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Thermal convection in a fluid saturating an anisotropic porous medium in local thermal nonequilibrium (LTNE) is investigated, with specific attention to the effect of variable viscosity on the onset of convection. Many fluids show a remarkable dependence of viscosity on temperature that cannot be neglected. For this reason, we take into account a fluid whose viscosity decreases exponentially with depth, according to Straughan (Acta Mech. 61:59–72, 1986), Torrance and Turcotte (J. Fluid Mech. 47(1):113–125, 1971). The novelty of this paper is to highlight how variable viscosity coupled with the LTNE assumption affects the onset of convection. A numerical procedure shows the destabilising effect of depth-dependent viscosity. Moreover, it comes out that the LTNE hypothesis makes the influence of viscosity more intense. Linear instability analysis of the conduction solution is carried out by means of the Chebyshev-tau method coupled to the QZ algorithm, which provides the critical Rayleigh number for the onset of convection in a straightforward way. The energy method is employed in order to study the nonlinear stability. The optimal result of coincidence between the linear instability threshold and the global nonlinear stability threshold is obtained. The influence of anisotropic permeability and conductivity, weighted conductivity ratio, and interaction coefficient on the onset of convection is highlighted.

The aim of this paper is to analyze the effect of a downward vertical net mass flow on the type of instability that occurs in a horizontal fluid-saturated porous layer that is heated from below. The strength of the downflow is modelled by [Formula: see text], the Péclet number. First, we prove the validity of the principle of exchange of stabilities; consequently, we perform a linear instability analysis of the basic steady flow to determine the critical Darcy–Rayleigh number for the onset of steady convective instability as a function of [Formula: see text]. Then, a weakly nonlinear stability analysis is performed to determine the smallest value of the Péclet number for which the onset of convection corresponds to a subcritical instability.

Thermal convection of a fluid filling an anisotropic porous medium, uniformly rotating about a vertical axis, with local thermal non equilibrium, is studied. The linear and nonlinear stability analysis are performed. In particular the coincidence between linear instability and nonlinear (global) stability thresholds is proved.

A reaction–diffusion ecoepidemic model of predator–prey type with a transmissible disease spreading among the predator species only is considered. The longtime behavior of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. Conditions guaranteeing the non existence of non-constant equilibria have been found. Linear and non-linear stability conditions for biologically meaningful equilibria are determined.

The stabilizing effect of Brinkmann viscosity (BV) in MHD convection in a horizontal porous layer L – filled by an electrically conducting fluid, heated from below and imbedded in a transverse magnetic field – is analyzed. The critical Rayleigh number of linear stability is found and – in closed forms – conditions for the onset of steady or oscillatory convection are obtained. Via the linearization principle given in [16] it is shown that unconditional nonlinear stability of thermal magnetic conduction solution is guaranteed by linear stability. The long-time behavior is characterized via the existence of L 2 -absorbing sets.

A reaction-diffusion system modeling cholera epidemic in a non-homogeneously mixed population is introduced. The interaction between population and toxigenic Vibrio cholerae concentration in contaminated water has been taken into account. The existence of biologically meaningful equilibria is investigated together with their linear and nonlinear stability. Using the data collected during the Haiti cholera epidemic, a numerical simulation is performed.

A reaction-diffusion system modeling the predation between two species is analyzed in the case in which the predators have to search, share and compete for food. The boundedness and uniqueness of the solutions is proved and conditions guaranteeing the global nonlinear asymptotic stability of the positive equilibrium point have been found. These conditions improve those ones present in the existing literature.

The instability of a vertical fluid motion (throughflow) in a binary mixture saturating a horizontal porous layer, uniformly heated from below, uniformly salted from below by one salt and permeated by an imposed uniform magnetic field H , normal to the layer, is analyzed. By employing the order-1 Galerkin weighted residuals method, the critical Rayleigh numbers for the onset of steady or oscillatory instability, have been determined.

Via the auxiliary system method (Rionero, 2012 [35] and Rionero, 2013 [36] , [37] ) the onset of convection in rotating porous layers in the presence of inertia is investigated. The effects of rotation and inertia are respectively measured through the Taylor number T and Vadasz number Va ( Section 2 ). For the tridimensional perturbations and the full non-linear problem, it is shown that: (a) there exists a critical Taylor number T c ≈ 1.53 such that for T ≤ T c the inertia has no effect on the onset of convection; (b) for T > T c there exists an associate critical Vadasz number V a ( c ) ( T ) ( > 0 ) such that, only for V a V a ( c ) ( T ) , the inertia has effect on the onset of convection, and only in this case the convection arises via an oscillatory motion (cf. Theorems 5.2 and 5.3); (c) subcritical instabilities do not exist; (d) the global non-linear stability is guaranteed by the linear stability; (e) also in the case { T > T c , V a V a ( c ) } the critical Rayleigh number can be given in closed form.

This paper deals with a reaction-diffusion SEIR model for infections. The longtime behaviour of the solutions is analyzed and, in particular, absorbing sets in the phase space are determined. By using a peculiar Lyapunov function, the nonlinear asymptotic stability of endemic equilibrium is investigated.

The paper is devoted to an extended Lotka–Volterra system of differential equations of predator–prey model. The extension is proposed with perturbation terms, which are null for the positive equilibrium state. In the original Lotka–Volterra system, the equilibrium state is not asymptotically stable due to the fact that perturbations are periodic in time. The aim of the paper is to characterize a form of perturbation terms guaranteeing the asymptotic stability or instability of equilibrium state. The reason of the proposed model is that for large time scale, the Lotka–Volterra model is too simple to be realistic. In the paper, the non-autonomous perturbations do not change the equilibrium state but introduce functions of time as well as for additional perturbed terms as for the main part of the equations modified from Lotka–Volterra model. Theorems are proposed in a renormalized form of the differential equations for time and the two variables. The key point of the paper comes from the use of a Liapunov function introduced in Section 2 which allows to obtain conditions for the asymptotic stability (Section 3) and instability (Section 4) by using a Cetaiev instability theorem following conditions on the renormalized coefficients in time of System (6). An appendix recalls the main results of the Liapunov Direct Method for non-autonomous binary systems of ordinary differential equations.

Penetrative convection in an anisotropic porous medium is simulated via internal heating. We consider both internal heat sources and sinks, where the porous medium has constant anisotropic thermal diffusivity and non-homogeneous permeability. Both linear instability and global nonlinear energy stability analyses are performed. By adopting a differential constraint approach, the nonlinear stability boundaries are shown to be sharp when compared with the linear instability thresholds. In contrast, it is shown that adopting a standard energy analysis does not yield sharp results.

Linear and nonlinear stability analises of vertical throughflow in a fluid layer, where the density is quadratic in temperature, are studied. To avoid the loss of key terms a weighted functional is used in the energy analysis. Both conditional and unconditional thresholds are derived. When the throughflow is ascending the linear and nonlinear boundaries show substantially agreement. The linear boundary remains close to the conditional nonlinear bioundary for descending throughflow, whilst the unconditional threshold begins to diverge.

The nonlinear global exponential pointwise stability of a vertical steady flow driven by a horizontal periodic temperature gradient in a porous layer is performed. It is shown that the stability threshold depends on the supremum of a quadratic functional, having non constant coefficients, and new in the literature on the convection problem. In solving the variational problem, a suitable functional transformation is used.

The long-time behaviour of the solutions of the Darcy–Oberbeck– Boussinesq system modeling fluid motion in horizontal porous layers, is investigated. The layer is supposed to be uniformly heated and salted from below, rotating around the vertical axis, showing large pores. Necessary and sufficient conditions guaranteeing the stability of a vertical constant throughflow are obtained. The non-linear, global, asymptotic \(L^2-\)stability of the throughflow solution, is investigated.

A nonlinear energy stability analysis of the onset of convection for fluids with viscosity convex nonincreasing function of temperature, is performed. It is shown that condition assuring linear stability, assures nonlinear (conditional) asymptotic stability too.

Convection problem in anisotropic and inhomogeneous porous media has been analysed. In particular, the effect of variable permeability and thermal diffusivity with respect to the vertical direction, has been studied. Linear and nonlinear stability analysis of the conduction solution have been performed.

The onset of penetrative convection in an anisotropic porous medium, for fluids with quadratic density law, is considered. In particular the effect of anisotropic permeability and thermal diffusivity, have been taken into account. The critical curves for the linear and nonlinear theories, global and local, are presented and compared.

The penetrative convection in a horizontal binary fluid mixture layer with an internal heating due to a quadratic concentration source, according to the Darcy-Oberbeque-Boussinesq model, is considered. Conditions guaranteeing the onset of penetrative convection are obtained.