KRM
A general consistent BGK model for gas mixtures
Alexander V. Bobylev Marzia Bisi Maria Groppi Giampiero Spiga Irina F. Potapenko
Kinetic & Related Models 2018, 11(6): 1377-1393 doi: 10.3934/krm.2018054

We propose a kinetic model of BGK type for a gas mixture of an arbitrary number of species with arbitrary collision law. The model features the same structure of the corresponding Boltzmann equations and fulfils all consistency requirements concerning conservation laws, equilibria, and H-theorem. Comparison is made to existing BGK models for mixtures, and the achieved improvements are commented on. Finally, possible application to the case of Coulomb interaction is briefly discussed.

keywords: Kinetic theory Boltzmann equation BGK models, gas mixtures gas mixtures exchange rates
KRM
Kinetic approach to deflagration processes in a recombination reaction
Fiammetta Conforto Maria Groppi Roberto Monaco Giampiero Spiga
Kinetic & Related Models 2011, 4(1): 259-276 doi: 10.3934/krm.2011.4.259
Steady one-dimensional flame structure is investigated in a binary gas mixture made up by diatomic molecules and atoms, which undergo an irreversible exothermic two--steps reaction, a recombination process followed by inelastic scattering (de-excitation). A kinetic model at the Boltzmann level, accounting for chemical encounters as well as for mechanical collisions, is proposed and its main features are analyzed. In the case of collision dominated regime with slow recombination and fast de-excitation, the model is the starting point for a consistent derivation, via suitable asymptotic expansion of Chapman-Enskog type, of reactive fluid-dynamic Navier-Stokes equations. The resulting set of ordinary differential equations for the smooth steady deflagration profile is investigated in the frame of the qualitative theory of dynamical systems, and numerical results for the flame eigenvalue and for the main macroscopic observables are presented and briefly commented on for illustrative purposes.
keywords: Kinetic theory; Irreversible chemical reactions; Deflagration waves.
MBE
Effects of predation efficiencies on the dynamics of a tritrophic food chain
Maria Paola Cassinari Maria Groppi Claudio Tebaldi
Mathematical Biosciences & Engineering 2007, 4(3): 431-456 doi: 10.3934/mbe.2007.4.431
In this paper the dynamics of a tritrophic food chain (resource, consumer, top predator) is investigated, with particular attention not only to equilibrium states but also to cyclic behaviours that the system may exhibit. The analysis is performed in terms of two bifurcation parameters, denoted by $p$ and $q$, which measure the efficiencies of the interaction processes. The persistence of the system is discussed, characterizing in the $(p,q)$ plane the regions of existence and stability of biologically significant steady states and those of existence of limit cycles. The bifurcations occurring are discussed, and their implications with reference to biological control problems are considered. Examples of the rich dynamics exhibited by the model, including a chaotic regime, are described.
keywords: stability tritrophic system bifurcations limit cycles.
KRM
Flame structure from a kinetic model for chemical reactions
Marzia Bisi Maria Groppi Giampiero Spiga
Kinetic & Related Models 2010, 3(1): 17-34 doi: 10.3934/krm.2010.3.17
Steady one-dimensional flame structure is investigated in a binary mixture made up by two components of the same chemical species undergoing binary irreversible exothermic reactive encounters. A kinetic model at the Boltzmann level, accounting for chemical transitions as well as for mechanical collisions, is proposed and its main features are analyzed. In the case of slow chemical reactions and collision dominated regime, the model is the starting point for a consistent derivation, via suitable asymptotic expansion of Chapman-Enskog type, of reactive Navier-Stokes equations at the fluid-dynamic scale. The resulting set of ordinary differential equations is investigated in the frame of the qualitative theory of dynamical systems, and numerical results are presented and briefly commented on for illustrative purposes.
keywords: Kinetic theory; Irreversible chemical reactions; Deflagration waves.
KRM
Qualitative analysis of the generalized Burnett equations and applications to half--space problems
Marzia Bisi Maria Paola Cassinari Maria Groppi
Kinetic & Related Models 2008, 1(2): 295-312 doi: 10.3934/krm.2008.1.295
The Generalized Burnett Equations, very recently introduced by Bobylev [3,4], are tested versus Fluid--Dynamic applications, considering the classical steady evaporation/condensation problem. By means of the methods of the qualitative theory of dynamical systems, comparison is made to other kinetic and hydrodynamic models, and indications on an appropriate choice of the disposable parameters are obtained.
keywords: Burnett equations; Hydrodynamics; Half-space Problems.
DCDS-B
Qualitative analysis of kinetic-based models for tumor-immune system interaction
Martina Conte Maria Groppi Giampiero Spiga
Discrete & Continuous Dynamical Systems - B 2018, 23(6): 2393-2414 doi: 10.3934/dcdsb.2018060

A mathematical model, based on a mesoscopic approach, describing the competition between tumor cells and immune system in terms of kinetic integro-differential equations is presented. Four interacting components are considered, representing, respectively, tumors cells, cells of the host environment, cells of the immune system, and interleukins, which are capable to modify the tumor-immune system interaction and to contribute to destroy tumor cells. The internal state variable (activity) measures the capability of a cell of prevailing in a binary interaction. Under suitable assumptions, a closed set of autonomous ordinary differential equations is then derived by a moment procedure and two three-dimensional reduced systems are obtained in some partial quasi-steady state approximations. Their qualitative analysis is finally performed, with particular attention to equilibria and their stability, bifurcations, and their meaning. Results are obtained on asymptotically autonomous dynamical systems, and also on the occurrence of a particular backward bifurcation.

keywords: Kinetic models Kinetic models macroscopic closures asymptotically autonomous systems stability forward and backward bifurcations

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