M. A. M. de Aguiar

Global patterns of speciation and diversity

In recent years, strikingly consistent patterns of biodiversity have been identified over space, time, organism type and geographical region. A neutral theory (assuming no environmental selection or organismal interactions) has been shown to predict many patterns of ecological biodiversity. This theory is based on a mechanism by which new species arise similarly to point mutations in a population without sexual reproduction. Here we report the simulation of populations with sexual reproduction, mutation and dispersal. We found simulated time dependence of speciation rates, species–area relationships and species abundance distributions consistent with the behaviours found in nature. From our results, we predict steady speciation rates, more species in one-dimensional environments than two-dimensional environments, three scaling regimes of species–area relationships and lognormal distributions of species abundance with an excess of rare species and a tail that may be approximated by Fisher’s logarithmic series. These are consistent with dependences reported for, among others, global birds and flowering plants, marine invertebrate fossils, ray-finned fishes, British birds and moths, North American songbirds, mammal fossils from Kansas and Panamanian shrubs. Quantitative comparisons of specific cases are remarkably successful. Our biodiversity results provide additional evidence that species diversity arises without specific physical barriers. This is similar to heavy traffic flows, where traffic jams can form even without accidents or barriers.

Semiclassical approximations in phase space with coherent states

We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Hellers thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

Spectral analysis and the dynamic response of complex networks

The eigenvalues and eigenvectors of the connectivity matrix of complex networks contain information about its topology and its collective behavior. In particular, the spectral density rho(lambda) of this matrix reveals important network characteristics: random networks follow Wigners semicircular law whereas scale-free networks exhibit a triangular distribution. In this paper we show that the spectral density of hierarchical networks follows a very different pattern, which can be used as a fingerprint of modularity. Of particular importance is the value rho(0), related to the homeostatic response of the network: it is maximum for random and scale-free networks but very small for hierarchical modular networks. It is also large for an actual biological protein-protein interaction network, demonstrating that the current leading model for such networks is not adequate.

Bifurcations of periodic trajectories in non-integrable Hamiltonian systems with two degrees of freedom: Numerical and analytical results

Numerical and analytical studies of the types of period n-upling bifurcations undergone by classsical periodic trajectories of non-intergrable Hamiltonians with two degrees of freedom are made. The Hamiltonians studied possess time reversal and reflexion symmetries and we found that these symmetries give rise to additional types of period n-upling bifurcations. The analytical study explains most of the numerically observed bifurcations.

Invasion and Extinction in the Mean Field Approximation for a Spatial Host-Pathogen Model

We derive the mean field equations of a simple spatial host-pathogen, or predator-prey, model that has been shown to display interesting evolutionary properties. We compare these equations, and the equations including pair-correlations, with the low-density approximations derived by other authors. We study the process of invasion by a mutant pathogen, both in the mean field and in the pair approximation, and discuss our results with respect to the spatial model. Both the mean field and pair correlation approximations do not capture the key spatial behaviors—the moderation of exploitation due to local extinctions, preventing the pathogen from causing its own extinction. However, the results provide important hints about the mechanism by which the local extinctions occur.

Chaotic signature in the motion of coupled carbon nanotube oscillators

The motion of coupled oscillators based on multiwalled carbon nanotubes is studied using rigid-body dynamics simulations. The results show the existence of chaotic and regular behaviours for a given total energy, indicating the manifestation of chaos in nanoscaled mechanical systems based on carbon nanotube oscillators. Different regular motions are observed for different total energies, and they can be obtained by appropriately choosing the initial conditions. This possibility can allow the construction of multi-functional nano-devices based on multiwalled carbon nanotube oscillators.

Reply to ‘Comment on ‘‘Semiclassical approximations in phase space with coherent states’’’

The Herman–Kluk (HK) formula was shown in (Baranger et al J. Phys. A: Math.Gen. 34 7227) not to be a correct semiclassical limit of an exact quantum mechanical formula. Two previous attempts to derive it using semiclassical arguments contain serious errors. These statements are left totally untouched by Herman and Grossmann’s comment. They argue that the formula which we found to be at fault is not the one that should be called the HK formula. However, the formula we criticized is definitely one of the steps, in fact the main step, in these two published derivations of the HK formula. Very recently, a new derivation was published by Miller. It is interesting, but it is not semiclassical. 1. Summary In this note of reply to Grossmann and Herman’s comment (GH) [Gro02], we sincerely hope that we can clear up the serious misunderstanding that has arisen between us. Perhaps we should begin by acknowledging now, rather than at the end of the paper, the efforts of the referee and the editor of this journal, who insiste dt hat this misunderstanding be aired out. We se et wo points of contention between GH and us. One is a fundamental point of physics. The other is a relatively trivial question of interpretation. Our first point is that the Herman–Kluk approximation (HK) [Her84], irrespective of its considerable other virtues, is not a semiclassical approximation in the strict sense of the term. About this we are certain. We have shown it already in our paper [Bar01] in great detail, and we are going to show it again in section 3 in at otally different way. Therefore, every statement by GH about HK being correctly semiclassical is misleading. If the H Ka pproximation is not semiclassical, then what

Mean-field approximation to a spatial host-pathogen model.

We study the mean-field approximation to a simple spatial host-pathogen model that has been shown to display interesting evolutionary properties. We show that previous derivations of the mean-field equations for this model are actually only low-density approximations to the true mean-field limit. We derive the correct equations and the corresponding equations including pair correlations. The process of invasion by a mutant type of pathogen is also discussed.

Chaos in a spin-boson system: classical analysis

Abstract We have found novel aspects of the spin-boson system in a fully classical analysis of the system. The finiteness of the spin phase space is shown to strongly influence the systematic behaviour of periodic orbits. We also give a detailed account of the consequences of the chaotic dynamics and of the superradiant phase transition.

Semiclassical propagation of wavepackets with complex and real trajectories

We consider a semiclassical approximation, first derived by Heller and coworkers, for the time evolution of an originally Gaussian wave packet in terms of complex trajectories. We also derive additional approximations replacing the complex trajectories by real ones. These yield three different semiclassical formulae involving different real trajectories. One of these formulae is Heller’s thawed Gaussian approximation. The other approximations are nonGaussian and may involve several trajectories determined by mixed initial–final conditions. These different formulae are tested for the cases of scattering by a hard wall, scattering by an attractive Gaussian potential and bound motion in a quartic oscillator. The formula with complex trajectories gives good results in all cases. The non-Gaussian approximations with real trajectories work well in some cases, whereas the thawed Gaussian works only in very simple situations.