A class of individual-based models


  • Miroslaw Lachowicz University of Warsaw, Faculty of Mathematics, Informatics and Mechanics, ul. Banacha 2, Warsaw, Poland




Individual--based models, Markov jump processes, Integro--differential equations, Stochastic semigroups, Stability


We discuss a class of mathematical models of biological systemsВ at microscopic level - i.e. at the level of interacting individuals of a population. The class leads to partially integral stochastic semigroups- [5]. We state general conditions guaranteeing the asymptotic stability.В  In particular under some ratherВ restrictive assumptions we observe that any, even non-factorized, initial probability density tends in the evolution to a factorized equilibrium probability density - [4]. We discuss possible applications of the general theory such as redistribution of individuals - [2], thermal denaturation of DNA [1], and tendon healing process - [3].


[1] M. Debowski, M. Lachowicz, and Z. Szymanska,В Microscopic description of DNA thermal denaturation, to appear.

В [2] M. Dolfin, M. Lachowicz, and A. Schadschneider, A microscopic model of redistribution of individuals inside an 'elevator', In Modern Problems in Applied Analysis, P. Drygas and S. Rogosin (Eds.), Bikhauser, Basel (2018), 77--86; DOI: 10.1007/978--3--319--72640-3.

[3] G. Dudziuk, M. Lachowicz, H. Leszczynski, and Z. Szymanska, A simple model ofВ collagen remodeling, to appear.

[4] M. Lachowicz, A class of microscopic individual models corresponding to the macroscopic logistic growth, Math. Methods Appl. Sci., 2017, on--line, DOI: 10.1002/mma.4680

[5] K. Pichor and R. Rudnicki, Continuous Markov semigroups and stability of transportВ equations, J. Math. Analysis Appl. 249, 2000, 668--685, DOI: 10.1006/jmaa.2000.6968






Original Articles