Mirosław Lachowicz

Methods of Small Parameter in Mathematical Biology

1 Small parameter methods - basic ideas.- 2 Introduction to the Chapman-Enskog method - linear models with migrations.- 3 Tikhonov-Vasilyeva theory.- 4 The Tikhonov theorem in some models of mathematical biosciences.- 5 Asymptotic expansion method in a singularly perturbed McKendrick problem.- 6 Diffusion limit of the telegraph equation.- 7 Kinetic model of alignment.- 8 From microscopic to macroscopic descriptions.- 9 Conclusion.

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.

Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions

This paper deals with the modeling, qualitative and numerical analysis, of welfare dynamics in societies viewed as complex evolutive systems subject to different policies of wealth distribution. A nonlinear model of wealth distribution is presented. The state of a population is modeled by a probability distribution over wealth classes and the dynamic of interaction is parameterized by a threshold, whose dynamics depends on an internal competition related to the wealth distribution. Therefore, the model is a system of equations in which the threshold is one of the dynamic variables. The approach contains the whole path from modeling to simulations, through a qualitative analysis of the initial value problem.

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.

A mathematical model for single cell cancer-Immune system dynamics

In this paper, we propose and analyse the model of competition between a single cell cancer and the immune system. The model is a system of integro-differential bilinear equations and it describes both very early stage of a solid tumor and all stages of leukemias.

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.

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

On the oort-hulst-safronov coagulation equation and its relation to the Smoluchowski equation

A connection is established between the classical Smoluchowski continuous coagulation equation and the Oort--Hulst--Safronov coagulation equation via generalized coagulation equations. Existence of solutions to the Oort--Hulst--Safronov coagulation equation is shown, and the large time behavior and the occurrence of gelation are studied as well. It is also shown that a compactly supported initial distribution propagates with finite speed.

Chaos for a class of linear kinetic models

Abstract In recent years it was observed that chaotic behaviour can occur in some infinite–dimensional linear systems. An example of this type, related to a kinetic model (death process), has been previously reported. In this paper we generalize these earlier results to the case of variable coefficients, showing that the property of being chaotic can be in a certain sense stable. On the other hand the ‘opposite’ birth process cannot be chaotic.

Nonlinear kinetic equations with dissipative collisions

This paper deals with the analysis and the asymptotic theory towards hydrodynamics of some nonlinear kinetic equations for a disparate mixture of spherical particles undergoing dissipative binary collisions.