# American Institute of Mathematical Sciences

May  2012, 17(3): 735-757. doi: 10.3934/dcdsb.2012.17.735

## Two theorems on singularly perturbed semigroups with applications to models of applied mathematics

 1 Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland 2 European Actuarial Services, Ernst & Young Business Advisory Sp. z o.o. i Wspólnicy sp.k., Rondo ONZ 1, 00-124, Warsaw, Poland

Received  May 2011 Revised  October 2011 Published  January 2012

We present two theorems on convergence of semigroups related to singularly perturbed abstract Cauchy problems, and apply them to some recent models of applied mathematics. The semigroups considered are related to piecewise deterministic Markov processes jumping between several copies of a rectangle in $\mathbb{R}^M$ and moving along deterministic integral curves of some ODEs between jumps. Our theorems describe limit behavior of the processes in the cases of frequent jumps and of fast motions in the direction of chosen variables. These results are motivated by Kepler--Elston's model of gene regulation and Lipniacki's model of gene expression. Application to other models, including those of mathematical economics, are also discussed.
Citation: Adam Bobrowski, Radosław Bogucki. Two theorems on singularly perturbed semigroups with applications to models of applied mathematics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 735-757. doi: 10.3934/dcdsb.2012.17.735
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