Giuseppe Mulone

A note on heroin epidemics

We show that the steady states of the White and Comiskey [E. White, C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci. 208 (2007) 312-324.] model of heroin epidemics are stable.

On the nonlinear stability of the rotating Bénard problem via the Lyapunov direct method

Abstract In this paper we use the Lyapunov direct method to study the nonlinear conditional stability of the Benard problem with rotation. In particular, for Prandtl numbers greater than or equal to one, and for Taylor numbers less than or equal to 80π4, we prove the coincidence between the linear and nonlinear critical stability parameters. We also give some values of the attracting radius for the conditionally stable disturbances of the basic motion.

MODELING BINGE DRINKING

We develop a two-stage (four component) model for youths with serious drinking problems and their treatment. The youths with alcohol problems are split into two classes, namely those who admit to having a problem and those who do not. It is shown that the model possesses two steady states, one where people have no alcohol problems and one where there is an endemic state involving those with an alcohol problem. The stability of these states is analyzed and a threshold established such that each state will be stable depending on whether the incidence rate is above or below the threshold. The model is analyzed in the context of actual data.

Stability of Epidemic Models with Evolution

Epidemic models are considered where the diffusion of individuals is influenced by intraspecific competition pressures and are weakly affected by different classes. The nonlinear stability of some of the models is proved by introducing new Liapunov functions.

Global stability in the Bénard problem for a mixture with superimposed plane parallel shear flows

The Lyapunov direct method is used to study the non-linear stability of parallel convective shear flows of a mixture heated and salted from below for any Schmidt and Prandtl numbers. Global non-linear exponential stability for small values of Reynolds number R is found and conditional stability results up to the criticality which are independent of R are given for rigid and stress-free boundaries. Copyright

Nonlinear Stability for Diffusion Models in Biology

The reduction method for studying optimal nonlinear stability of constant solutions to some ecological models with diffusion, which include the Cantrell–Cosner and the May–Leonard systems, is given. A new canonical energy (Lyapunov function) is introduced, and it is proved that the regions of linear and nonlinear stability coincide with a known radius of attraction for the initial data. Attention is focused on a May–Leonard system with circular symmetry, an asymmetric May–Leonard system with diffusion, and a system for aggregation of glia in the brain.

Bidispersive-inclined convection

A model is presented for thermal convection in an inclined layer of porous material when the medium has a bidispersive structure. Thus, there are the usual macropores which are full of a fluid, but there are also a system of micropores full of the same fluid. The model we employ is a modification of the one proposed by Nield & Kuznetsov (2006 Int. J. Heat Mass Transf. 49, 3068–3074. (doi:10.1016/j.ijheatmasstransfer.2006.02.008)), although we consider a single temperature field only.

Laminar hydromagnetic flows in an inclined heated layer

In this paper we investigate, analytically, stationary laminar flow solutions of an inclined layer filled with a hydromagnetic fluid heated from below and subject to the gravity field. In particular we describe in a systematic way the many basic solutions associated to the system. This extensive work is the basis to linear instability and nonlinear stability analysis of such motions.

Global analysis of a stage-structured model with population diffusion

The population diffusions are incorporated into the prey–predator model with the stage structure for predators. The threshold is established above which the predators are permanent and below which the predators become extinct. The necessary and sufficient conditions for the local stability of the positive steady state are obtained by analysing eigenvalues. With the Lyapunov direct method, explicit conditions for the global stability of the positive equilibrium are derived.

Some results in the nonlinear stability for rotating Bénard problem with rigid boundary condition

The scope of this article is to expose the stabilizing properties of rotation and solute gradient for the Benard problem with (at least one-sided) rigid boundary conditions. We perform a linear investigation of the critical threshold for the rotating Benard problem with a binary fluid, and we also make an investigation with a Lyapunov function for the particular problem of a rotating single fluid. In all the these cases an increase of the Taylor number has stabilizing effects.