Benoît B. Mandelbrot (1924–2010)

This obituary may be quite surprising to many, if not most, readers of the Journal of Plankton Research as both the man himself and the word “fractal” he created to describe the complexity of forms and shapes occurring in nature might still be quite unfamiliar to most plankton biologists and ecologists. The Web of Science (accessed 3 January 2011) returns 17202 and 39 855 papers including the word “fractal” in their title and topic, respectively. Fractals are indeed a prolific topic, and have found applications in nearly all scientific fields, including plankton research.

BENOîT B. MANDELBROT, THE FATHER OF FRACTALS

The man behind the word “fractal”, Benoît B. Mandelbrot (Fig. 1), a Polish-born French mathematician, often referred to as the father of fractals, died on 14 October 2010 at the age of 85. He dedicated his life to study the complexity of patterns and processes arising in fields as diverse as geology, medicine, cosmology, engineering and finance. B. B. Mandelbrot wrote more than 200 publications, reaching nearly 13 000 citations and an H-index of 47; a list of publications selected for their potential interest to a wide audience is given in Table I.

Table I:

B. B. Mandelbrot, selected list of publications

Mandelbrot, B. B. (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156, 636–638

Mandelbrot, B. B. and van Ness, J. W. (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev., 10, 422–437

Mandelbrot, B. B. and Wallis, J. R. (1969) Some long run properties of geophysical records. Water Resour. Res., 5, 321–340

Mandelbrot, B. B. (1975) Les Objects Fractals: Forme, Hasard et Dimension. Flammarion

Mandelbrot, B. B. (1977) Fractals. Form, Chance, and Dimension. Freeman

Mandelbrot, B. B. (1983) The Fractal Geometry of Nature. Freeman

Mandelbrot, B. B. (1985) Self-affine fractals and fractal dimension. Phys. Scripta, 32, 257–260

Mandelbrot, B. B. (1986) Self-affine fractal sets. In: Pietronero, L. and Tosatti, E. (eds.), Fractals in Physics. North-Holland

Mandelbrot, B. B. (1988) An introduction to multifractal distribution functions. In: Stanley, H. E. and Ostrowsky, N. (eds), Fluctuations and Pattern Formation. Kluwer, 345–360

Mandelbrot, B. B. (1989) Multifractal measures, especially for the geophysicist. Pure Appl. Geophys., 131, 5–42

Mandelbrot, B. B. (2001) Gaussian Self-Affinity and Fractals. Springer

Frame, M. and Mandelbrot, B. B. (2002) Fractals, Graphics, and Mathematics Education. The Mathematical Foundation of America

Mandelbrot, B. B. (2004) Fractals and Chaos: The Mandelbrot Set and Beyond. Springer

Mandelbrot, B. B. and Hudson, R. L. (2006) The Misbehavior of Markets: A Fractal View of Financial Turbulence. Basic Books

Mandelbrot, B. B. (2010) Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer

Fig. 1.

Benoît B. Mandelbrot in 1997 (Source: http://www.math.yale.edu/mandelbrot/). A colour version of this figure is available online.

After graduating from the prestigious Ecole Polytechnique then in Paris (1947), B. B. Mandelbrot received a Master in Science (1948) and an engineering degree (1949) from the California Institute of Technology. He later earned a doctorate in Mathematical Sciences from the University of Paris (1952), before receiving a prestigious scholarship from the Rockefeller Foundation to work with John von Neumann (a pioneering mathematician in game theory, economics, computer science, hydromechanics and quantum mechanics) as a post-doctoral fellow in the School of Mathematics of the Institute for Advanced Study at Princeton. He subsequently held positions in a range of prestigious institutions, including The Centre National de la Recherche Scientifique in Paris, IBM Research in New York, Harvard University, the Massachusetts Institute of Technology, and Yale University where he retired in 2005.

Mandelbrot never published on plankton nor on any marine-related topic. However, with a little bit of bad faith, his seminal work, entitled “How long is the coast of Britain? Statistical self-similarity and fractional dimensions” (Mandelbrot, 1967), might be thought as a marine contribution. In late annotations to this stepping-stone paper, Mandelbrot stated that this paper “was intended to be a Trojan horse allowing a bit of mathematical esoteric to infiltrate surreptitiously, hence near-painlessly, the investigation of the messiness of raw nature”. He was right, as this work planted the fractal seed and after more than a decade of self-proclaimed “immature aimlessness” with work published on information and game theories, linguistic, thermodynamics, noise and econometrics, Mandelbrot subsequently considered his work as an “increasingly acute single-mindedness” which gave rise to a few “Fractal Bibles” such as Fractals: Form, Chance, and Dimension (1977) and the self-explanatory The Fractal Geometry of Nature (1983), both translated into countless languages. His work inspired generations of scientists, helped both by its nearly universal fundamental importance and by the mesmerizing beauty of the pictures generated by fractal algorithms, such as the Mandelbrot and the Julia sets (e.g. Peitgen and Richter, 1986; Peitgen and Saupe, 1988; Briggs, 1992; Pickover, 1995, 2001; Lu, 1997; Frame and Mandelbrot, 2002; Peitgen et al., 2004; Stevens, 2005; Lesmoir-Gordon, 2010).

In 1967, Mandelbrot defined the basis of what will be formally coined “fractal geometry” a decade later (Mandelbrot, 1977, 1983), and introduced a new concept that has rapidly provided a unifying and cross-disciplinary basis for the description of nature's complexity. Many natural phenomena have a nested irregularity and may look similarly complex under different resolution (e.g. turbulent water flow or clouds; Fig. 2). While this nested structure, referred to as scale-invariant or self-similar (i.e. each portion can be considered a reduced-scale image of the whole) could be thought of as an additional source of complexity, it becomes a source of simplicity in the framework of fractal geometry. Hence, the degree of complexity of a given pattern or process can be described by a dimension D_F, the so-called fractal dimension. In contrast to conventional (Euclidean) dimensions, a fractal dimension is fractional, hence describing the degree of complexity and tortuosity of an object. For instance, the dimensions of a straight line, a circle and a cube are, respectively, D= 1, D= 2 and D= 3. However, what is the dimension of the meandering trajectory of a ciliate, a copepod or a fish larvae? And what is the dimension of marine snow particles? These a priori naïve questions would not have found an answer without the help of fractal geometry (Logan and Wilkinson, 1990; Coughlin et al., 1992; Bundy et al., 1993; Kilps et al., 1994; Li et al., 1998; Jonsson and Johansson, 1997; Dowling et al., 2000; Uttieri et al., 2005, 2007, 2008; Seuront et al., 2004a, b, c; Seuront, 2006, 2010, 2011). Eleven papers directly related to fractals, however, have been published in the Journal of Plankton Research (Table II).

Table II:

Publications using fractals that appeared in the Journal of Plankton Research

Ascioti, F. A., Beltrami, E., Carroll, T. O. and Wirick, C. (1993) Is there chaos in plankton dynamics? J. Plankton Res., 15, 603–617.

Dur, G., Souissi, S., Schmitt, F., Cheng, S.-H. and Hwang, J.-S. (2010) The different aspects in motion of the three reproductive stages of Pseudodiaptomus annandalei (Copepoda: Calanoida). J. Plankton Res., 32, 423–440.

Lovejoy, S., Currie, W. J. S., Claereboudt, M. R., Bourget, E., Roff, J.-C. and Schertzer, D. (2001) Universal multifractals and ocean patchiness: phytoplankton, physical fields and coastal heterogeneity. J. Plankton Res., 23, 117–141.

Ludovisi, A., Todini, C., Pandolfi, P. and Toticchi, M. I. (2008) Scale patterns of diel distribution of the copepod Cyclops abyssorum Sars in a regulated lake: the importance of physical and biological factors. J. Plankton Res., 30, 495–509.

Pascual, M., Ascioti, F. A. and Caswell, H. (1995) Intermittency in the plankton: a multifractal analysis of zooplankton biomass variability. J. Plankton Res., 17, 1209–1232.

Popova, E. E., Fasham, M. J. R., Osipov, A. V. and Ryabchenko, V. A. (1997) Chaotic behavior of an ocean ecosystem model under seasonal external forcing. J. Plankton Res., 19, 1495–1515.

Seuront, L. (2006) Effect of salinity on the swimming behavior of the estuarine calanoid copepod Eurytemora affinis. J. Plankton Res., 28, 805–813.

Seuront, L. and Lagadeuc, Y. (1998) Spatio-temporal structure of tidally mixed coastal waters: variability and heterogeneity. J. Plankton Res., 20, 1387–1401.

Seuront, L. and Lagadeuc, Y. (2001) Multiscale patchiness of the calanoid copepod Temora longicornis in a turbulent coastal sea. J. Plankton Res., 23, 1137–1145.

Seuront, L., Schmitt, F., Lagadeuc, Y., Schertzer, D. and Lovejoy, S. (1999) Universal multifractal analysis as a tool to characterize multiscale intermittent patterns: example of phytoplankton distribution in turbulent coastal waters. J. Plankton Res., 21, 877–922.

Uttieri, M., Nihongi, A., Mazzocchi, M. G., Strickler, J. R. and Zambianchi, E. (2007) Pre-copulatory swimming behaviour of Leptodiaptomus ashlandi (Copepoda: Calanoida): a fractal approach. J. Plankton Res., 29(Suppl. I), i17–i26.

Fig. 2.

Illustration of the nested structure perceptible in the geometry of clouds (A) and the geometry of two theoretical fractal objects, the Sierpinskin carpet (B) and the Sierpinski gasket (C); modified from Seuront (Seuront, 2009). A colour version of this figure is available online.

FRACTALS AND PLANKTON RESEARCH

Papers including fractals, and related concepts such as chaos theory, have been published in the Journal of Plankton Research (Table II) to describe (i) the predictability of phytoplankton and zooplankton biomass times series (Ascioti et al., 1993), (ii) the distribution of passive scalars (i.e. temperature, salinity), phytoplankton and zooplankton biomass (Pascual et al., 1995; Seuront and Lagadeuc, 1997, 1998, 2001; Seuront et al., 1999; Lovejoy et al., 2001; Ludovisi et al., 2008), (iii) the emerging properties of an ocean ecosystem model (Popova et al., 1997) and (iv) the complexity of zooplankton swimming behaviour (Seuront, 2006; Uttieri et al., 2007; Dur et al., 2010).

The first fractal paper published in the Journal of Plankton Research, entitled “Intermittency in the plankton: a multifractal analysis of zooplankton biomass variability” (Pascual et al., 1995), went well beyond the concept of fractal itself. Multifractals can indeed be considered as a generalization of fractal geometry initially introduced to describe the relationship between a given quantity and the scale at which it is measured. While fractal geometry describes the structure of a given pattern with the help of only one parameter (the fractal dimension D_F), multifractals characterize its detailed variability by an infinite number of sets, each with its own fractal dimensions. An intuitive approach specifically intended to demystify the complexity of multifractality and make it readily usable by ecologists has been introduced on the basis of the spatial structure of modern cities (Seuront, 2010). Consider a city viewed from above, it can be considered as a succession of black and white objects: in black are buildings and in white are streets and parks. The only information one can get is the distribution of the built and the unbuilt areas. This is the geometric support of the city. Now, change the angle of vision by taking a position not directly above the city, but from the side. The black and white city is now a set of buildings with different heights. This is the measure we are now interested in. It could also have been the colour, the width or the age of the building. It is now possible to estimate the distribution of a wide range of building heights. Each height will (eventually) be characterized by a fractal dimension, thus the concept of multifractals. Multifractals were subsequently used, in a more general form, to describe the intermittency in time series and transects of passive scalars, phytoplankton and zooplankton biomass (Seuront et al., 1999; Lovejoy et al., 2001; Seuront and Lagadeuc, 2001), and to characterize the stochastic properties of the successive displacements of the calanoid copepod Pseudodiaptomus annandalei (Dur et al., 2010). Specifically, in plankton behavioural ecology, multifractals uniquely provide a direct, objective and quantitative tool to thoroughly identify models of motion behaviour, such as Brownian motion (i.e. normal diffusion), fractional Brownian motion, ballistic motion, Lévy flight/walk and multifractal random walk (Fig. 3). Fractals have also been used to describe the space–time patterns of phytoplankton and zooplankton communities (Seuront and Lagadeuc, 1998; Ludovisi et al., 2008), and more recently to provide new insights into the behavioural ecology of copepods, through investigations of the fractal properties of two- and three-dimensional swimming paths (Seuront, 2006, 2011; Uttieri et al., 2007).

Fig. 3.

Illustration of the fundamental strength of the multifractal function ζ(q) in the identification of a model of movement from empirical behavioural data, typically (x, y) or (x, y, z) coordinates. ζ(q) is a continuous function of the statistical order of moment q (q = 1 and q = 2 correspond to the mean and the variance). The function ζ(q) is linear for fractional Brownian motion [ζ(q) = qH], with the limits ζ(q) = 0 and ζ(q) = 1 corresponding, respectively, to confinement and localization and ballistic motion. For Brownian motion (i.e. normal diffusion; dotted line), ζ(q) = q/2. When H>1/2, the motion is superdiffusive, whereas when H<1/2, it is subdiffusive. For a Lévy flight (black dots), ζ(q) = q/μ and ζ(q) = 1 for q < μ and q = μ respectively; μ (1 < μ ≤3) is an exponent characterizing the tail of the probability distribution function of successive displacements, with μ = 2 characterizing optimal Lévy flights for the location of stationary targets that are randomly and sparsely distributed, and once visited are not depleted but instead remain targets for future searches (Viswanathan et al., 1999). For a multifractal random walk, the function ζ(q) is non-linear and convex (continuous line); see Seuront (Seuront, 2010) and Viswanathan et al. (Viswanathan et al., 2011) for further details.

TOWARDS A MORE FRACTAL PLANKTON WORLD?

Despite demonstrations that fractal dimensions, and more generally fractal metrics, are much more sensitive descriptors than conventional ones to describe both plankton distribution patterns (Seuront et al., 2002; Seuront, 2004, 2005) and zooplankton behavioural properties (Seuront et al., 2004a, c; Seuront, 2006, 2010, 2011), the Web of Science (accessed 3 January 2011) returns only 31 papers including the words “fractal” and “plankton” in their topic. This indicates that plankton research is still poorly fractally coloured, hence the potential for fractally inspired approaches is significant as fractals have been successfully applied to the structure of coral reefs (Bradbury et al., 1984) and rocky reefs (Frost et al., 2005; Kostylev et al., 2005; Warfe et al., 2008), marine snow dynamics (Logan and Wilkinson, 1990; Grout et al., 2001; Ploug et al., 2008), the structural complexity of mussel beds (Snover and Commito, 1998; Commito and Rusignuolo, 2000; Kostylev and Erlandson, 2001), the spatial distribution of intertidal benthic communities (Azovsky et al., 2000), the behaviour of freshwater invertebrates (Seuront et al., 2004a; Uttieri et al., 2007), marine invertebrates (Erlandson and Kostylev, 1995; Seuront et al., 2004b, 2007) and marine vertebrates (Dowling et al., 2000; Mouillot and Viale, 2001; Seuront, 2011), species diversity (Frontier, 1987, 1994), size spectra (Vidondo et al., 1997), zooplankton patchiness (Tsuda, 1995), phytoplankton patchiness (Seuront and Lagadeuc, 1997; Waters and Mitchell, 2002; Waters et al., 2003) and microphytobenthos patchiness (Seuront and Spilmont, 2002). The interested reader is referred to, for example, Barnsley (Barnsley, 1993, 2000), Falconer (Falconer, 1985, 1993), Feder (Feder, 1988), Hastings and Sugihara (Hastings and Sugihara, 1993), Tricot (Tricot, 1995), Gouyet (Gouyet, 1996), Schroeder (Schroeder, 1991, 2009), Seuront and Strutton (Seuront and Strutton, 2004), Seuront (Seuront, 2010) and Viswanathan et al. (Viswanathan et al., 2011) for further reading on fractals, multifractals and related topics.

Turbulence, one of the most relevant processes for planktonic life, is known to have fractal and multifractal properties (Meneveau and Sreenivasan, 1991; Frish, 1996; Wijesekera, 1996). As such, patterns and processes directly related to turbulence are likely to exhibit a fractal/multifractal behaviour. In addition, fractals and multifractals have the desirable properties of (i) being independent of measurement scale and (ii) describing in simple terms the extreme complexity of nature. Hence they challenge the long standing and widely held belief that turbulence-related processes are extremely difficult to describe, due to their intrinsic complexity, many degrees of freedom and the chaotic nature of turbulent flows. In this context, it is believed and hoped that our journey to understand the zooplankton ecosystem from a fractally coloured angle is still at its early stage, and will add a plankton component to the unique legacy of Benoît B. Mandelbrot, the father of fractals.