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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.

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.