Exotic marginal fractals from hierarchical random deposition

Indekeu, Joseph; Fleerackers, Gunther; Posazhennikova, Anna; Bervoets, E

doi:10.1016/s0378-4371(00)00277-6

Cited by 10 publications

(14 citation statements)

References 9 publications

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“…This correspondence holds as long as there are no "overhangs" in the biofilm, and thus works for towering pillar structures but not for mushroom-shaped ones [3]. The similarity between the three-dimensional characteristic landscapes generated in this model, and the structure of its horizontal cross-section [6] (see Fig.6 in that reference), and images of vertical and horizontal sections of a towering pillar biofilm, as obtained by confocal laser scanning microscopy, is remarkable [7]. This suggests that a hierarchical model may be a reasonable first approximation for describing biofilm growth.…”

Section: Introduction Of the Modelmentioning

confidence: 78%

Hierarchical population model with a carrying capacity distribution for bacterial biofilms

Indekeu

Sznajd-Weron

2003

Phys. Rev. E

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A time-and space-discrete model for the growth of a rapidly saturating local biological population N (x, t) is derived from a hierarchical random deposition process previously studied in statistical physics. Two biologically relevant parameters, the probabilities of birth, B, and of death, D, determine the carrying capacity K. Due to the randomness the population depends strongly on position, x, and there is a distribution of carrying capacities, Π(K). This distribution has selfsimilar character owing to the imposed hierarchy. The most probable carrying capacity and its probability are studied as a function of B and D. The effective growth rate decreases with time, roughly as in a Verhulst process. The model is possibly applicable, for example, to bacteria forming a "towering pillar" biofilm. The bacteria divide on randomly distributed nutrient-rich regions and are exposed to random local bactericidal agent (antibiotic spray). A gradual overall temperature change away from optimal growth conditions, for instance, reduces bacterial reproduction, while biofilm development degrades antimicrobial susceptibility, causing stagnation into a stationary state.

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Section: Introduction Of the Modelmentioning

confidence: 78%

Hierarchical population model with a carrying capacity distribution for bacterial biofilms

Indekeu

Sznajd-Weron

2003

Phys. Rev. E

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“…Thus, the addition modulo p of two real numbers u, v is a particular instance of a generalized bitwise arithmetic operator b q ( 2 R p ; u, v) with q = p and with the magma 2 R p having code R given by Eq. (20). We note that our expression Eq.…”

Section: Digital Calculus: the Digit Function Discrete Operatorsmentioning

confidence: 85%

Fractal surfaces from simple arithmetic operations

Garcı́a-Morales

2016

Physica A: Statistical Mechanics and its Applications

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Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent $H$ that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.Comment: 15 pages, 6 figures, minor corrections. Published in Physica

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“…One can see that the number of squares is proportional to the inverse impact cross-section. In this model [2,3] a logarithmic fractal law was found for the surface (length) of the resulting landscape.…”

Section: Hierarchical Depositionmentioning

confidence: 99%

“…A richer variety of phenomena appears when level sets of the deposition model are studied [3]. We define a "sea level" in the landscape and study the geometry of the coastal points in D = 1 (or coastlines in D = 2) that remain.…”

Section: Coastal Points and Non-universalitymentioning

confidence: 99%

“…One can study level sets for random hierarchical deposition on a Euclidean substrate of dimension D and the resulting collection of points or contour lines, which are, respectively, named coastal points and coastlines for D = 1 or D = 2 [3]. This nomenclature stems from the context of islands, which result when the landscape is floaded up to a certain level, and is useful for describing the geometry of their jagged beaches.…”

Section: Introductionmentioning

confidence: 99%

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Coastlines and percolation in a model for hierarchical random deposition

Berx

Bervoets

Giuraniuc

et al. 2021

Physica A: Statistical Mechanics and its Applications

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We revisit a known model in which (conducting) blocks are hierarchically and randomly deposited on a d-dimensional substrate according to a hyperbolic size law with the block size decreasing by a factor λ > 1 in each subsequent generation. In the first part of the paper the number of coastal points (in D = 1) or coastlines (in D = 2) is calculated, which are points or lines that separate a region at "sea level" and an elevated region. We find that this number possesses a non-universal character, implying a Euclidean geometry below a threshold value P c of the deposition probability P , and a fractal geometry above this value. Exactly at the threshold, the geometry is logarithmic fractal. The number of coastlines in D = 2 turns out to be exactly twice the number of coastal points in D = 1. We comment briefly on the surface morphology and derive a roughness exponent α. In the second part, we study the percolation probability for a current in this model and two extensions of it, in which both the scale factor and the deposition probability can take on different values between generations. We find that the percolation threshold P c is located at exactly the same value for the deposition probability as the threshold probability of the number of coastal points. This coincidence suggests that exactly at the onset of percolation for a conducting path, the number of coastal points exhibits logarithmic fractal behaviour.a Given his role as Editor of this journal, Joseph O. Indekeu had no involvement in the peer-review of articles for which he was an author and had no access to information regarding their peer-review. Full responsibility for the peer-review process for this article was delegated to another Editor.

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Exotic marginal fractals from hierarchical random deposition

Cited by 10 publications

References 9 publications

Hierarchical population model with a carrying capacity distribution for bacterial biofilms

Hierarchical population model with a carrying capacity distribution for bacterial biofilms

Fractal surfaces from simple arithmetic operations

Coastlines and percolation in a model for hierarchical random deposition

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