P.L. Antonelli

Seismic rays as Finsler geodesics

We prove that, in general, for anisotropic nonuniform continua, seismic rays are geodesics in Finsler geometry. In particular, for separable velocity functions, the geometry is Wagnerian. We provide concrete examples with theoretical discussions and introduce the seismic Finsler metric.

Theories and models in symbiogenesis

Two modern theories of biological evolution, one by Carl Woese, and the other by Lynn Margulis, are modelled with Volterra–Hamilton systems. Their predictions are evaluated and compared within this modelling framework. For example, Woese’s theory turns out to su8er from instability in its chemical exchanges processes, whereas Margulis’ does not. An introduction to the mathematical and biological ideas is included. ? 2003 Elsevier Science Ltd. All rights reserved.

Fundamentals of Finslerian Diffusion with Applications

Introduction. 1. Finsler Spaces. 2. Introduction to Stochastic Calculus on Manifolds. 3. Stochastic Development on Finsler Spaces. 4. Volterra-Hamilton Systems of Finsler Type. 5. Finslerian Diffusion and Curvature. 6. Diffusion on the Tangent and Indicatrix Bundles. A. Diffusion and Laplacian on the Base Space. B. Two-Dimensional Constant Berwald Spaces. Bibliography. Index.

A Transient-State Analysis of Tyson's Model for the Cell Division Cycle by Means of KCC-Theory

The transient-state stability analysis for the trajectories of Tysons equations for the cell-division cycle is given by the so-called KCC-Theory. This is the differential geometric theory of the variational equations for deviation of whole trajectories to nearby ones. The relationship between Lyapunov stability of steady-states and limit cycles is throughly examined. We show that the region of stability (where, in engineering parlance, the system is “hunting”) encloses the Tyson limit cycle, while outside this region the trajectories exhibit aperiodic behaviour.

The Differential Geometry of Lagrangians which Generate Sprays

The geometry of regular Lagrangians provides useful differential geometric models for a variety of fields, including variational calculus, electromagnetic theory, general relativity and relativistic optics, [MA]. Although the general theory of Lagrange differential geometry has been fully developed, only the so-called almost Finsler Lagrangians have been studied for purposes of applications until now. In the present paper, another class of Lagrangians, which arise in biology, are studied from a purely geometrical point-of-view, [MA].

Volterra-Hamilton production models with discounting: general theory and worked examples

with Gi jk being n 3 functions positively homogeneous of degree zero in yi and with smooth initial conditions (t0; xi 0; y i 0): This system plays an important role in electrical engineering where xi; i = 1; : : : ; m; is interpreted as electric charge and for i = m + 1; : : : ; n; as position coordinate of a moving part of a rotating electrical machine. Thus, dxi=dt = yi; are electrical currents and velocities with ki = 1 in (i). This model was extensively developed by G. Kron in the 1930s and afterwards won him the prestigious Monte6ore Prize. Kron used the geometry of T.Y. Thomas, L. Eisenhart,

Modelling density-dependent aggregation and reproduction in certain terrestrial and marine ecosystems: A comparative study☆

Abstract This paper presents a new model of the lynx/hare/plant-system and the sea otter/red sea urchin/kelp-system of Canada. The mathematical model is motivated by previous work of the authors on aggregation induced cycles in starfish/coral-and chemically mediated plant/ herbivore-systems. Three first-order nonlinear, coupled, ordinary differential equations with constant coefficients are postulated. The herbivore equation exhibits a cooperative term of the form γF p +1 where parameter p (fixed throughout) is related to potential for aggregation or reproductivity of the F -population, and γ is the variable aggregation coefficient and Hopf bifurcation parameter. This term represents density-dependent effects. The main technical result is that under natural hypothesis, limit cycles can occur for p p ⩾ 1. For the lynx/hare/plant-system a stable limit cycle results whose period is modulated by food quality of the plant, while for the otter/red urchin/kelp-system there are two equilibria, but no limit cycle. Some comments are made about R. Endeans hypothetical giant triton/ starfish/coral-system. In particular, it is shown that Charonia tritonis is a keystone predator if and only if starfish, Acanthaster planci , is neither highly aggregative nor highly fecund (i.e. p > 1), in the context of the mathematical model.

Corals and starfish waves on the great barrier reef: Analytical trophodynamics and 2-patch aggregation methods

In this paper, singular perturbation theory is applied to the Antonelli/Kazarinoff limit cycle model of the Crown-of-Thorns starfish (COTS) predation of corals on the Great Barrier Reef. At the microscale of individual reefs, the previously published 2-patch dynamics based on aggregation of individual behaviour is extended to include social interactions, spatial diffusion, and advection currents. As a consequence, the parameters which characterize the well-known wave solutions (i.e., the Reichelt starfish waves) are reinterpreted in terms of individual behaviour parameters. Likewise, the analytical trophodynamics of coral community production (without starfish), involving geometric invariants of second-order ODEs is defined at the microscale by direct manipulation of the reefal structure. The aggregation method leads to new insights into Liapunov stability of production of the coral community as a whole.

A New Class Of Spray-Generating Lagrangians

Geodesics are solutions of