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In the present paper, double-diffusive convection, taking into account Coriolis effects, in a horizontal layer of Brinkman-anisotropic bi-disperse porous medium is analysed. Via linear instability analysis, we found that convection can set in through stationary or oscillatory motions and the critical Rayleigh numbers for the onset of stationary secondary flow (steady convection) and overstability (oscillatory convection) are determined.

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.

The onset of double-diffusive convection in horizontal porous layers for the thermo-diffusive Soret phenomenon in the case of a generalized Darcy model including inertia term is investigated. Via a linearization principle recently introduced in Rionero (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur 28:21–47, 2017) the coincidence between linear and nonlinear (global) stability thresholds of the thermo-solutal conduction solution, is proved. Necessary and sufficient conditions guaranteeing the onset of steady or oscillatory convection in a closed algebraic form are obtained.

The onset of thermal convection in an electrically conducting fluid saturating a porous medium, uniformly heated from below, salted by one chemical and embedded in an external transverse magnetic field is analyzed. The critical Rayleigh thermal numbers at which steady and Hopf convection can occur, are determined. Sufficient conditions guaranteeing the effective onset of convection via steady or oscillatory state are provided.

An intraguild predator–prey model with a carrying capacity proportional to the biotic resource, is generalized by introducing a Holling type II functional response. The longtime behavior of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. The existence of biologically meaningful equilibria (boundary and internal equilibria) has been investigated. Linear and nonlinear stability conditions for biologically meaningful equilibria are performed. Finally, numerical simulations on different regimes of coexistence and extinction of the involved populations have been shown.

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.

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.

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.

MHD convection in a horizontal porous-layer filled by a plasma, imbedded in a transverse magnetic field and heated from below, is investigated. The critical Rayleigh number is found and, in simple algebraic closed forms, conditions necessary and sufficient for the onset of steady and oscillatory convection are obtained. It is shown that the stabilizing effect of the magnetic field grows with $$Q^2$$ , Q being the Chandrasekhar number. The linearization principle (Rionero, Rend Lincei Mat Appl 25:368, 2014): “Decay of linear energy for any initial data implies decay of nonlinear energy at any instant” continues to hold also in the case at stake and allows to recover for the global nonlinear stability the conditions of linear stability.

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.

This paper deals with a reaction-diffusion SEIR model for infections under homogeneous Neumann boundary conditions. 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.

A system modeling fluid motions in horizontal porous layers, uniformly heated from below and salted from above by one salt, is analyzed. The definitely boundedness of solutions (existence of absorbing sets) is proved. Necessary and sufficient conditions ensuring the linear stability of a vertical constant throughflow have been obtained via a new approach. Moreover, conditions guaranteeing the global non-linear asymptotic stability are determined.

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.

Non-linear stability of vertical throughflows in porous layers, uniformly heated and salted from below, is analyzed. The definitively boundedness of the solutions (existence of absorbing sets in the phase space) is proved. Conditions guaranteeing global non-linear asymptotic stability have been found. In closed form, the critical Rayleigh number has been found.

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 long-time behavior of triply fluid mixtures saturating horizontal porous layers uniformly rotating around the vertical axis, according to the Brinkman law (holding for large pores), is investigated. The most destabilizing case of layers heated from below and salted from above by two salts, is considered. The ultimately boundedness of the solutions (existence of absorbing sets) is shown. A necessary and sufficient condition guaranteeing the global non-linear asymptotic L 2 -stability of the conduction solution is obtained.