Discussion

The failure of ecologists to use the term "self-affine" to describe fractal behavior appears to be limited to a few authors publishing in plant and forest ecology. Microbial ecologists (XXX) and marine ecologists (Seuront XXXX) seem to have a better understanding of self-affinity and its consequences.

Closing in on Self-Affinity from the Metabolic Scaling Theory

The misuse of the term "self-similar" for describing self-affine processes in ecology has gone largely unreported. Interestingly, some authors appear to be approaching the issue from different perspectives. Both Bentley et al. (2013) and Smith et al. (2014) report deviations from symmetry in branching, better predicting metabolism, hydraulics, and light interception in plants. Yet, neither explicitly refers to this behavior as "self-affine," although their results are equivalent to how a self-affine object operates. For instance, Bentley et al. (2013) observed differences in scaling exponents at different-sized branching nodes within individual trees. Similarly, Smith et al. (2014) showed that path fractions (the ratio of minimum to maximum twig-to-trunk path lengths) varied from symmetric (self-similar) to highly asymmetric (self-affine).

Demonstrations of Self-Affine Fractals

Demonstrations of self-affine fractals resembling trees and plants, such as Barnsley's Fern (XX), have further solidified the concept of self-affinity in branching networks. The study of lacunarity, as proposed by Plotnick et al. (1996), supports the notion of self-affinity in hierarchical networks. Plotnick et al. found that tree seedlings exhibited clumping behavior around parent trees instead of a binomial random pattern, which aligns with the predictions of lacunarity in self-affine fractals.

A Fractal Theory of Ecology

Seely and Macklem (2012) proposed two null hypotheses regarding fractal variability in nature:

1. Complex dissipative systems, such as the circulatory systems within animals and capillary networks in plants, emerge as self-optimized systems that dissipate energy in ways that maximize entropy production.
2. The degree of variability in the system is proportional to the ratio of its maximum work output to its resting work output, reflecting the system’s adaptability to change.

These hypotheses relate closely to other discussions in ecology concerning fractals and the thermodynamics of metabolism in organisms (e.g., MST). MST also theorizes that hierarchical branching networks are optimized to maximize the delivery of substrates to active sites where metabolism takes place. The optimization of fractal networks allows organisms to balance metabolic demand with mass and energy flow, ultimately leading to predictable fractal dimensions.

In contrast to Rubner’s (1883) "surface rule," which suggests that an organism’s body scales its metabolism proportionally to its surface area with a ⅔ exponent, Kleiber (1932) showed that the surface rule was invalid. Instead, organisms scale their metabolism to body mass with a \(3/4\) exponent, as observed across 23 orders of magnitude in body size. This \(3/4\) exponent reflects the fractal nature of the metabolic scaling law proposed by WBE, which remains robust when applied to vascular organisms.

Fractal Dimensions of Non-Vascular Organisms

Colonies of single-celled organisms, such as bacteria, exhibit diffusion-limited aggregation (DLA), a self-similar fractal pattern. When substrate levels are not limiting, these colonies grow as uniform advancing fronts, following a fractal model described by the Eden Model and the Kardar-Parisi-Zhang (KPZ) equation. The body forms of primitive non-vascular land plants, such as liverworts, hornworts, and mosses, exhibit similar DLA fractal patterns. Without hierarchical branching supply networks, these organisms are self-limited in their ability to deliver substrates over long distances, relying instead on diffusion.

Aquatic organisms without vascular structures, such as sponges and xenophyophores, have evolved hyperbolic geometries to maximize substrate delivery. These organisms bypass the need for hierarchical networks by leveraging turbulent mixing in aquatic environments, which allows them to compete effectively despite lacking vascularity. The fractal dimension of such organisms often aligns with the fractal geometry of turbulence (Kolmogorov turbulence, \( \beta = -\frac{5}{3} \)).

Extended Applications: Explaining Why Vascular Life Forms Dominate

The dominance of vascular organisms in terrestrial ecosystems can be explained by their ability to utilize turbulent and laminar fluid flows within their internal networks, enhancing substrate transport. The hypothesis proposed by Seely and Macklem (2012) that vascularity maximizes entropy production seems self-evident when comparing vascular and non-vascular organisms. Vascular organisms increase their rate of metabolism by facilitating and controlling convective forces, moving substrates to points where diffusion and enzyme kinetics take over.

While the evolution of fractal forms is influenced by natural selection, the maximization of entropy production appears to be a fundamental mechanism underlying the self-affine fractal dimensions observed in organisms. Organisms with lower entropy production are typically constrained by external factors, such as gravity and resource availability, leading to a diversity of body forms that optimize resource utilization under specific environmental conditions.

In 1985, Sernetz et al. suggested considering the organism as a "bioreactor"—a volume-area hybrid where fractal surfaces maximize exchange through turbulent and diffusive motions. This view aligns with MST and supports the idea that fractal geometry in organisms reflects an optimization of mass transfer mechanisms. Although Mandelbrot (1982) suggested that self-affinity lacked a universally valid reduction to basic principles, the fundamental processes of fractional Brownian motion, Fickian diffusion, and convective motion by pulsatile or capillary action provide sufficient explanations for the emergence of self-affine fractal geometry in life forms.

Toward a Unified Theory

In ecology, the Zipf-Mandelbrot law has been used to describe relative abundance distributions (RAD) and species abundance distributions (SAD). Ecologists considering fractal analyses should be aware of the distinctions between self-similar and self-affine fractals, ensuring the appropriate methods are used for deriving affine fractal dimensions. The techniques used in this study, such as differential box counting and lacunarity analysis, are well-suited for spatial fractal dimensions in grayscale images, but other techniques (e.g., power spectrum analysis, detrended fluctuation analysis, rescaled range analysis) may offer further insights into fractal behavior in ecosystems.

Further exploration of the fractal geometry of organisms may help to resolve questions about the role of entropy production in determining fractal dimensions. The authors believe that a unification of thermodynamics, information theory, the reduction law of metabolism, and Darwinian evolution could provide a comprehensive framework for understanding the fractal nature of life.