Sander C. Hille

Ergodic decompositions associated with regular Markov operators on Polish spaces

For any regular Markov operator on the space of nite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrisation of the ergodic probability measures associated to this operator in terms of particular subsets of the state space. We use this parametrisation to prove an integral decomposition of every invariant probability measure in terms of the ergodic probability measures and give an ergodic decomposition of the state space. This extends results by Yosida (Functional analysis, Chapter XIII.4), Hern andez-Lerma and Lasserre (Acta Appl. Math. 54, 99{119) and Zaharopol (Acta. Appl. Math. 104, 47{81), who considered the setting of locally compact separable metric spaces.

Maximal Degenerate Representations, Berezin Kernels and Canonical Representations

In this work we present a new context of the canonical representations which have been introduced by Berezin, Gel’fand, Graev and Vershik for simple Lie groups G of Hermitian type. We discuss maximal-degenerate representations of the complexification of G and the decomposition of the canonical representations into irreducible parts.

Modelling the dynamics of polar auxin transport in inflorescence stems of Arabidopsis thaliana

Highlight An experimental and mathematical approach to polar auxin transport results in a model based on an extended general advection–diffusion equation including auxin immobilization and surrounding tissue exchange that accounts for crucial observations.

Modeling biological gradient formation: combining partial differential equations and Petri nets

Both Petri nets and differential equations are important modeling tools for biological processes. In this paper we demonstrate how these two modeling techniques can be combined to describe biological gradient formation. Parameters derived from partial differential equation describing the process of gradient formation are incorporated in an abstract Petri net model. The quantitative aspects of the resulting model are validated through a case study of gradient formation in the fruit fly.

Limit theorems for some Markov chains

Abstract The exponential rate of convergence and the Central Limit Theorem for some Markov operators are established. These operators were efficiently used in some biological models (see Hille, Horbacz & Szarek (2015) [8] ), which generalize the cell cycle model given by Lasota & Mackey (1999) [12] .

Law of the iterated logarithm for some Markov operators

The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by A. Lasota and M.C. Mackey, J. Math. Biol. (1999).

Modelling with measures : approximation of a mass-emitting object by a point source

We consider a linear diffusion equation on Ω: = R(2) \ Ω[Symbol: see text], where Ω[Symbol: see text] is a bounded domain. The time-dependent flux on the boundary Γ: = ∂ Ω[Symbol: see text] is prescribed. The aim of the paper is to approximate the dynamics by the solution of the diffusion equation on the whole of R(2) with a measure-valued point source in the origin and provide estimates for the quality of approximation. For all time t, we derive an L(2)([0,t];L2(Γ))-bound on the difference in flux on the boundary. Moreover, we derive for all t > 0 an L(2)(Ω)-bound and an L2([0,t];H(1)(Ω))-bound for the difference of the solutions to the two models.

REVIEW OF MATHEMATICAL MODELLING IN THE STUDY OF COLLECTIVE MOVEMENT OF DICTYOSTELIUM DISCOIDEUM

We give a quick review of mathematical models used to elucidate the biological mechanisms behind the collective behaviour of the amoeba Dictyostelium discoideum. Moreover, we identify still-open questions of biophysical nature for which mathematical modelling, analysis and simulation may help in their answering. 1. The purpose of modelling: biological questions

Modeling Oxygen Consumption in Germinating Seeds

The consumption of oxygen by a germinating seed is assumed to be a good indicator of seed vitality and can potentially be used to predict the germination time. With the current availability of relatively simple single-seed respiration measurement methods and more oxygen consumption data opportunities emerge for detailed analysis of the underlying mechanisms relating respiration to germination processes. Due to the complex (structural and physiological) nature of seeds experimental analysis alone is very difficult. Mathematical modeling may provide an insight into the relationship between the germination of seeds and respiration. We have approached this problem by considering the population dynamics of mitochondria in seeds subject to limited oxygen supply and present a simple but rigorous and easily testable mathematical model that can handle large amounts of data and is interpretable in terms of the effective biological parameters of the seeds.

ON METRIZATION OF UNIONS OF FUNCTION SPACES ON DIFFERENT INTERVALS

Report MI-2010-16 Abstract. Let (S;ds) be a metric space. We define a metric on the space C s which consists of the disjoint union of continuous functions defined on dierent intervals (0; ), > 0, with values in the metric space S. From this, we define another metric space, C N s , where elements are sequences of functions from the space C s . We study the traditional properties of these metric spaces like separability and completeness. The abstract mathematical problem that we discuss here is motivated by the study of a stochastic model in population dynamics. That is, we consider a deterministic system, that is subject to interventions at discrete points in time randomly, in which the deterministic system jumps to a new position, also randomly. However, we feel that the results discussed here are of separate, mathematical interest.