Possibility of Turbulent Crystals

The possibility for the occurrence in crystals of a phenomenon, resembling turbulence, is discussed. This phenomenon, called {\it heterophase turbulence}, is manifested by the fluctuational appearance inside a crystalline sample of disordered regions randomly distributed in space. The averaged picture for such a turbulent solid is exemplified by an exactly solvable lattice-gas model. The origin of heterophase turbulence is connected with stochastic instability of quasi-isolated systems.Comment: Latex file, 20 pages, no figure

Network analysis of complex biological systems : boundedness of weakly reversible chemical reaction networks and conditions for synchronisation of coupled oscillators

Imperial Users onl

Selected topics on reaction-diffusion-advection models from spatial ecology

We discuss the effects of movement and spatial heterogeneity on population dynamics via reaction-diffusion-advection models, focusing on the persistence, competition, and evolution of organisms in spatially heterogeneous environments. Topics include Lokta-Volterra competition models, river models, evolution of biased movement, phytoplankton growth, and spatial spread of epidemic disease. Open problems and conjectures are presented

Asymptotic behaviour for a class of non-monotone delay differential systems with applications

The paper concerns a class of $n$-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.Comment: 26 pages, J Dyn Diff Equat (2017

Mathematical general relativity: a sampler

We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.Comment: Some updates. A shortened version, to appear in the Bulletin of the AMS, is available online at http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2010-01304-