Luisa Arlotti

GENERALIZED KINETIC (BOLTZMANN) MODELS: MATHEMATICAL STRUCTURES AND APPLICATIONS

This paper deals with the development of suitable general mathematical structures including a large variety of Boltzmann type models. The contents are organized in three parts. The first part is devoted to modeling the above general framework. The second part to the development of specific models of interest in applied sciences. The third part develops a critical analysis towards research perspectives both on modeling and analytic problems.

Kinetic equations modelling population dynamics

Abstract This paper deals with the analysis of a class of models of population dynamics with competition and kinetic interactions. Several models are proposed to describe the dynamics of large populations of individuals undergoing kinetic (stochastic) interactions which modify the states of the interacting pair. Models are characterized by time and space structure, and are motivated by recent research activity in mathematical immunology. The evolution equations are stated in terms of nonlinear integrodifferential equations which are similar to the Boltzmann equation. This paper deals with modelling and qualitative analysis of the related Cauchy problem.

Solution of a New Class of Nonlinear Kinetic Models of Population Dynamics

Abstract This paper refers to the mathematical analysis of a new class of kinetic type models applicable to population dynamics. Modelling of the evolution equations with multiple interactions and qualitative analysis of the solutions to the Cauchy problem are dealt with.

A Kinetic Model of Tumor/Immune System Cellular Interactions

In this paper, a model of cellular tumor dynamics in competition with the immune system is proposed. The characteristic scale of the phenomenon is the cellular one and the model is developed with probabilistic methods analogous to those of the kinetic theory. The interacting individuals are the cells of the populations involved in the competition between the tumor and the immune system. Interactions can change the activation state of the tumor and cause tumor proliferation or destruction. The model is expressed in terms of a bi-linear system of integro-differential equations. Some preliminary mathematical analysis of the model as well as computational simulations are presented.

From the Jager and Segel model to kinetic population dynamics nonlinear evolution problems and applications

This paper deals with the analysis of a new class of models of population dynamics with competition and kinetic interactions. The content is organized in three parts. The first one refers to modelling in the framework of the so-called generalized Boltzmann models. The second part deals with qualitative analysis of the initial and initial boundary value problems. The third part of the paper provides a survey of applications and develops an analysis of some open problems.

On a class of integro-differential equations modeling complex systems with nonlinear interactions

Abstract This work deals with the qualitative analysis of the initial value problem for a class of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. The contents are specifically focused on the case where the system interacts with the outer environment and the entities are subject to nonlinearly additive interactions.

Population dynamics with stochastic interaction

Abstract This paper deals with the mathematical modelling of population dynamics with stochastic individual interaction. The model provides a general framework for the prediction of general physical behaviours: from social behaviours to the dynamics of immunological defences. The qualitative analysis is developed in the last part of the paper.

Generalized kinetic models in applied sciences : lecture notes on mathematical problem

From the Boltzmann Equation to the Averaged Boltzmann Equation - On the Cauchy Problem for the Averaged Boltzmann Equation - Asymptotic Theory for the Averaged Boltzmann Equation - Kinetic (Boltzmann) Models: Modeling and Analytic Problems - Critical Analysis and Research Perspectives

Qualitative analysis of a nonlinear integrodifferential equation modeling tumor-host dynamics

This paper deals with the qualitative analysis of the behavior of a kinetic model, proposed in [1], of the interactions among tumor, host environment, and immune system. It is shown that for a particular choice of the parameters of the model, the basic information is contained in the corresponding macroscopic model. The analysis is first developed for the general model. Then, two simplified models are studied in details. The first model deals with the tumor growth generated by the interactions between the tumor cells and those of a carcenogenic environment. The second one also includes interactions between pairs of tumor cells. In both cases, conditions for blow-up/decay of the tumor are described.

On Perturbed Substochastic Semigroups in Abstract State Spaces

The object of this paper is twofold: In the first part, we unify and extend the recent developments on honesty theory of perturbed substochastic semigroups (on