986;
Kauffman & Johnsen 1991; Bak & Sneppen 1993;
Solé et al. 1997; Chaline et al. 1999; Nottale et al.
2002), leading Stuart Kauffman (1993, p. 78) tohypothesize that evolution constitutes a“marriageof selection and self-organization”.
Within ecology proper, there is also a clear trend
toward greater use of fractal tools, and many
fruitful applications have already been identified(Wiens 1989; Sugihara & May 1990; Halley 1996;
Solé et al. 1999; Seuront & Strutton 2004; Seuront2010). It is impossible to touch upon all of them
here, but among these, fractal statistics have beenused to understand structural hierarchies in coral
reef (Bradbury & Reichelt 1983), succession
dynamics in forest ecosystems (Hastings et al.
1982), various properties of food webs (Sugihara etal. 1989), and variability in animal population
densities (Pimm & Redfearn 1988). But one area inwhich the application of fractal statistics hasflourished, perhaps even to the point of becoming amainstream approach, and the one which is mostclosely related to my own interests, is in the field ofanimal movement ecology. Three generalapproaches have been undertaken in this area, andthese are described in the following sections,beginning with the one that has garnered thegreatest appeal: the Lévy flight foraging hypothesis.
3.1 The Lévy Flight Foraging Hypothesis
In order to understand what has come to be
known as the Lévy flight foraging (LFF)
hypothesis, we need to first think about movement
as a reaction-diffusion process centered aroundbiological encounters; animals diffuse via
movements through their respective environments
in search of relevant encounters, be they food ormates, or in avoidance of others such as predator
encounterss, at which point they react in someappropriate manner before returning again to search

Figure 3: Typical trajectories taken by random walkers with step length distributions exhibiting(A) Gaussian versus (B) Cauchy (Lévy) statistics. In both cases, turning angles were
pulled from a random distribution. Notice the clear differences in coverage generated
by the two walkers, despite that both mean step length and number of iterations are anorder of magnitude (10x) larger in the Gaussian walk. Lévy walks are among those that
produce anomalous diffusion; more specifically they are super-diffusive with varianceincreasing as a power-law function of time. The search strategies suggested by thesecontrasting figures can also be thought of as intensive versus extensive, and it’s easy tosee how the Lévy walk presents a more energy-efficient alternative. A growing bodyof literature suggests that the patterns shown in B may more generally reflect observed
patterns of animal search (Viswanathan et al. 2011). These simulations were run usinga model from O'Sullivan & Perry (2013) implemented in NetLogo® (Wilensky 1999).

The Fractal Primate: Interdisciplinary Science and the Math behind the Monkey(Viswanathan et al. 2008; Viswanathan et al. 2011).
Put this way, animal movement falls within thegeneral area of reaction-diffusion processes, thus
lending itself well to examination using existing
mathematical concepts, e.g. random walk models(Codling et al. 2008; Raposo et al. 2009; Smouse etal. 2010). In normal diffusion processes, the meansquared displacement of a particle, i.e. the extent towhich it has‘explored’its environment, is a linear
function of time (Fig. 3A). However, in much of
the animal movement literature, the diffusion
process is instead anomalous, in which the meansquared displacement displays a non-linear
relationship with time, i.e. exhibits a power-law
function (Fig. 3B). Examples from across theanimal kingdom abound, from wandering albatross(Viswanathan et al. 1996) to spider monkeys (Boyeret al. 2004; Ramos-Fernández et al. 2004; Boyer et
al. 2006) to a range of marine predators such as
sharks, sea turtles and penguins (Sims et al. 2008),
and even to humans (Bertrand et al. 2005; Brown etal. 2007; Raichlen et al. 2013). The significance of
this seeming ubiquity of complex search patterns is
encapsulated by the LFF hypothesis, which predicts
the emergence of complex, non-linear movement
patterns (Lévy flights, Lévy walks, Lévy motions,
all of which tend to be used interchangeably despite
their actual differences in meaning) as an adaptation
to the typically heterogeneous nature of biologicalencounters in both space and time (Viswanathan etal. 2011).
It is beyond the scope of this essay to delve
deeply into theory and practice concerning the LFF
hypothesis, so I direct readers instead to thenumerous reviews and critical assessments thathave already done just that (Bartumeus et al. 2005;
Reynolds 2006; Bartumeus 2007; Benhamou 2007;
Edwards et al. 2007; Viswanathan et al. 2008;
Raposo et al. 2009; Reynolds 2009; Reynolds &Rhodes 2009; Giuggioli & Bartumeus 2010;
Benichou et al. 2011; James et al. 2011; Gautestad

72012). What is important to note here, however, isthat studies of Lévy patterns and processes inanimal diffusion are only beginning to movebeyond the identification of models that best
describe species-typical movement trajectories, e.g.
asking who does the Lévy walk, and into theirunderlying causes (Viswanathan et al. 2011). It isalso clear that the observation of Lévy patterns inanimal search does not imply that a Lévy process isnecessarily at work, as such patterns can begenerated through species-environment interactions
following a mixture of simpler rules (Benhamou2007). For example, the many works of Andrew
Reynolds an colleagues present various
mechanisms by which scale-free movements canarise, including avoidance of conspecific odour
trails (Reynolds 2007), randomly reorienting atcues left by animals following more classicaldiffusion processes (correlated random walks)(Reynolds 2010), and general use of chemotaxis inprey location (Reynolds 2008). And, studies haveshown that Lévy walks are particularly common
when the resource base is heterogeneous, or inother words is itself Lévy-distributed (Boyer et al.
2006; Humphries et al. 2010). Indeed it is inprecisely this context in which Lévy patterns are
expected to confer fitness advantages (Reynolds
2006; 2009; Humphries et al. 2012; Reynolds2012).
Concerning primate studies, Lévy statistics havebeen found to characterize movement in somespecies (Boyer et al. 2004; Ramos-Fernández et al.
2004; Boyer et al. 2006; Vandercone et al. 2013)
but not in others (Sueur 2011), or a combination of
Lévy and other statistics has been found in various
components of primate diffusion (Schreier & Grove2010; Sueur et al. 2011a). But there are major
methodological issues inherent in many of theseprimate studies, particularly with regards to data
sampling. For example, studies have used locational
points collected at arbitrary temporal resolutions set

8Andrew James Jonathan MacIntosh

by the observer which are unlikely to reflect natural
timescales relevant to their subjects’movements(Plank & Codling 2009). Such arbitrary set points
actually preclude our ability to measure true step
lengths, turning angles and waiting times (3 key
elements of the reaction-diffusion process). Twoother limitations, which have occurred in
combination or in isolation, further exacerbate
problems interpreting the results of these studies.
One is that spatial recording devices have typically
been attached not to the animals but to the observer,
meaning that the latter’s location must be used as aproxy for that of their subjects. The other is thatmeasurements have not always been at theindividual level but rather at the level of the socialgroup. Group-level movement patterns may reflect
individual movements in a coarse way, but only at
the largest of scales. Not only do suchmeasurements preclude any analysis of finer scale
movement patterns, they also eliminate all
possibility of assessing variation across individuals.
But even at large scales, it is not always so simple
to take a location for an entire group, particularly
when group spread is large. Finally, using grouplocations will invariably have a smoothing function
that, particularly if dispersal is large relative torealized range size, might introduce further bias
into the results. Despite the associated difficulties,
researchers interested in pursuing such work withprimates should consider using animal-attacheddevices to bring the quality of the data up to current
standards. No doubt we have much yet to learn
about multi-scale spatial patterns in primate
movement, particularly with reference to therampant habitat disturbance and impending climatechange that are threatening the majority of species
in existence.
3.2 Fractal Dimension, Tortuosity and Scaledependence
The second fractal approach surfaced in the late1980’s and early 1990’s in an attempt to understandthe importance of landscape scales and perceptionof habitat heterogeneity on animal movement,
dispersal and distributional properties (Dicke &Burrough 1988; Crist et al. 1992; Johnson et al.
1992; Wiens 1993; Wiens et al. 1995). A key
concept in this approach is tortuosity or sinuosity,or‘wiggliness’to quote Dicke & Burrough (1988),
which refers to the degree to which a curve consistsof frequent turns with large turning angles orotherwise‘fills its space’. Tortuous paths diverge
from bee lines along a continuous gradient of
increasing complexity. In fractal dimension terms,
curves range from the straight lines characteristicof the Euclidian first dimension (D=1) to those so
convoluted and folded back onto themselves so as
to fill the entire plane, nearly overlapping thesecond Euclidian dimension (D=2).
Unlike the studies testing the LFF hypothesis,
which use distributional characteristics of step
lengths, turning angles and waiting times, fractal
dimension estimates can be roughly thought of as acomposite of all of these parameters measuring
global properties of the movement path itself, i.e.
tortuosity and self-similarity. This approach thus
has the desirable trait of being able to characterize
complex systems such as those involving diffusion
using a single parameter (Seuront 2010). In
practice, however, single dimension estimates havebeen criticized in cases in which scaling has notbeen rigorously demonstrated (Turchin 1996),
though this is also true for studies testing the LFF
hypothesis (Edwards et al. 2007).
The presence of multiple scaling regions itself,
however, offers further insight into organismspecific fundamental scales and responses tochanging scales within or across landscapes (Fritz
et al. 2003; Seuront et al. 2004a; Garcia et al. 2005;
Nams & Bourgeois 2005). For example, fractal
dimension estimates were shown to change withscale in wandering albatross (Diomedea exulans)

The Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

and were suggested to reflect scale-dependent
movement patterns in response to wind currents at
small scales, food search at medium scales andlong-distance movements at large scales (Fritz etal. 2003). Scale-dependence has also been used to
identify salience in landscape features; while
microhabitat heterogeneity featured strongly inshaping American marten (Martes Americana)
movements, grazing ewes responded only tolandscape scales above a certain threshold (Garcia
et al. 2005). Given the size differences andrespective ecological roles of these species, thismay seem intuitive, but determining ecologicallyrelevant scales can be a complex matter andremains an active area of research in ecology with
important implications for conservation andmanagement strategies.
Note that to my knowledge, no studies have yet
investigated fractal dimension in primate movement
paths. Fractal dimension has been used in at least
two primate studies, but in both cases analyses
were performed in the temporal rather than spatial
domain. I will discuss these studies and the
motivation behind them below, after first providing
some context for the study of ‘fractal time’in
animal behavior.
3.3 Fractal Time
What we have to observe from the start is that
there has been an interesting, yet perhaps
unnecessary, dichotomy in the application of fractal
analysis to animal behavior. On the one hand, thestudies in movement ecology presented above havefocused largely on presenting and interpreting
patterns of animal movement in relation totheoretically optimal statistical patterns of search,which has culminated in the LFF hypothesis, or
responses to landscape heterogeneity. On the other
hand, studies of fractal time have largely beeninterested in using fractal analysis as a diagnostic
tool to distinguish between the behaviors of animals

9

in various physiological conditions (see next
section). As a result, these latter studies have been
exceptionally interested in individual differences inpatterns of behavior, though perhaps too restrictedin scope and ignoring other individual factors of
ecological significance, such as those related toanimal life history trajectories and even to someextent landscape-level variables. In contrast,
interest in individual-level variation seems to belacking in the spatial literature, particularly thatconcerning the LFF hypothesis. One notable
exception, however, can be found in the study bySueur et al. (2011a), which showed individual
differences in the ways in which Tonkean macaques(Macaca tonkeana) diffused in relation to their ages
and sex. Furthermore, the anecdotal account of ayoung basking shark (Cetorhinus maximus)
producing aberrant diffusion statistics in Sims et al.
(2008) provides an example of how individual
variation can lead to insight into possible
developmental processes involved in searchstrategies. It is therefore clear that much future
work addressing individual differences in searchpatterns must now be done.
The concept of fractal time in animal behavior
itself first appeared in the literature two decadesago, and while some authors were simply
introducing a novel model of behavior to capture
statistical properties in animal search, i.e. scale-free
distributions in behavior sequences (Shimada et al.
1993; Cole 1995; Shimada et al. 1995), others wereusing this approach to determine the effects ofpotential stressors on animal behavior (Motohashi
et al. 1993; Escos et al. 1995; Alados et al. 1996;
Alados & Weber 1999). It is these studies which set
the stage for the discussion in the next section. But
before moving on, it is important to first put thesestudies into the broader context of animal search.
Like the movement ecology literature, studies offractal time assume that there must be some optimal
range of complexity within which behavior

10

Andrew James Jonathan MacIntosh

sequences of normally-functioning individuals
should fall, forming the basis for interpretation of
deviations from this range. Rather than assessing
this range using spatial data, these studies instead
examine the sequential distributions of animal
behavior across time, i.e. behavior time series, thus
operating in the temporal rather than spatial
domain. Such studies are critically dependent on
the ability to clearly define and measure specificbehaviors, as these form the basis of the time series
upon which the analysis is performed. For
movement trajectories, in contrast, there is often
little or no consideration of what the animal isactually doing at each time step, thoughexamination of intermittence has provided key
insight into the importance of specific behaviors
and behavioral modes in animal movement(Benichou et al. 2006; Bartumeus & Levin 2008;
Bartumeus 2009; Benichou et al. 2011; Reynolds
2011), as has the examination of scale-dependence
in movement paths as shown above.
But also like the movement ecology literature,studies of fractal time have contributed to ourunderstanding of animal-environment interactions,
supporting the general tenet of the LFF hypothesisthat complex patterns are required to navigate lesspredictable environments. Taking an example frommy own work, sequential distributions of behavior
in Japanese macaques (Macaca fuscata yakui)
showed a kind of behavior-environment complexity
matching (MacIntosh et al. 2011). To elaborate on
this, movement sequences were significantly more
complex in arboreal contexts, which are much morediscontinuous across all three dimensions than are
terrestrial contexts across two. Conversely, foraging
sequences were significantly more complex interrestrial contexts, in which foraging problems (i.e.
pursuit of cryptic food resources such as fallen
seeds and insects) are clearly more difficult to solve
than in arboreal contexts, in which the primaryresources include visible and predictable resources

such as fruits and leaves. That foraging on insect
prey involved performance of significantly more
complex sequences than did foraging on fruits andleaves further supports this hypothesis.
Furthermore, studies on the foraging behavior offruit flies (Drosophila melanogaster) (Shimada etal. 1995) and Japanese quail (Coturnix coturnix)(Kembro et al. 2009b), and on the vigilance
behavior of domestic hens (Rutherford et al. 2003),
all suggest that animals in novel environments or
those otherwise motivated to explore theirenvironments display increased complexity in their
behavioral patterns. Analogously, the foraging
trajectories of clownfish larvae (Amphiprion
perideraion) (Coughlin et al. 1992) and foraging
time series of Japanese macaques (MacIntosh et al.
2011) both display smoothing as individuals age,
becoming less complex over time. Results of thenovel environment tests in Shimada et al. (1995)
suggest that individuals can adapt to new conditions
and alter their behavior accordingly to best exploit
the resources therein. A similar process may
underlie the observed age differences in complexity,
but we might also speculate that immature animals
are primed through natural selection to behave
more erratically in search contexts if such behavior
is indeed more effective in cases in which targets
are not a priori known. It would be interesting totest the degree to which complexity is induced in
adults relocated to novel environments, andwhether this approximates levels of complexity
seen in immature animals or whether behavior
patterns smoothed with age and experience cannot
or simply need not return to pre-maturation levels.
Once again, these studies suggest the existenceof optimal ranges of complexity which may be finetuned through selection either alone or in
combination with experience to the particulars ofeach organism’s typical environmental context. In
the next section I address deviations from thisoptimal range, perturbations to the system that

The Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

result in behavior patterns that, in severe cases, canbe diagnostic of some underlying pathological or atleast physiologically challenging state.
4. Toward the Diagnostic Fractal
To begin with a slight digression, interesting
examples abound illustrating the utility of fractal
geometry as a diagnostic tool. For example,
Physicist Richard Taylor discovered in the late1990’s that Jackson Pollock paintings could be
described using fractal dimension statistics (Taylor
et al. 1999). Remarkably, this work showed that thefractal dimensions of his works increased as thepainter aged. Because of this, when a cache ofalleged Pollock paintings was found in 2005, art
authorities turned to Taylor to apply his fractal
geometry to the new paintings so as to‘diagnose’
them as authentic or not (Abbott 2006; Taylor et al.
2007). Though there is debate about theapplicability of this work (Jones-Smith & Mathur
2006; Taylor et al. 2006; Jones-Smith et al. 2009),
there does seem to exist a strong connection
between art, aesthetics and fractal geometry,
inclusive of human perception of what is beautiful(Spehar et al. 2003; Taylor et al. 2011). Whentested, human subjects seem to prefer Pollock
paintings with fractal dimension estimates in therange of ca. 1.5, though natural landscapes closer todimension 1.3 were preferred in the same set oftests.
The relationship between fractal geometry andaesthetics may yet have much tell us about humanperception, but studies originating in humanphysiology and medicine may serve as better
examples for our purposes here; it is from this workthat stemmed the idea of ‘complexity loss’in
biological systems. As we have seen from the
movement ecology example above, complexity
seems to emerge as the result of an adaption to
general diffusion problems, which need not be

11

restricted to animal movement ecology. The
mammalian lung provides a perfect example of an
adaptive solution to a diffusion problem in finite
space, and it perhaps comes as no surprise then thatits architecture is best characterized using fractal
statistics (Nelson et al. 1990; Shlesinger & West1991). Indeed, the same can be said of various
physiological structures, including the heart,
intestines and nervous system (West & Goldberger1987).
But complexity is also thought to be adaptive
because it is error tolerant, i.e. able to cope withsudden changes and stresses arising from both
intrinsic and extrinsic factors (West 1990). In thiscontext, if we consider a complexity optimum
falling somewhere along a spectrum from random
to invariably cyclic, tendencies toward either pole
may be indicative of a breakdown in optimalfunctionality (Goldberger et al. 2002a; Goldberger
et al. 2002b). In a classic example of this, while it
was once thought that regularity in the inter-beatintervals of a healthy heart were necessary tomaintain homeostasis, fractal and other non-linear
analyses have clearly shown this not to be the case;the inter-beat intervals of a healthy heart fluctuate
more chaotically than do those of a diseased heart(West & Goldberger 1987; Goldberger et al. 1990;
Peng et al. 1995). The results of such experiments
appear so robust as to allow some researchers tosuggest that fractal analysis can be used to diagnose
underlying heart conditions.
Similarly, the application of fractal analysis tohuman gait dynamics has also shown that healthy
subjects display fractal-like patterns in their strideto-stride intervals (Hausdorff et al. 1995; Hausdorff
et al. 1997; Hausdorff et al. 2001). In contrast to
the heart rate example, however, irregularity in gaitdynamics appears to increase with age and theadvancement of neuromuscular disorder to thepoint of virtually unpredictable fluctuation(Hausdorff et al. 2001). Thus, it is important to note12

Andrew James Jonathan MacIntosh

that the fractal dimension of a biological system
may increase or decrease with impairment,
depending on the parameters of the system underinvestigation, but that altered complexity signatures
are nonetheless indicative of compromisedfunctionality.
Further evidence of the existence of optimal
complexity ranges can be found in dynamical
properties of neural activity (Abasolo et al. 2008;
Montez et al. 2009), lung architecture (Mishima etal. 1999; Boser et al. 2005) and even respiration,
the latter of which can become more erratic inelderly men (Peng et al. 2002). Even outside ofphysiology, the growth of plants and trees, whichbears striking resemblance to the branching patternsof the mammalian lung, also exhibits altered fractal
geometry when subjected to stressful conditions(Alados et al. 1994; Escos et al. 1995; Alados et al.
1999). Scaling up from individual plants, Kefi et al.
(2007) demonstrated the vegetation patch structure
in entire arid ecosystems is apparently fractal, andthat the loss of such structure, for example understrong grazing pressure, is strongly predictive of
imminent desertification. All of these studies havelaid a solid foundation for investigating further therole of complexity and complexity loss in otherbiological systems such as, for instance, animal
behavior.
4.1 Complexity Loss and Animal Behavior
As mentioned above, studies demonstratingcomplexity loss in animal behavior have invariably
used temporal rather than spatial behavior
sequences. This is likely to change in the near
future, however, because (a) the use of fractal
statistics are now commonplace in spatial ecology
and (b) understanding how spatial distributions ofvarious species are affected by, for example,
climate change or more direct forms ofanthropogenic disturbance is of paramountimportance. Indeed, studies showing how livestock

production systems can affect fractal characteristics
of vegetation structure within an ecosystem have
led to significant insights into ecosystem health andhave allowed for improved predictive modeling of
the effects of continued landscape abuse (Alados etal. 2002; Kefi et al. 2007; Alados & El Aich 2008).
Such an approach could also be applied to animal
movement behavior, e.g. in response to theconstruction of a road through important corridor
habitat of species A or the selective logging ofpreferred feeding trees of species B. But in theabsence of such studies, let us draw our attention to
those that have examined the distribution of
behaviors as they occur through time, and how it
can change under stress.
First, it is worth explaining the rationale behind
applying novel analytical tools such as fractal
analysis to behavioral data in the context ofdiagnostics. Behavioral health monitoring programs
are common in animal welfare contexts, largely
because stressors tend to manifest in particularbehavioral phenotypes (Cooper 1998; Dawkins
1998; 2003; Weary et al. 2009). Clinical signs ofhealth impairment, while obvious, tend to be rare
and/or difficult to observe under natural conditions(Hart 1988; 1990; Huffman 1997; Krief et al.
2005). In their absence, in situ behavioral measures
are often too coarse to detect the various subtle
changes that can present under mild or chronic
stress (Rutherford et al. 2004). Fractal analyses, tothe extent to which they have been applied in thiscontext, can thus enhance our understanding of
behavioural variation and how it relates to stressand disease (Rutherford et al. 2004; Asher et al.
2009). Indeed, various studies have demonstrated
cases in which fractal analyses were able to detectbehavioural consequences of various stressors
whereas traditional, frequency-based approaches
were not (Motohashi et al. 1993; Alados et al.
1996; Rutherford et al. 2004; Rutherford et al.
2006; Kembro et al. 2009a; Kembro et al. 2009b).

The Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

Rutherford et al. (2004) therefore stress that fractal
analysis can be a useful tool in behavioral
assessments of animal welfare because of its ability
to objectively measure qualitative differences in
temporal sequences.
The first study to use fractal time in animal
behavior as an indicator variable was probably thatby Motohashi et al. (1993), who showed that rats
exposed to Tetrachloroethylene, a common organic
solvent of public health concern used in drycleaning and degreasing, displayed significantly
increased slopes of fluctuation in the power
spectrum, deviating from the observed pretreatment optimal range which approximated 1/
f-noise, a model of optimality, almost perfectly
over 24-hr cycles. There has since been some
application of this approach in toxicologicalstudies, including the demonstrations that: (1) lead
exposure during critical developmental periods canhave lasting impacts on the expression andcomplexity of mating rituals in adult male fatheadminnows (Pimephales promelas) (Alados & Weber
1999); (2) bioactive essential oils lead to increased
complexityinmosquito
larvae(Culexquinquefasciatus) locomotion patterns (Kembro etal. 2009a); and (3) complexity signatures inzooplankton (Centropages hamatus) swimming
behavior change in the presence of hydrocarbon
contamination (Seuront & Leterme 2007).
Another common context within which fractal
time has been investigated is animal welfare(reviewed in Asher et al. 2009). As was mentioned
above, behavior has long held an important place inanimal welfare assessment, and studies have now
seemingly added fractal analysis to the welfare
assessment toolkit. In this context, perhaps thework by Kenneth Rutherford and colleagues has
been the most extensive (Rutherford et al. 2003;
2004; Rutherford et al. 2006). These studies
showed that various livestock behaviors responded
to stressors induced by typical housing and13

management conditions. For example, theorganization of postural behaviors in pigs exposed
to chronic social stressors, repeated aggressive
social interactions, was less complex andsignificantly more structured than that of controls(Rutherford et al. 2006). Conversely, hens subjected
to an acute stress, exposure to a novel environment,
exhibited increased complexity in the temporal
dynamics of their vigilance behavior (Rutherford etal. 2003). Furthermore, three studies tested theeffects of food limitation on behavior sequences ofdomestic fowl. In the first, chickens starved for 12hours before observation displayed reduced
ambulatory complexity signatures when compared
to controls (Maria et al. 2004). In the second,
Japanese quail (Coturnix japonica) starved for 3
hours before observation exhibited the opposite
effect (Kembro et al. 2009b). These seemingly
conflicting results can be explained, however, by acouple of key differences between the studies, i.e.
the lengths of food restriction (12 versus 3 hours)
and the additional scattering of food across theground to stimulate active foraging in the second
study. A third study found no effect of foodrestriction on complexity of behavioral patterns in
broiler chickens (Hocking et al. 2007).
But investigation of the impacts that stressors
can have on fractal time in animal behavior is mostlacking in the area of wildlife health monitoring. To
date there have been only a handful of studies thathave used this approach with animals in natura(Alados et al. 1996; Alados & Huffman 2000;
MacIntosh et al. 2011; Seuront & Cribb 2011;
Cottin et al. 2014). Much of this work was inspiredby the early work of Concepción Alados, whose
many studies applying fractal tools in diagnostic
ecology included the first example of complexity
loss in wild animal behavior, in which pregnancy
and external parasitism were correlated withcomplexity loss in foraging and vigilance behaviorof Spanish ibex (Capra pyrenaica) (Escos et al.

14

Andrew James Jonathan MacIntosh

1995; Alados et al. 1996). A second study showedthat chimpanzee (Pan troglodytes schweinfurthii)social behavior became less complex and morestructured through time in individual femalespresenting clinical signs of impaired health (Alados& Huffman 2000). Clinical health impairment,along with advanced reproductive state and,weakly, intestinal parasitism were also shown
through my own work to correlate with features ofcomplexity loss in Japanese macaques (MacIntosh
et al. 2011). A pair of studies have also
demonstrated complexity loss in marine predators
exposed to certain stressors, i.e. Indo-Pacificbottlenose dolphins (Tursiops aduncus) exposed toboat traffic (Seuront & Cribb 2011) and Adélie
penguins (Pygoscelis adeliae) exposed to increased
concentrations of circulating glucocorticoids (Cottin
et al. 2014). It is worth highlighting that the latterstudy was the first and currently only study to haveexperimentally manipulated a physiologicalparameter and then measured complexity signaturesin a free-ranging animal.
My own work with Japanese macaques on
Yakushima (MacIntosh et al. 2011) perhaps
provided the most comprehensive study of fractal
time in animal behavior to date, not only addressing
the impact of the aforementioned stressors on
behavioral complexity but also the roles of various

Figure 4: Detrended fluctuation analysis (DFA) of dive sequences collected from two little penguins (Eudyptula
minor) using animal-attached time-depth recorders. (A, D) Binary sequences (z(i)) generated from thediving (black bars) and surface behaviour at 1 s intervals. (B, E) Integrated sequences (y(t)) generated by
the accumulation of z(i). (C, F) Log-log plots of the average fluctuation F(n) on the y-axes as a function ofscale (n) on the x-axes. The α -exponent is the slope of the regression line; the lower α -exponent reflects
greater complexity. In this case, the sequence illustrated in D-F presents a more complex structure throughtime than that illustrated in A-C. This can easily be discerned by comparing the integrated time series in B
and E, the latter presenting greater fluctuation throughout the sequence than the former. Note that only the
points in black were used to fit the regression line, in accordance with the region of best scaling (MacIntosh
et al. 2013), to avoid biases in calculation of the exponent caused by deviations at small (<10 s) and large(sequence length/10) scales. Data used in this figure were provided by Yan Ropert-Coudert, Akiko Katoand Andre Chiaradia.

The Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

life history, demographic and environmental
parameters, some of which were highlighted in
Section 3.3. But even still, this study also
exemplifies at least three of the major shortcomings
of previous studies applying fractal analysis toanimal behavior, namely; (1) relatively short
behavioral sequence lengths; (2) insufficientevaluation of scaling behavior; and (3) reliance ona single method to estimate complexity, in this casedetrended fluctuation analysis or DFA (Peng et al.
1995) (see Fig. 4 for a description of the approach).
The first is not always avoidable in studies of freeranging animals, but the second two can beaddressed easily enough.
For my own part, I was later able to address eachof these concerns using data collected via biologging from little penguins (Eudyptula minor)
foraging at sea in Australia (MacIntosh et al. 2013).
This study addressed a number of key
methodological concerns, showing for example thatfractal scaling is indeed present in temporalsequences of animal behavior, though does not
persist across the entire range of scales so one must
be careful in interpreting scaling exponents
generated from the entire set. We were also able toshow that changing the sampling resolution does
not always affect estimation of scaling exponents,
contrary to what was demonstrated by Plank &Codling (2009) in their analyses of Lévy walks. It
would be a great boon indeed if binary time series
such as those used in our study are generally
resilient to potential sampling biases, though thisshould be examined critically in future studies
before jumping to any ill-fated conclusions. Finally,
our study also employed 4 fractal estimators, all ofwhich correlated very well with one another.
Similar agreement across a different set of
estimators was found in our study of Adélie
penguins (Cottin et al. 2014), adding weight to ourgeneral conclusions about fractal scaling in penguin
behavior.

15

In sum, these studies show that fractal analysis
appears to be a rather useful tool for behavioral
toxicology, biological conservation and animal
welfare, but they also illustrate a flexibility to
fractal time that is not captured by the animal
movement ecology literature: almost any behavior,insofar as it produces a suitable time series, can be
examined through the fractal lens. A brief scan of
the above paragraphs reveals a number ofpossibilities beyond movement trajectories,
including foraging, vigilance, diving, postural shifts
and even social interactions such as grooming, playand mating displays. Undoubtedly this list is not
exhaustive, and I for one look forward to seeing
novel applications of this approach continue toappear in the future.
4.2 Complexity Loss Revisited: the Bigger Picture
The above section illustrates the apparent
robustness of the fractal approach in a diagnostic
sense, but it also makes it abundantly clear thatcomplexity loss, while perhaps once thought toreflect a change in the complexity of behavioralorganization in a single direction, i.e. toward morestructured or stereotypical behavior, is a rather
complicated matter. Though admittedly somewhat
problematic in terms of making predictions andavoiding ad hoc explanations, we may now define‘complexity loss’operationally as a change ineither direction, i.e. toward lesser or greaterroughness, along a spectrum of scaling exponents
with random fluctuation at the center and greater
structuring, either persistent or anti-persistent, atboth ends (Fig. 5). The optimal complexity range,
which we might think of as an attractor like thosefound in SOC, would first need to be identified inorder to determine when and whether complexity
loss has taken place. This might be done via
comparisons of free-ranging individuals in various
physiological conditions or exposed to various
environmental stressors (Alados et al. 1996; Alados

16

Andrew James Jonathan MacIntosh

Figure 5: Schematic showing hypothetical optimal
complexity range within which we expect to findscaling exponents describing sequences of animal
behavior. Deviations from this optimal scaling
range may indicate underlying pathologicalconditions. The x-axis illustrates the Hurst
estimator, a typical scaling exponent metric used
to indicate fractal dimension. Note that this is ahypothetical scenario only, and optimal scaling
ranges may occur at any point along this spectrum
including, perhaps in some cases, white noise.
See Section 4.1 for further explanation.

& Huffman 2000; MacIntosh et al. 2011; Seuront& Cribb 2011) or through direct manipulation of
the system to bring forth desired outcomes (Alados& Weber 1999; Rutherford et al. 2003; Maria et al.
2004; Seuront et al. 2004b; Kembro et al. 2008;
Kembro et al. 2009a; Seuront 2011; Cottin et al.
2014). As always, the key challenges concern
eliminating confounding factors and naturalvariability in the first case (observational) andassuming natural behavior as a starting point underunnatural conditions in the second (experimental).
Perhaps a good compromise can be found in thefield experimentation approach undertaken by
Cottin et al. (2014), though this is not always
desirable nor feasible.
Such comparative and correlational works canusefully discriminate between individuals, but adesirable outcome of all of this would certainly be
the ability to use complexity signatures asstandalone indicators of animal or environmental‘quality’, in every sense of the word (sensu Seuront& Leterme 2007). Studies in animal welfare would
clearly benefit from this ability, particularly in thecase of developing early-indicators of impending
disease, as would domestic livestock production
systems and test battery systems used inpharmacologicalstudies.
Conservation,
management and environmental impact assessment
programs could also all benefit from this approachif fractal behavior patterns could be used as such anindicator. Therefore, the potential applications of
this approach are numerous and diverse, and clearlyjustify its continued investigation.
However, we have not yet established in the
majority of cases which factors contribute to theemergence of and variation in observed complex
fractal patterns in animal behavior. In studies of the
dynamics of complex systems, and presumably
those wanting to use the diagnostic capacity ofcomplexity signatures as indicators, a central
question concerns how the patterns observed arecontrolled through endogenous behavior-initiating
mechanisms (Maye et al. 2007). From thisperspective, external factors that cause further
fluctuations in these systems, such as the effect ofbreathing (Perakakis et al. 2009) or activity level(Billat et al. 2009) on fractal patterns of heart-rate
variability, or hydrodynamic disturbances onaquatic microorganism movements (Seuront et al.
2004c) for that matter, are considered‘noise’andneed to be controlled for. So to determine theimportance of intrinsic factors responsible formediating complex patterns in animal behavior,such noise or in this case environmental variability,
must also be accounted for. Simply put, if such
patterns arise as a result of animal-environment
interactions centered around biological encountersin which the types, locations and accessibilities ofsought resources are all salient features, this will
always raise the question of whether changes inThe Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

fractal behavior patterns occur because of someinternal condition related to stress or disease or
simply that the type of resource sought has changed
and therefore the search protocols have alsochanged. Either case is interesting, but suggestiveof very different mechanisms at play. As a result,current interpretation remains complicated andfurther work addressing these issues is now
necessary.
5. Conclusion
I have attempted to show how studies ofcomplexity, specifically fractality, in animal
behavior can lead to novel conclusions about
ecological patterns and processes. Moreover, by
thinking about and investigating behavior in thisway, we are approaching it at a more fundamental
level than, say, when we extract time budgets fromblocks of observation. At least in theory fractal
properties are scale-invariant, meaning that we canget a real sense of how animals behave acrossmultiple spatial and temporal scales, though as wehave seen even the scale-dependence in fractal
dimension estimation is an extremely useful featureof this approach. While I do not wish to diminish
the utility of classical measures of behavior,frequency-based approaches tend to convey
secondary rather than fundamental information thatis highly scale-dependent, i.e. the length ofobservation is a critical feature, and we are typically
restricted to quantifying behavior (asking questionsof how much?) rather than truly qualifying behavior(asking questions of how?). And, until now, it has
been rather difficult to find objective measures of
behavioral qualities.
There are also numerous extensions of the ideas
presented in these pages, other contexts in which
complexity in behavior is thought to be adaptive.
For example, an idea proposed decades ago wasthat of proteanism in animal behavior. Humphries

17& Driver (1970) first used this term, which refers to
unpredictable or highly variable patterns of
behavior, to describe defense strategies in preyanimals when encountering a predator, e.g. inevasive movements, color changes or variation in
the fight or flight response. These authors argued
that proteanism conferred a selective advantage toprey individuals, insofar as the presentation of
unfamiliar escape patterns to predators allows themto avoid being eaten. Following this idea, Geoffrey
Miller’s ‘Protean Primates’ (Miller 1997)
hypothesis posits that being random or otherwise
unpredictable in social contexts might be an
adaptive strategy associated with successful
maintenance of high rank; by keeping opponents
off balance the protean primate will always be at asocial advantage over those whose behavior can bereadily predicted, and thus exploited. In a recent
study, Martin et al. (2014) showed via computer
touch panel tasks that chimpanzees exhibit virtuallyunpredictable choice patterns across trials incompetitive social interactions, very closely
approximating Nash equilibrium across a variety of
pay-off structures and thus performing in an evenmore game-theoretic manner than humans. The
robust theoretical framework presented in this essay
provides a context within which these and otherphenomena can be further examined andinterpreted.
Still other avenues in which complexity inbehavioral organization has recently become
paramount in primate studies include social
networks (Brent et al. 2011; Sueur et al. 2011b) andcollective phenomena (King & Sueur 2011). The
rising interest in social networks, for example, hasbeen fueled by the realization that analysis of thecomplex interaction patterns within primate andother animal societies, many of which are nonlinear, allow for deeper understanding of various
social processes, including but not limited to
instabilities in social groups (McCowan et al. 2008;

18

Andrew James Jonathan MacIntosh

Beisner et al. 2011), transmission of infectious
disease agents (Griffin & Nunn 2011; MacIntosh etal. 2012; Rushmore et al. 2013) and of course therole of social structure in information transfer andcollective decision-making (Sueur & Petit 2008;
King et al. 2011). And, it has been suggested that
not only do scale-free properties in nonhumanprimate societies exist (Hill et al. 2008), they may
confer clear advantages in the speed and accuracy
with which collective decisions are reached (Sueuret al. 2013).
Moving forward, then, studies in behavioral
ecology will only become increasingly quantitativeand complex (Seuront 2010; Bradbury &Vehrencamp 2014). However, this means that wewill increasingly be able to identify both finer- andbroader-scale phenomena in animal behavior andits relation to various ecological processes.
Behavior is part of an adaptive, complex systemdesigned to facilitate, mediate and mitigate
biological encounters. Complexity science hassince its inception been interested in identifying
universal properties in just such systems, and istherefore poised to take us that much further on our
way to a deeper understanding of the math behindthe monkey.
6. Acknowledgments
My interests in fractal analyses of animal
behavior were kindled in the early days of my PhD
work at the Primate Research Institute of Kyoto
University. For this and a great many other things I
owe many thanks to my advisor, Dr. Michael
Huffman. Dr. Alex Hernandez also provided muchneeded encouragement early on. I express heartfelt
thanks to fellow philosophizer Dr. Chris Martin, forcountless and often endless stimulating discussions
about these ideas and a great many more. I am also
grateful to Dr. Inma Alados, whose guidance kept
me grounded during my initial forays into thefractal world. For introducing me to new horizons
and rekindling my passion for discovery, but also
for their relentless support and friendship, I also
thank Dr. Yan Ropert-Coudert and Dr. Akiko Kato.
I acknowledge the Cooperative Research Programof the Wildlife Research Center of Kyoto
University for facilitating my work in Japan,
members of CICASP and the Department of
Ecology and Social Behavior at KUPRI and all of
my collaborators that made my own endeavors
discussed in this essay a reality. Finally, I deeply
thank the Primate Society of Japan and Kyouei
Steel, LtD for conferring upon me this prestigious
award, the Takashima Prize, which formed the basis
for my invitation to submit this work.
7. Literature Cited
Abasolo D, Hornero R, Escudero J, Espino P 2008: A
study on the possible usefulness of detrended
fluctuation analysis of the electroencephalogram
background activity in Alzheimer's disease. IEEE Trans
Biomed Eng 55:2171-2179.
Abbott A 2006: Fractals and art: In the hands of a master.
Nature 439:648-650.
Alados CL, El Aich A 2008: Stress assessment of argan(Argania spinosa (L.) Skeels) in response to land uses
across an aridity gradient: translational asymmetry andbranch fractal dimension. J Arid Environ 72:338-349.
Alados CL, Escos J, Emlen JM 1994: Scale asymmetry: atool to detect developmental instability under thefractal geometry scope. In Novak MM (ed): Fractals in
the natural and applied sciences, pp 25-36.
Alados CL, Escos J, Emlen JM, Freeman DC 1999:
Characterization of branch complexity by fractal
analyses. Int J Plant Sci 160:S147-S155.
Alados CL, Escos JM, Emlen JM 1996: Fractal structureof sequential behaviour patterns: an indicator of stress.
Anim Behav 51:437-443.
Alados CL, Giner ML, Dehesa L, Escos J, Barroso FG,
Emlen JM, Freeman DC 2002: Developmental
instability and fitness in Periploca laevigata
experiencing grazing disturbance. Int J Plant Sci
163:969-978.
Alados CL, Huffman MA 2000: Fractal long-range
correlations in behavioural sequences of wild
chimpanzees: a non-invasive analytical tool for theThe Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

evaluation of health. Ethology 106:105-116.
Alados CL, Weber DN 1999: Lead effects on thepredictability of reproductive behavior in fatheadminnows (Pimephales promelas): a mathematicalmodel. Environ Toxicol Chem 18:2392-2399.
Asher L, Collins LM, Ortiz-Pelaez A, Drewe JA, Nicol
CJ, Pfeiffer DU 2009: Recent advances in the analysis
of behavioural organization and interpretation as
indicators of animal welfare. J Royal Soc Interface
6:1103-1119.
Bak P, Sneppen K 1993: Punctuated equilibrium andcriticality in a simple model of evolution. Phys Rev
Lett 71:4083-4086.
Bak P, Tang C, Wiesenfeld K 1987: Self-organizedcriticality: An explanation of the 1/f noise. Phys Rev
Lett 59:381-384.
Bartumeus F 2007: Levy processes in animal movement:
an evolutionary hypothesis. Fractals 15:151-162.
Bartumeus F 2009: Behavioral intermittence, Levy
patterns, and randomness in animal movement. Oikos
118:488-494.
Bartumeus F, Da Luz MGE, Viswanathan GM, Catalan J
2005: Animal search strategies: a quantitative randomwalk analysis. Ecology 86:3078-3087.
Bartumeus F, Levin SA 2008: Fractal reorientation
clocks: linking animal behavior to statistical patterns ofsearch. Proc Natl Acad Sci U S A 105:19072-19077.
Beisner BA, Jackson ME, Cameron AN, McCowan B
2011: Detecting Instability in Animal Social Networks:
Genetic Fragmentation Is Associated with Social
Instability in Rhesus Macaques. PLoS ONE 6.
Benhamou S 2007: How many animals really do the Levy
walk? Ecology 88:1962-1969.
Benichou O, Loverdo C, Moreau M, Voituriez R 2006:
Two-dimensional intermittent search processes: An
alternative to Levy flight strategies. Physical Review E
74.
Benichou O, Loverdo C, Moreau M, Voituriez R 2011:
Intermittent search strategies. Reviews of Modern
Physics 83.
Beran J 1994: Statistics for long-memory processes.
Chapman and Hall/CRC, Boca Raton, FL, 315 pp.
Bertrand S, Burgos JM, Gerlotto F, Atiquipa J 2005: Levy
trajectories of Peruvian purse-seiners as an indicator of
the spatial distribution of anchovy (Engraulis ringens).
ICES J Mar Sci 62:477-482.
Billat VL, Mille-Hamard L, Meyer Y, Wesfreid E 2009:
Detection of changes in the fractal scaling of heart rateand speed in a marathon race. Physica a-Statistical
Mechanics and Its Applications 388:3798-3808.
Boser SR, Park H, Perry SF, Menache MG, Green FHY
2005: Fractal geometry of airway remodeling in human19

asthma. Am J Respir Crit Care Med 172:817-823.
Boyer D, Miramontes O, Ramos-Fernández G, Mateos
JL, Cocho G 2004: Modeling the searching behavior ofsocial monkeys. Physica a-Statistical Mechanics and
Its Applications 342:329-335.
Boyer D, Ramos-Fernández G, Miramontes O, Mateos
JL, Cocho G, Larralde H, Ramos H, Rojas F 2006:
Scale-free foraging by primates emerges from theirinteraction with a complex environment. Proc R Soc
Lond B Biol Sci 273:1743-1750.
Bradbury JW, Vehrencamp SL 2014: Complexity andbehavioral ecology. Behav Ecol.
Bradbury RH, Reichelt RE 1983: Fractal dimension of acoral reef at ecological scales. Mar Ecol Prog Ser
10:169-171.
Brent LJN, Lehmann J, Ramos-Fernández G 2011: Social
network analysis in the study of nonhuman primates: A
historical perspective. Am J Primatol 73:720-730.
Brown C, Liebovitch L, Glendon R 2007: Lévy Flights in
Dobe Ju/’hoansi Foraging Patterns. Hum Ecol 35:129138.
Chaline J, Nottale L, Grou P 1999: Is the evolutionary
tree a fractal structure? Comptes Rendus de l’Académiedes Sciences - Series IIA - Earth and Planetary Science
328:717-726.
Codling EA, Plank MJ, Benhamou S 2008: Random walkmodels in biology. J Royal Soc Interface 5:813-834.
Cole BJ 1995: Fractal time in animal behavior: themovement activity of Drosophila. Anim Behav
50:1317-1324.
Cooper JE 1998: Minimally invasive health monitoringof wildlife. Anim Welf 7:35-44.
Cottin M, MacIntosh AJJ, Kato A, Takahashi A, Debin M,
Raclot T, Ropert-Coudert Y 2014: Corticosterone
administration leads to a transient alteration of foraging
behaviour and complexity in a diving seabird. Mar
Ecol Prog Ser 496:249-262.
Coughlin DJ, Strickler JR, Sanderson B 1992: Swimmingand search behavior in clownfish, Amphiprion
perideraion, larvae. Anim Behav 44:427-440.
Crist TO, Guertin DS, Wiens JA, Milne BT 1992: Animal
movement in heterogeneous landscapes: an experimentwith Eleodes beetles in shortgrass prairie. Funct Ecol
6:536-544.
Dawkins MS 1998: Evolution and animal welfare. Q Rev
Biol 73:305-328.
Dawkins MS 2003: Behaviour as a tool in the assessmentof animal welfare. Zoology 106:383-387.
Dicke M, Burrough PA 1988: Using fractal dimensions
for characterizing tortuosity of animal trails. Physiol
Entomol 13:393-398.
Edwards AM, Phillips RA, Watkins NW, Freeman MP,

20

Andrew James Jonathan MacIntosh

Murphy EJ, Afanasyev V, Buldyrev SV, da Luz MGE,
Raposo EP, Stanley HE, Viswanathan GM 2007:
Revisiting Levy flight search patterns of wandering
albatrosses, bumblebees and deer. Nature 449:1044U5.
Eldridge N, Gould SJ 1974: Punctuated equilibria: analternative to phyletic gradualism. In Schopf TJM (ed):Models in paleobiology. Freeman, Cooper and Co., San
Francisco.
Eldridge N, Gould SJ 1977: Punctuated equilibria: thetempo and mode of evolution reconsidered.
Paleobiology 3:115-151.
Escos JM, Alados CL, Emlen JM 1995: Fractal structuresand fractal functions as disease indicators. Oikos
74:310-314.
Fritz H, Said S, Weimerskirch H 2003: Scale-dependenthierarchical adjustments of movement patterns in along-range foraging seabird. Proc R Soc Lond B BiolSci 270:1143-1148.
Garcia F, Carrere P, Soussana JF, Baumont R 2005:
Characterisation by fractal analysis of foraging pathsof ewes grazing heterogeneous swards. Appl Anim
Behav Sci 93:19-37.
Gautestad AO 2012: Brownian motion or Lévy walk?
Stepping towards an extended statistical mechanics foranimal locomotion. J Royal Soc Interface.
Giuggioli L, Bartumeus F 2010: Animal movement,
search strategies and behavioural ecology: a crossdisciplinary way forward. J Anim Ecol 79:906-909.
Gleick J 1987: Chaoes: making a new science. Penguin,
New York, 384 pp.
Goldberger AL, Amaral LAN, Hausdorff JM, Ivanov PC,
Peng CK, Stanley HE 2002a: Fractal dynamics inphysiology: alterations with disease and aging. Proc
Natl Acad Sci U S A 99:2466-2472.
Goldberger AL, Peng CK, Lipsitz LA 2002b: What is
physiologic complexity and how does it change withaging and disease? Neurobiol Aging 23:23-26.
Goldberger AL, Rigney DR, West BJ 1990: Chaos andfractals in human physiology. Sci Am 262:43-49.
Griffin RH, Nunn CL 2011: Community structure and the
spread of infectious disease in primate social networks.
Evol Ecol 26:779-800.
Halley JM 1996: Ecology, evolution and 1/f-noise. Trends
Ecol Evol 11:33-37.
Halley JM, Hartley S, Kallimanis AS, Kunin WE, Lennon
JJ, Sgardelis SP 2004: Uses and abuses of fractal
methodology in ecology. Ecol Lett 7:254-271.
Hart BL 1988: Biological basis of the behavior of sickanimals. Neurosci Biobehav Rev 12:123-137.
Hart BL 1990: Behavioral adaptations to pathogens andparasites: 5 strategies. Neurosci Biobehav Rev 14:273-

294.
Hastings HM, Pekelney R, Monticciolo R, Vun Kannon
D, Del Monte D 1982: Time scales, persistence andpatchiness. Biosystems 15:281-289.
Hausdorff JM, Ashkenazy Y, Peng CK, Ivanov PC,
Stanley HE, Goldberger AL 2001: When humanwalking becomes random walking: fractal analysis andmodeling of gait rhythm fluctuations. Physica
a-Statistical Mechanics and Its Applications 302:138147.
Hausdorff JM, Mitchell SL, Firtion R, Peng CK,
Cudkowicz ME, Wei JY, Goldberger AL 1997: Alteredfractal dynamics of gait: reduced stride interval
correlations with aging and Huntington's disease. J
Appl Physiol 82:262-269.
Hausdorff JM, Peng CK, Ladin Z, Wei JY, Goldberger AL
1995: Is walking a random walk? evidence for longrange correlations in stride interval of human gait J
Appl Physiol 78:349-358.
Hill RA, Bentley RA, Dunbar RIM 2008: Network
scaling reveals consistent fractal pattern in hierarchicalmammalian societies. Biol Lett 4:748-751.
Hocking PM, Rutherford KMD, Picard M 2007:
Comparison of time-based frequencies, fractal analysis
and T-patterns for assessing behavioural changes inbroiler breeders fed on two diets at two levels of feed
restriction: A case study. Appl Anim Behav Sci 104:3748.
Huffman MA 1997: Current evidence for self-medicationin primates: a multidisciplinary perspective. Yearb
Phys Anthropol 40:171-200.
Humphries DA, Driver PM 1970: Protean defence by
prey animals. Oecologia 5:285-302.
Humphries NE, Queiroz N, Dyer JRM, Pade NG, Musyl
MK, Schaefer KM, Fuller DW, Brunnschweiler JM,
Doyle TK, Houghton JDR 2010: Environmental
context explains Lévy and Brownian movement
patterns of marine predators. Nature 465:1066-1069.
Humphries NE, Weimerskirch H, Queiroz N, Southall EJ,
Sims DW 2012: Foraging success of biological Lévy
flights recorded in situ. Proc Natl Acad Sci U S A.
James A, Plank MJ, Edwards AM 2011: Assessing Lévy
walks as models of animal foraging. J Royal Soc
Interface 8:1233-1247.
Johnson AR, Wiens JA, Milne BT, Crist TO 1992: Animal
movements and population dynamics in heterogeneouslandscapes. Landsc Ecol 7:63-75.
Jones-Smith K, Mathur H 2006: Fractal Analysis:
Revisiting Pollock's drip paintings. Nature 444:E9E10.
Jones-Smith K, Mathur H, Krauss LM 2009: Drip
paintings and fractal analysis. Physical Review E

The Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

79:046111.
Kauffman SA 1993: The origins of order: selforganization and selection in evolution. Oxford
University Press, Oxford.
Kauffman SA, Johnsen S 1991: Coevolution to the edgeof chaos: Coupled fitness landscapes, poised states, andcoevolutionary avalanches. J Theor Biol 149:467-505.
Kefi S, Rietkerk M, Alados CL, Pueyo Y, Papanastasis
VP, ElAich A, de Ruiter PC 2007: Spatial vegetation
patterns and imminent desertification in Mediterraneanarid ecosystems. Nature 449:213-217.
Kembro JM, Marin RH, Zygaldo JA, Gleiser RM 2009a:
Effects of the essential oils of Lippia turbinata andLippia polystacha (Verbenaceae) on the temporal
pattern of locomotion of the mosquito Culex
quinquefasciatus (Diptera: Culicidae) larvae. Parasitol
Res 104:1119-1127.
Kembro JM, Perillo MA, Pury PA, Satterlee DG, Marin
RH 2009b: Fractal analysis of the ambulation patternof Japanese quail. Br Poult Sci 50:161-170.
Kembro JM, Satterlee DG, Schmidt JB, Perillo MA,
Marin RH 2008: Open-field temporal pattern ofambulation in Japanese quail genetically selected forcontrasting adrenocortical responsiveness to brief
manual restraint. Poult Sci 87.
King A, Sueur C 2011: Where Next? Group Coordination
and Collective Decision Making by Primates. Int J
Primatol 32:1245-1267.
King AJ, Sueur C, Huchard E, Cowlishaw G 2011: A
rule-of-thumb based on social affiliation explains
collective movements in desert baboons. Anim Behav
82:1337-1345.
Krief S, Huffman MA, Sevenet T, Guillot J, Bories C,
Hladik CM, Wrangham RW 2005: Noninvasivem o n i t o r i n g o f t h e h e a l t h o f P a n t ro g l o d y t e sschweinfurthii in the Kibale National Park, Uganda. Int
J Primatol 26:467-490.
Lorenz K 1981: The foundations of ethology. SpringerVerlag New York, Inc,, New York, 380 pp.
MacIntosh AJ, Alados CL, Huffman MA 2011: Fractal
analysis of behaviour in a wild primate: behaviouralcomplexity in health and disease. J Royal Soc Interface
8:1497-1509.
MacIntosh AJJ, Jacobs A, Garcia C, Shimizu K, Mouri K,
Huffman MA, Hernandez AD 2012: Monkeys in the
middle: parasite transmission through the social
network of a wild primate. PLoS ONE 7:e51144.
MacIntosh AJJ, Pelletier L, Chiaradia A, Kato A, RopertCoudert Y 2013: Temporal fractals in seabird foraging
behaviour: diving through the scales of time. Sci. Rep.
3.
Mandelbrot BB 1967: How long is the coast of Britain?

21

Statistical self-similarity and fractional dimension.
Science 156:636-638.
Mandelbrot BB 1977a: The fractal geometry of nature. W.
H. Freeman and Company, New York, 468 pp.
Mandelbrot BB 1977b: Fractals: form, chance anddimension. W. H. Freeman & Company, San Francisco,
365 pp.
Maria GA, Escos J, Alados CL 2004: Complexity of
behavioural sequences and their relation to stressconditions in chickens (Gallus gallus domesticus): a
non-invasive technique to evaluate animal welfare.
Appl Anim Behav Sci 86:93-104.
Martin CF, Bhui R, Bossaerts P, Matsuzawa T, Camerer C
2014: Experienced chimpanzees behave more gametheoretically than humans in simple competitive
interactions. Sci Rep 4:5182; DOI:10.1038/srep05182
Maye A, Hsieh CH, Sugihara G, Brembs B 2007: Orderin Spontaneous Behavior. PLoS ONE 2.
McCowan B, Anderson K, Heagarty A, Cameron A 2008:
Utility of social network analysis for primate
behavioral management and well-being. Appl Anim
Behav Sci 109:396-405.
Miller GF 1997: Protean primates: The evolution ofadaptive unpredictability in competition and courtship
In Whiten A, Byrne RW (eds): Machiavellian
intelligence II: extensions and evaluations. Cambridge
University Press, Cambridge, pp 312-340.
Mishima M, Hirai T, Itoh H, Nakano Y, Sakai H, Muro S,
Nishimura K, Oku Y, Chin K, Ohi M, Nakamura T,
Bates JHT, Alencar AM, Suki B 1999: Complexity ofterminal airspace geometry assessed by lung computed
tomography in normal subjects and patients withchronic obstructive pulmonary disease. Proc Natl Acad
Sci U S A 96:8829-8834.
Montez T, Poil SS, Jones BF, Manshanden I, Verbunt
JPA, van Dijk BW, Brussaard AB, van Ooyen A, StamCJ, Scheltens P, Linkenkaer-Hansen K 2009: Alteredtemporal correlations in parietal alpha and prefrontal
theta oscillations in early-stage Alzheimer disease.
Proc Natl Acad Sci U S A 106:1614-1619.
Motohashi Y, Miyazaki Y, Takano T 1993: Assessment of
behavioral effects of tetrachloroethylene using a set oftime series analyses. Neurotoxicol Teratol 15:3-10.
Nams VO, Bourgeois M 2005: Fractal analysis measures
habitat use at different spatial scales: an example with
American marten. Can J Zool 82:1738-1747.
Nelson TR, West BJ, Goldberger AL 1990: The fractal
lung: universal and species-related scaling patterns.
Experientia 46:251-254.
Nottale L, Chaline J, Grou P 2002: On the fractal
structure of evolutionary trees. In Losa G, Merlini D,
Nonnenmacher T, Weibel E (eds): Fractals in Biology

22

Andrew James Jonathan MacIntosh

and Medicine. Birkhäuser Basel, pp 247-258.
O'Sullivan D, Perry GLW 2013: Random walks andmobile entities. Spatial Simulation. John Wiley &Sons, Ltd, pp 97-131.
Peitgen H, Jurgens H, Saupe D 2004: Chaos and fractals.
Springer, New York, 864 pp.
Peng CK, Havlin S, Stanley HE, Goldberger AL 1995:
Quantification of scaling exponents and crossoverphenomena in nonstationary heartbeat time-series.
Chaos 5:82-87.
Peng CK, Mietus JE, Liu YH, Lee C, Hausdorff JM,
Stanley HE, Goldberger AL, Lipsitz LA 2002:
Quantifying fractal dynamics of human respiration: age
and gender effects. Ann Biomed Eng 30:683-692.
Perakakis P, Taylor M, Martinez-Nieto E, Revithi I, Vila J
2009: Breathing frequency bias in fractal analysis ofheart rate variability. Biol Psychol 82:82-88.
Pimm SL, Redfearn A 1988: The variability of population
densities. Nature 334:613-614.
Plank MJ, Codling EA 2009: Sampling rate andmisidentification of Levy and non-Levy movement
paths. Ecology 90:3546-3553.
Raichlen DA, Wood BM, Gordon AD, Mabulla AZP,
Marlowe FW, Pontzer H 2013: Evidence of Lévy walk
foraging patterns in human hunter-gatherers. Proc Natl
Acad Sci U S A.
Ramos-Fernández G, Mateos JL, Miramontes O, Cocho
G, Larralde H, Ayala-Orozco B 2004: Levy walkpatterns in the foraging movements of spider monkeys(Ateles geoffroyi). Behav Ecol Sociobiol 55:223-230.
Raposo EP, Buldyrev SV, da Luz MGE, Viswanathan
GM, Stanley HE 2009: Levy flights and random
searches. Journal of Physics a-Mathematical andTheoretical 42.
Raup D 1986: Biological extinction in earth history.
Science 231:1528-1533.
Reynolds AM 2006: Optimal scale-free searchingstrategies for the location of moving targets: New
insights on visually cued mate location behaviour in
insects. Phys Lett A 360:224-227.
Reynolds AM 2007: Avoidance of conspecific odour trails
results in scale-free movement patterns and theexecution of an optimal searching strategy. EPL 79.
Reynolds AM 2008: Deterministic walks with inversesquare power-law scaling are an emergent property of
predators that use chemotaxis to locate randomly
distributed prey. Physical Review E 78.
Reynolds AM 2009: Scale-free animal movement
patterns: Levy walks outperform fractional Brownian
motions and fractional Levy motions in random search
scenarios. Journal of Physics a-Mathematical andTheoretical 42.

Reynolds AM 2010: Animals that randomly reorient atcues left by correlated random walkers do the Levy
walk. Am Nat 175:607-613.
Reynolds AM 2011: On the origin of bursts and heavy
tails in animal dynamics. Physica A: Statistical
Mechanics and its Applications 390:245-249.
Reynolds AM 2012: Fitness-maximizing foragers can use
information about patch quality to decide how tosearch for and within patches: optimal Levy walksearching patterns from optimal foraging theory. J
Royal Soc Interface 9:1568-1575.
Reynolds AM, Rhodes CJ 2009: The Levy flightparadigm: random search patterns and mechanisms.
Ecology 90:877-887.
Rushmore J, Caillaud D, Matamba L, Stumpf RM,
Borgatti SP, Altizer S 2013: Social network analysis ofwild chimpanzees provides insights for predicting
infectious disease risk. J Anim Ecol 82:976-986.
Rutherford KMD, Haskell MJ, Glasbey C, Jones RB,
Lawrence AB 2003: Detrended fluctuation analysis ofbehavioural responses to mild acute stressors indomestic hens. Appl Anim Behav Sci 83:125-139.
Rutherford KMD, Haskell MJ, Glasbey C, Jones RB,
Lawrence AB 2004: Fractal analysis of animal
behaviour as an indicator of animal welfare. Anim Welf
13:S99-S103.
Rutherford KMD, Haskell MJ, Glasbey C, Lawrence AB
2006: The responses of growing pigs to a chronicintermittent stress treatment. Physiol Behav 89:670680.
Schreier AL, Grove M 2010: Ranging patterns ofhamadryas baboons: random walk analyses. Anim
Behav 80:75-87.
Seuront L 2010: Fractals and multifractals in ecology andaquatic science. Taylor and Francis, LLC, USA.
Seuront L 2011: Behavioral fractality in marine copepods:
endogenous rhythms versus exogenous stressors.
Physica a-Statistical Mechanics and Its Applications
390:250-256.
Seuront L, Brewer M, Strickler JR 2004a: Quantifying
zooplankton swimming behavior: the question of scale.
In Seuront L, Strutton PG (eds): Handbook of scaling
methods in aquatic ecology. CRC Press, Boca Raton,
pp 333-359.
Seuront L, Cribb N 2011: Fractal analysis reveals
pernicious stress levels related to boat presence andtype in the Indo-Pacific bottlenose dolphin, Tursiops
aduncus. Physica a-Statistical Mechanics and Its
Applications 390:2333-2339.
Seuront L, Hwang JS, Tseng LC, Schmitt FG, Souissi S,
Wong CK 2004b: Individual variability in theswimming behavior of the sub-tropical copepod

The Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

Oncaea venusta (Copepoda : Poecilostomatoida).
Marine Ecology-Progress Series 283:199-217.
Seuront L, Leterme S 2007: Increased zooplankton
behavioral stress in response to short-term exposure tohydrocarbon contamination. The Open Oceanography
Journal 1:1-7.
Seuront L, Strutton PG, (eds) 2004. Handbook of scling
methods in aquatic ecology. CRC Press, Boca Raton.
Seuront L, Yamazaki H, Souissi S 2004c: Hydrodynamic
disturbance and zooplankton swimming behavior. Zool
Stud 43:376-387.
Shimada I, Kawazoe Y, Hara H 1993: A temporal modelof animal behavior based on a fractality in the feedingof Drosophila melanogaster. Biol Cybern 68:477-481.
Shimada I, Minesaki Y, Hara H 1995: Temporal fractal in
the feeding behavior of Drosophila melanogaster. J
Ethol 13:153-158.
Shlesinger MF, West BJ 1991: Complex fractal dimensionof the bronchial tree. Phys Rev Lett 67:2106-2108.
Sims DW, Southall EJ, Humphries NE, Hays GC,
Bradshaw CJA, Pitchford JW, James A, Ahmed MZ,
Brierley AS, Hindell MA, Morritt D, Musyl MK,
Righton D, Shepard ELC, Wearmouth VJ, Wilson RP,
Witt MJ, Metcalfe JD 2008: Scaling laws of marine
predator search behaviour. Nature 451:1098-1102.
Smouse PE, Focardi S, Moorcroft PR, Kie JG, Forester
JD, Morales JM 2010: Stochastic modelling of animal
movement. Philosophical Transactions of the Royal
Society B-Biological Sciences 365:2201-2211.
Solé RV, Manrubia SC, Benton M, Bak P 1997: Selfsimilarity of extinction statistics in the fossil record.
Nature 388:764-767.
Solé RV, Manrubia SC, Benton M, Kauffman S, Bak P
1999: Criticality and scaling in evolutionary ecology.
Trends Ecol Evol 14:156-160.
Spehar B, Clifford CWG, Newell BR, Taylor RP 2003:
Universal aesthetic of fractals. Computers & Graphics
27:813-820.
Sueur C 2011: A non-Levy random walk in chacma
baboons: what does it mean? PLoS ONE 6.
Sueur C, Briard L, Petit O 2011a: Individual analyses of
Levy walk in semi-free ranging Tonkean macaques(Macaca tonkeana). PLoS ONE 6.
Sueur C, Jacobs A, Amblard F, Petit O, King AJ 2011b:
How can social network analysis improve the study ofprimate behavior? Am J Primatol 73:703-719.
Sueur C, King A, Pelé M, Petit O 2013: Fast and accurate
decisions as a result of scale-free network properties in
two primate species. In Gilbert T, Kirkilionis M,
Nicolis G (eds): Proceedings of the European
Conference on Complex Systems 2012. Springer
International Publishing, pp 579-584.

23

Sueur C, Petit O 2008: Organization of group members at
departure is driven by social structure in Macaca. Int J
Primatol 29:1085-1098.
Sugihara G, May RM 1990: Applications of fractals in
ecology. Trends Ecol Evol 5:79-86.
Sugihara G, Schoenly K, Trombla A 1989: Scaleinvariance in food web properties. Science 245:48-52.
Taylor R, Spehar B, Hagerhall C, Van Donkelaar P 2011:
Perceptual and physiological responses to Jackson
Pollock’s fractals. Frontiers in Human Neuroscience 5.
Taylor RP, Guzman R, Martin TP, Hall GDR, Micolich
AP, Jonas D, Scannell BC, Fairbanks MS, Marlow CA
2007: Authenticating Pollock paintings using fractal
geometry. Pattern Recognition Letters 28:695-702.
Taylor RP, Micolich AP, Jonas D 1999: Fractal analysis
of Pollock's drip paintings. Nature 399:422-422.
Taylor RP, Micolich AP, Jonas D 2006: Fractal analysis:
revisiting Pollock's drip paintings (Reply). Nature
444:E10-E11.
Turchin P 1996: Fractal analyses of animal movement: A
Critique. Ecology 77:2086-2090.
Vandercone R, Premachandra K, Wijethunga GP, Dinadh
C, Ranawana K, Bahar S 2013: Random walk analysisof ranging patterns of sympatric langurs in a complex
resource landscape. Am J Primatol 75:1209-1219.
Vicsek T, (ed) 2001. Fluctuations and scaling in biology.
Oxford University Press, New York, 256 pp.
Vicsek T, Shlesinger MF, Matsushita M, (eds) 1994.
Fractals in natural sciences. World Scientific,
Singapore.
Viswanathan GM, Afanasyev V, Buldyrev SV, Murphy
EJ, Prince PA, Stanley HE 1996: Levy flight searchpatterns of wandering albatrosses. Nature 381:413-415.
Viswanathan GM, Da Luz MGE, Raposo EP, Stanley HE
2011: The physics of foraging. Cambridge University
Press, Cambridge, 178 pp.
Viswanathan GM, Raposo EP, da Luz MGE 2008: Levy
flights and superdiffusion in the context of biologicalencounters and random searches. Physics of Life
Reviews 5:133-150.
Weary DM, Huzzey JM, von Keyserlingk MAG 2009:
Using behavior to predict and identify ill health inanimals. J Anim Sci 87:770-777.
West BJ 1990: Physiology in fractal dimensions: error
tolerance. Ann Biomed Eng 18:135-149.
West BJ, Goldberger AL 1987: Physiology in fractal
dimensions. Am Sci 75:354-365.
Wiens JA 1989: Spatial scaling in ecology. Funct Ecol
3:385-397.
Wiens JA 1993: On quantifying insect movements.
Environ Entomol 22:709.
Wiens JA, Crist TO, With KA, Milne BT 1995: Fractal

24

Andrew James Jonathan MacIntosh

patterns of insect movement in microlandscapemosaics. Ecology 76:663-666.
Wilensky U 1999: Netlogo. http://ccl.northwestern.edu/
netlogo/. Evanston, IL: Center for Connected Learning
and Computer-Based Modeling, Northwestern
University.

The Fractal Primate: Interdisciplinary Science and the Math behind the Monkey

25

（要約）

The Fractal Primate: Interdisciplinary Science and the Math behind theMonkeyAndrew James Jonathan MacIntosh 1), 2)
1)

2)

Kyoto University Wildlife Research Center, Kyoto, Japan
Center for International Collaboration and Advanced Studies in Primatology(CICASP), Kyoto University Primate Research Institute, Inuyama, Japan

Drawing upon a large body of literature comprising various applications of fractal theory and complexity
science in ecology and evolution, I illustrate that animal behavior can be characterized usefully by its
fundamentally fractal nature in both time and space. I begin with an introduction to fractal geometry and thenprovide examples of how and why its analysis has been employed in biological/ecological studies. I examine thethree main approaches used to model animal behavior through the fractal lens, and show how (i) the Lévy flightforaging hypothesis, (ii) spatial fractal dimension estimates and (iii) fractal time have each complemented more
traditional investigations of animal behavior and provided unique insights into ecological processes. I then focus
on illustrating how fractal analysis can be used as a sensitive indicator of behavioral or environmental ‘quality’.I
argue that complex behavior sequences both result from and are necessary to cope with environmental
heterogeneity. This body of work supports the hypotheses that complexity is biologically adaptive and thatcomplexity loss, i.e. greater periodicity or stereotypy, is not only indicative of altered or impaired condition but
may in addition carry with it significant fitness consequences. Throughout this essay, I attempt to emphasize thatby extending beyond our traditional disciplinary boundaries, behavioral ecologists stand to gain unprecedented
insight into complex phenomena in the lives and activities of organisms all around us. It is my sincere hope thatthe ideas presented herein encourage future interdisciplinary research into the role of complexity in behavioralorganization, and what that might mean for ecological processes.
Keywords: fractal geometry; chaos theory; complexity science; behavioral organization; interdisciplinary studies
Andrew J. J. MacIntosh
Center for International Collaboration and Advanced Studies in Primatology (CICASP)
Kyoto University Primate Research InstituteInuyama, Aichi, Japan 484-8506
email: andrew.j.jmacintosh@gmail.com