Analysis of trophic networks: an optimisation approach.

We introduce a methodology to study the possible matter flows of an ecosystem defined by observational biomass data and realistic biological constraints. The flows belong to a polyhedron in a multi dimensional space that may make statistical exploration difficult in practice; instead, we propose to solve a convex optimization problem. Seven criteria based on ecological network indices have been selected to be used as convex goal functions. Numerical results show that the method is fast and can be used for large systems. Minimum flow solutions are analyzed using flow decomposition in paths and circuits. Their consistency is also tested by introducing a system of differential equations for the biomasses and examining the stability of the biomass fixed point. The method is illustrated and explained throughout the text on an ecosystem toy model. It is also applied to realistic food models.

Stopping a reaction-diffusion front.

We revisit the problem of pinning a reaction-diffusion front by a defect, in particular by a reaction-free region. Using collective variables for the front and numerical simulations, we compare the behaviors of a bistable and monostable front. A bistable front can be pinned as confirmed by a pinning criterion, the analysis of the time independent problem, and simulations. Conversely, a monostable front can never be pinned, it gives rise to a secondary pulse past the defect and we calculate the time this pulse takes to appear. These radically different behaviors of bistable and monostable fronts raise issues for modelers in particular areas of biology, as for example, the study of tumor growth in the presence of different tissues.

Nonlinear waves in networks: model reduction for the sine-Gordon equation.

To study how nonlinear waves propagate across Y- and T-type junctions, we consider the two-dimensional (2D) sine-Gordon equation as a model and examine the crossing of kinks and breathers. Comparing energies for different geometries reveals that, for small widths, the angle of the fork plays no role. Motivated by this, we introduce a one-dimensional effective model whose solutions agree well with the 2D simulations for kink and breather solutions. These exhibit two different behaviors: a kink crosses if it has sufficient energy; conversely a breather crosses when v>1-ω, where v and ω are, respectively, its velocity and frequency. This methodology can be generalized to more complex nonlinear wave models.

Reaction-diffusion front crossing a local defect.

The interaction of a Zeldovich-Frank-Kamenetsky reaction-diffusion front with a localized defect is studied numerically and analytically. For the analysis, we start from conservation laws and develop simple, collective variable, ordinary differential equations for the front position and width. Their solutions are in good agreement with the solutions of the full problem. Finally, using this reduced model, we explain the pinning of the front on a large defect and obtain a quantitative criterion.

Vortex polarity switching by a spin-polarized current.

The spin-transfer effect is investigated for the vortex state of a magnetic nanodot. A spin current is shown to act similarly to an effective magnetic field perpendicular to the nanodot. Then a vortex with magnetization (polarity) parallel to the current polarization is energetically favorable. Following a simple energy analysis and using direct spin-lattice simulations, we predict the polarity switching of a vortex. For magnetic storage devices, an electric current is more effective to switch the polarity of a vortex in a nanodot than the magnetic field.

Reduced Maxwell-Duffing description of extremely short pulses in nonresonant media.

The propagation of extremely short pulses of an electromagnetic field (electromagnetic spikes) is considered in the framework of a model wherein the material medium is represented by anharmonic oscillators with cubic nonlinearities (Duffing model) and waves can propagate only in the right direction. The system of reduced Maxwell-Duffing equations admits two families of exact analytical solutions in the form of solitary waves. These are bright spikes propagating on a zero background, and bright and dark spikes propagating on a nonzero background. We find that these steady-state pulses are stable in terms of boundedness of the Hamiltonian. Direct simulations demonstrate that these pulses are very robust against perturbations. We find that a high-frequency modulated electromagnetic pulse evolves into a breather-like one. Conversely, a low frequency pulse transforms into a quasiharmonic wave.

Importance of the internal shape mode in magnetic vortex dynamics.

We investigate the motion of a nonplanar vortex in a circular easy-plane magnet with a rotating in-plane magnetic field. Our numerical simulations of the Landau-Lifshitz equations show that the vortex tends to a circular limit trajectory, with an orbit frequency which is lower than the driving field frequency. To describe this we develop a new collective variable theory by introducing additional variables which account for the internal degrees of freedom of the vortex core, strongly coupled to the translational motion. We derive the evolution equations for these collective variables and find limit-cycle solutions whose characteristics are in qualitative agreement with the simulations of the many-spin system.