Ryszard Rudnicki

Long-time behaviour of a stochastic prey-predator model

We consider a system of stochastic equations which models the population dynamics of a prey-predator type. We show that the distributions of the solutions of this system are absolutely continuous. We analyse long-time behaviour of densities of the distributions of the solutions. We prove that the densities can converge in L1 to an invariant density or can converge weakly to a singular measure.

Global stability in a delayed partial differential equation describing cellular replication

Here we consider the dynamics of a population of cells that are capable of simultaneous proliferation and maturation. The equations describing the cellular population numbers are first order partial differential equations (transport equations) in which there is an explicit temporal retardation as well as a nonlocal dependence in the maturation variable due to cell replication. The behavior of this system may be considered along the characteristics, and a global stability condition is proved.

Markov Semigroups and Their Applications

Some recent results concerning asymptotic properties of Markov operators and semigroups are presented. Applications to diffusion processes and to randomly perturbed dynamical systems are given.

Model of phenotypic evolution in hermaphroditic populations

We consider an individual based model of phenotypic evolution in hermaphroditic populations which includes random and assortative mating of individuals. By increasing the number of individuals to infinity we obtain a nonlinear transport equation, which describes the evolution of phenotypic distribution. The main result of the paper is a theorem on asymptotic stability of trait distribution. This theorem is applied to models with the offspring trait distribution given by additive and multiplicative random perturbations of the parental mean trait.

A DISCRETE MODEL OF EVOLUTION OF SMALL PARALOG FAMILIES

We introduce and analyze a simple probabilistic model of genome evolution. It is based on three fundamental evolutionary events: gene loss, duplication and accumulated change. We are mainly interested in asymptotic size distribution of small paralogous gene families in a genome. This is motivated by previous works which consisted in fitting the available genomic data into, what is called, paralog distributions. This formalism is described as a discrete-time Markov chain. The formulas for equilibrium paralog family sizes are derived. Moreover, we show that when probabilities of gene removal and duplication are small and close to each other, then the resulting distribution is close to logarithmic distribution. Some empirical results for microbial genomes are presented.

SIZE DISTRIBUTION OF GENE FAMILIES IN A GENOME

We consider a probabilistic model of genome evolution. We are interested in size distribution of gene families. The model is based on three fundamental evolutionary events: gene loss, duplication and accumulated change. We assume that the probability of gene loss and duplication is constant and the probability of gene mutation mi depends on the size i of a family. We prove that size distribution of paralogous gene families in a genome converges to the equilibrium as time goes to infinity. Moreover, we show how this equilibrium depends on the sequence (mi). Theoretical results are compared with the available genomic data.

MARKOV SEMIGROUPS AND STABILITY OF THE CELL MATURITY DISTRIBUTION

A model of the maturity-structured cell population is considered. This model is described by a partial differential equation with a transformed argument. Using the theory of Markov semigroups we establish a new criterion for asymptotic stability of such equations.

A Case Study of Genome Evolution: From Continuous to Discrete Time Model

We introduce and analyse a simple model of genome evolution. It is based on two fundamental evolutionary events: gene loss and gene duplication. We are mainly interested in asymptotic distributions of gene families in a genome. This is motovated by previous work which consisted in fitting the available genomic data into, what is called paralog distributions. Two approaches are presented in this paper: continuous and discrete time models. A comparison of them is presented too – the asymptotic distribution for the continuous time model can be seen as a limit of the discrete time distributions, when probabilities of gene loss and gene duplication tend to zero. We view this paper as an intermediate step towards mathematically settling the problem of characterizing the shape of paralog distribution in bacterial genomes.

GLOBAL SOLVABILITY OF A FRAGMENTATION-COAGULATION EQUATION WITH GROWTH AND RESTRICTED COAGULATION

We consider a fragmentation-coagulation equation with growth, where the nonlinear coagulation term, introduced in O. Arino and R. Rudnicki [2], is designed to model processes in which only a part of particles in the aggregates is capable of coalescence. We introduce various growth models, describing both biological and inorganic processes, and discuss their effect on the generation of the linear growth-fragmentation semigroup. Once its existence has been established, the solution of the full nonlinear problem follows by showing that the coagulation term is globally Lipschitz.

Piecewise deterministic Markov processes in biological models

We present a short introduction into the framework of piecewise deterministic Markov processes. We illustrate the abstract mathematical setting with a series of examples related to dispersal of biological systems, cell cycle models, gene expression, physiologically structured populations, as well as neural activity. General results concerning asymptotic properties of stochastic semigroups induced by such Markov processes are applied to specific examples.