Jacek Banasiak

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.

Singularly perturbed telegraph equations with applications in the random walk theory

In the paper we analyze singularly perturbed telegraph systems applying the newly developed compressed asymptotic method and show that the diffusion equation is an asymptotic limit of singularly perturbed telegraph system of equations. The results are applied to the random walk theory for which the relationship between correlated and uncorrelated random walks is explained in asymptotic terms.

On the application of substochastic semigroup theory to fragmentation models with mass loss

A linear integro-differential equation modelling multiple fragmentation with inherent mass loss is investigated by means of substochastic semigroup theory. The existence of a semigroup is established and, under natural conditions on certain coefficients, the generator of this semigroup is identified. This yields, in particular, a validation of the formal mass-loss rate equation for the model.

Global strict solutions to continuous coagulation–fragmentation equations with strong fragmentation

In this paper we give an elementary proof of the unique, global-in-time solvability of the coagulation-(multiple) fragmentation equation with polynomially bounded fragmentation and particle production rates and a bounded coagulation rate. The proof relies on a new result concerning domain invariance for the fragmentation semigroup which is based on a simple monotonicity argument.

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.

CONSERVATIVE AND SHATTERING SOLUTIONS FOR SOME CLASSES OF FRAGMENTATION MODELS

Equations describing processes of cluster fragmentation have received considerable attention in recent years due to their importance in modelling e.g. the polymer degradation, droplet break-up, or rock crushing. In this paper we shall consider two mathematically most interesting features of them. One of these is the so-called shattering fragmentation, that is, the decrease of the total mass of the system that is formally conservative. The other is the existence of multiple solutions which indicates that in some cases the equations do not give the full picture of the dynamics of the model. Shattering fragmentation, that is considered to be an analogous but opposite process to a better understood gelation in coagulation models, has been investigated in several papers whose interest was, however, restricted to coefficients of a particular form that allowed either explicit solutions or yielded to a probabilistic approach. There have been few attempts to provide a deeper analysis of the existence of multiple solutions; most authors confined themselves to noticing them. In the present paper we present a systematic approach to both problems using the semigroup theory, and link them to the characterization of the generator of the solution semigroup of the fragmentation equation. As a result we provide criteria for both existence and absence of shattering fragmentation as well as of multiple solutions for large classes of coefficients that include those considered in all earlier works.

Inelastic scattering models in transport theory and their small mean free path analysis

In this paper we perform an asymptotic analysis of a singularly perturbed linear Boltzmann equation with inelastic scattering operator in the Lorentz gas limit, when the parameter corresponding to the mean free path of particles is small. The physical model allows for two-level field particles (ground state and excited state), so that scattering test particles trigger either excitation or de-excitation processes, with corresponding loss or gain of kinetic energy. After examining the main properties of the collision mechanism, the compressed Chapman–Enskog expansion procedure is applied to find the asymptotic equation when the collisions are dominant. A peculiarity of this inelastic process is that the collision operator has an infinite dimensional null-space. On the hydrodynamic level this is reflected in the small mean free path approximation being rather a family of diffusion equations than a single equation, as is the case in classical transport theory. Also the appropriate hydrodynamic quantity turns out to be different from the standard macroscopic density. Copyright

Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems

One of the aims of systems biology is to build multiple lay- ered and multiple scale models of living systems which can efficiently describe phenomena occurring at various level of resolution. Such mod- els should consist of layers of various microsystems interconnected by a network of pathways, to form a macrosystem in a consistent way; that is, the observable characteristics of the macrosystem should be, at least asymptotically, derivable by aggregation of the appropriate features of the microsystems forming it, and from the properties of the network. In this paper we consider a general macromodel describing a population consisting of several interacting with each other subgroups, with the rules of interactions given by a system of ordinary differential equations, and we construct two different micromodels whose aggregated dynamics is approximately the same as that of the original macromodel. The mi- cromodels offer a more detailed description of the original macromodels dynamics by considering an internal structure of each subgroup. Here, each subgroup is represented by an edge of a graph with diffusion or transport occurring along it, while the interactions between the edges are described by interface conditions at the nodes joining them. We prove that with an appropriate scaling of such models, roughly speak- ing, with fast diffusion or transport combined with slow exchange at the nodes, the solutions of the micromodels are close to the solution to the macromodel.

ASYMPTOTIC ANALYSIS FOR A PARTICLE TRANSPORT EQUATION WITH INELASTIC SCATTERING IN EXTENDED KINETIC THEORY

In this paper we perform the asymptotic analysis for a linear transport equation for test particles in an absorbing and inelastically scattering background, when the excited species can be considered as non-participating. This model is derived in the frame of extended kinetic theory and rescaled with the Knudsen number ∊. After examining the main properties of the collision model and of the scattering operator in the case with an infinite interval of energy as well as the case with a finite interval, the modified (compressed) Chapman–Enskog expansion procedure is applied to find the asymptotic equation for small mean free path. A specific feature of this model is that the collision operator has an infinite-dimensional null-space. The main result is that in the small mean free path approximation on level we obtain a free molecular flow for a suitable hydrodynamic quantity, rather than the diffusion which is typical for linear transport problems.

MATHEMATICAL PROPERTIES OF INELASTIC SCATTERING MODELS IN LINEAR KINETIC THEORY

In this paper we analyse properties of collision operators which occur in linear Boltzmann–Maxwell models with inelastic scattering. In particular, we prove the solvability of the Cauchy problem for such models. These results form a basis for further applications of these models and, in particular, for their asymptotic analysis and the derivation of the drift–diffusion approximation.