Rhombic Tilings and Primordia Fronts of Phyllotaxis

We introduce and study properties of phyllotactic and rhombic tilings on the cylin- der. These are discrete sets of points that generalize cylindrical lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical system S that models plant pattern formation by stacking disks of equal radius on the cylinder. This system has the advantage of allowing several disks at the same level, and thus multi-jugate config- urations. We provide partial results toward proving that the attractor for S is entirely composed of rhombic tilings and is a strongly normally attracting branched manifold and conjecture that this attractor persists topologically in nearby systems. A key tool in understanding the geometry of tilings and the dynamics of S is the concept of pri- mordia front, which is a closed ring of tangent disks around the cylinder. We show how fronts determine the dynamics, including transitions of parastichy numbers, and might explain the Fibonacci number of petals often encountered in compositae

Convergence in a Disk Stacking Model on the Cylinder

We study an iterative process modeling growth of phyllotactic patterns, wherein disks are added one by one on the surface of a cylinder, on top of an existing set of disks, as low as possible and without overlap. Numerical simulations show that the steady states of the system are spatially periodic, lattices-like structures called rhombic tilings. We present a rigorous analysis of the dynamics of all configurations starting with closed chains of 3 tangent, non-overlapping disks encircling the cylinder. We show that all these configurations indeed converge to rhombic tilings. Surprisingly, we show that convergence can occur in either finitely or infinitely many iterations. The infinite-time convergence is explained by a conserved quantity

Lagrangian Systems on Hyperbolic Manifolds

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler-Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry-Mather theory of twist maps and the Hedlund-Morse-Gromov theory of minimal geodesies on closed surfaces and hyperbolic manifolds

A Dynamical System for Plant Pattern Formation: A Rigorous Analysis

We present a rigorous mathematical analysis of a discrete dynamical system modeling plant pattern formation. In this model, based on the work of physicists Douady and Couder, fixed points are the spiral or helical lattices often occurring in plants. The frequent occurrence of the Fibonacci sequence in the number of visible spirals is explained by the stability of the fixed points in this system, as well as by the structure of their bifurcation diagram. We provide a detailed study of this diagram

Phyllotaxis: a remarkable example of developmental canalization in plants

Why living forms develop in a relatively robust manner, despite various sources of internal or external variability, is a fundamental question in developmental biology. Part of the answer relies on the notion of developmental constraints: at any stage of ontogenenesis, morphogenetic processes are constrained to operate within the context of the current organism being built, which is thought to bias or to limit phenotype variability. One universal aspect of this context is the shape of the organism itself that progressively channels the development of the organism toward its final shape. Here, we illustrate this notion with plants, where conspicuous patterns are formed by the lateral organs produced by apical meristems. These patterns, called phyllotaxis, traditionally fall into two broad categories, spiral or whorled that present striking symmetries and regularities. These properties suggest that plant development is strongly canalized and cannot escape specific attraction patterns. Since the early 19th century, researchers have looked for biological or physical explanations for such amazing and specific form "attractors". Thanks to this collective and sustained effort, we now have gained much insight on this self-organizing process, and uncovered important parts of the mystery. This paper aims to provide an easy-to-read overview of the main concepts that have been developed to explain phyllotaxis and to make clear their connections in a step-by-step progression, while keeping the mathematics light. We suggest that altogether a view emerges where phyllotaxis appears as a remarkable example of how shapes may be canalized during development

Extracellular DNAses Facilitate Antagonism and Coexistence in Bacterial Competitor-Sensing Interference Competition

Over the last 4 decades, the rate of discovery of novel antibiotics has decreased drastically, ending the era of fortuitous antibiotic discovery. A better understanding of the biology of bacteriogenic toxins potentially helps to prospect for new antibiotics. To initiate this line of research, we quantified antagonists from two different sites at two different depths of soil and found the relative number of antagonists to correlate with the bacterial load and carbon-to-nitrogen (C/N) ratio of the soil. Consecutive studies show the importance of antagonist interactions between soil isolates and the lack of a predicted role for nutrient availability and, therefore, support an in situ role in offense for the production of toxins in environments of high bacterial loads. In addition, the production of extracellular DNAses (exDNases) and the ability to antagonize correlate strongly. Using an in domum-developed probabilistic cellular automaton model, we studied the consequences of exDNase production for both coexistence and diversity within a dynamic equilibrium. Our model demonstrates that exDNase-producing isolates involved in amensal interactions act to stabilize a community, leading to increased coexistence within a competitor-sensing interference competition environment. Our results signify that the environmental and biological cues that control natural-product formation are important for understanding antagonism and community dynamics, structure, and function, permitting the development of directed searches and the use of these insights for drug discovery

Periodic orbits for Hamiltonian systems in Cotangent Bundles

Abstract: We prove the existence of at least cl(M) periodic orbits for certain time dependant Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold M. These Hamiltonians are not necessarily convex but they satisfy a certain boundary condition given by a Riemannian metric on M. We discretize the variational problem by decomposing the time 1 map into a product of “symplectic twist maps”. A second theorem deals with homotopically non trivial orbits in manifolds of negative curvature.

Convergence in a disk stacking model on the cylinder

International audienceWe study an iterative process modelling growth of phyllotactic patterns, wherein disks are added one by one on the surface of a cylinder, on top of an existing set of disks, as low as possible and without overlap. Numerical simulations show that the steady states of the system are spatially periodic, lattices-like structures called rhombic tilings. We present a rigorous analysis of the dynamics of all configurations starting with closed chains of 3 tangent, non-overlapping disks encircling the cylinder. We show that all these configurations indeed converge to rhombic tilings. Surprisingly, we show that convergence can occur in either finitely or infinitely many iterations. The infinite-time convergence is explained by a conserved quantity

Ghost Circles for Twist Maps

Completely ordered invariant circles are found for the gradient of the energy flow in the state space, containing the critical sets corresponding to the Birkhoff orbits of all rotation number. In particular, these ghost circles contain the Aubry-Mather sets and map-invariant circles as completely critical sets when these exist. We give a criterion for a sequence of rational ghost circles to converge to a completely critical one