A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence

Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form $\beta I_p S_q$ for the case p<1 are constructed. Global stability of the models is thereby established

Global asymptotic properties for a Leslie-Gower food chain model

We study global asymptotic properties of a continuous time Leslie-Gower food chain model. We construct a Lyapunov function which enables us to establish global asymptotic stability of the unique coexisting equilibrium state.Comment: 5 Pages, 1 figure. Keywords: Leslie-Gower model, Lyapunov function, global stabilit

Long-term coexistence for a competitive system of spatially varying gradient reaction-diffusion equations

Spatial distribution of interacting chemical or biological species is usually described by a system of reaction-diffusion equations. In this work we consider a system of two reaction diffusion equations with spatially varying diffusion coefficients which are different for different species and with forcing terms which are the gradient of a spatially varying potential. Such a system describes two competing biological species. We are interested in the possibility of long-term coexistence of the species in a bounded domain. Such long-term coexistence may be associated either with a periodic in time solution (usually associated with a Hopf bifurcation), or with time-independent solutions. We prove that no periodic solution exists for the system. We also consider some steady-states (the time-independent solutions) and examine their stability and bifurcations

Cluster formation for multi-strain infections with cross-immunity

Many infectious diseases exist in several pathogenic variants, or strains, which interact via cross-immunity. It is observed that strains tend to self-organise into groups, or clusters. The aim of this paper is to investigate cluster formation. Computations demonstrate that clustering is independent of the model used, and is an intrinsic feature of the strain system itself. We observe that an ordered strain system, if it is sufficiently complex, admits several cluster structures of different types. Appearance of a particular cluster structure depends on levels of cross-immunity and, in some cases, on initial conditions. Clusters, once formed, are stable, and behave remarkably regularly (in contrast to the generally chaotic behaviour of the strains themselves). In general, clustering is a type of self-organisation having many features in common with pattern formation

Spherical thermal waves in laser plasmas

The thermal wave produced in a uniform plasma, when energy is absorbed on a spherical surface such that convection is negligible, is analyzed using an integral method, which is very accurate. The curvature speeds up (slightly slows down) the inner (outer) wave front and does not affect the temperature maximum