Positive semigroups and perturbations of boundary conditions

We present a generation theorem for positive semigroups on an $L^1$ space. It provides sufficient conditions for the existence of positive and integrable solutions of initial-boundary value problems. An application to a two-phase cell cycle model is given

A model of seasonal savanna dynamics

We introduce a mathematical model of savanna vegetation dynamics. The usual approach of nonequilibrium ecology is extended by including the impact of wet and dry seasons. We present and rigorously analyze a model describing a mixed woodland-grassland ecosystem with stochastic environmental noise in the form of vegetation biomass losses manifesting fires. Both, the probability of ignition and the strength of these losses depend on the current season (as well as vegetation growth rates etc.). Formally it requires an introduction and analysis of a system that is a piecewise deterministic Markov process with parameters switching between given constant periods of time. We study the long time behavior of time averages for such processes.Comment: 22 pages, 3 figure

A model for random fire induced tree-grass coexistence in savannas

Tree-grass coexistence in savanna ecosystems depends strongly on environmental disturbances out of which crucial is fire. Most modeling attempts in the literature lack stochastic approach to fire occurrences which is essential to reflect their unpredictability. Existing models that actually include stochasticity of fire are usually analyzed only numerically. We introduce a new minimalistic model of treegrass coexistence where fires occur according to a stochastic process. We use the tools of the linear semigroup theory to provide a more careful mathematical analysis of the model. Essentially we show that there exists a unique stationary distribution of tree and grass biomasses

The Mathematical Legacy of Andrzej Lasota

Professor Andrzej Lasota (1932–2006) was a Polish mathematician with wide ranging interests in dynamical systems, probability theory and ergodic theory who saw the inter-relationships between all three and who successfully synthesized these apparently disparate fields. He used that synthesis to both further mathematical research as well as to investigate problems in biology. One of his over-riding interests was the way in which seemingly “random” or “probabilistic” processes (in a mathematical sense) could actually be thought of as equivalently coming from deterministic dynamics. How did we each come to know him and his work? Michael C. Mackey met Lasota in Cracow in 1977 through his collaborator Dr. MariaWazewska-Czyzewska, a hematologist and daughter of Professor Tadeusz Wazewski. That meeting blossomed into an almost 30-year-long friendship and collaboration in biomathematics. Marta Tyran-Kaminska met Lasota during her mathematical studies at the University of Silesia in Katowice in 1992 and did her Ph.D. under his supervision. Hans-Otto Walther met Lasota during a year at Michigan State University, 1979–1980, where Pavel Brunovský was also visiting, and they had all been brought together by Shui-Nee Chow

Stochastic semigroups and their applications to biological models

Some recent results concerning generation and asymptotic properties of stochastic semigroups are presented. The general results are applied to biological models described by piecewise deterministic Markov processes: birth-death processes, the evolution of the genome, genes expression and physiologically structured models

Convergence to L\'evy stable processes under some weak dependence conditions

For a strictly stationary sequence of random vectors in $\mathbb{R}^d$ we study convergence of partial sum processes to L\'evy stable process in the Skorohod space with $J_1$-topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing.Comment: Change of the title. Minor revision

Effects of Noise on Entropy Evolution

We study the convergence properties of the conditional (Kullback-Leibler) entropy in stochastic systems. We have proved very general results showing that asymptotic stability is a necessary and sufficient condition for the monotone convergence of the conditional entropy to its maximal value of zero. Additionally we have made specific calculations of the rate of convergence of this entropy to zero in a one-dimensional situations, illustrated by Ornstein-Uhlenbeck and Rayleigh processes, higher dimensional situations, and a two dimensional Ornstein-Uhlenbeck process with a stochastically perturbed harmonic oscillator and colored noise as examples. We also apply our general results to the problem of conditional entropy convergence in the presence of dichotomous noise. In both the single dimensional and multidimensional cases we are to show that the convergence of the conditional entropy to zero is monotone and at least exponential. In the specific cases of the Ornstein-Uhlenbeck and Rayleigh processes as well as the stochastically perturbed harmonic oscillator and colored noise examples, we have the rather surprising result that the rate of convergence of the entropy to zero is independent of the noise amplitude.Comment: 23 page

Substochastic semigroups and densities of piecewise deterministic Markov processes

Necessary and sufficient conditions are given for a substochastic semigroup on $L^1$ obtained through the Kato--Voigt perturbation theorem to be either stochastic or strongly stable. We show how such semigroups are related to piecewise deterministic Markov process, provide a probabilistic interpretation of our results, and apply them to fragmentation equations.Comment: 26 pages; corrected typo