A class of individual-based models

Abstract

We discuss a class of mathematical models of biological systems 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.  In particular under some rather 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, Microscopic description of DNA thermal denaturation, to appear.  [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 of 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 transport equations, J. Math. Analysis Appl. 249 , 2000, 668--685, DOI: 10.1006/jmaa.2000.6968