Abstract
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].В [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
Highlights
In the present paper we review the general class of individual–based models in Biology developed in Ref. [15] — see [3], [14], [16] and references therein
We show that the class corresponds to the partially integral stochastic semigroups and under some more restrictive assumptions leads to the stability result
(Section II) we show that exp tΛ t≥0 is a partially integral stochastic semigroup and, under some additional assumptions, leads to a stability result
Summary
In the present paper we review the general class of individual–based models in Biology developed in Ref. [15] — see [3], [14], [16] and references therein. We consider the general equation that defines the evolution of a number N of individuals of biological populations — cf Refs. The transition into state v of an n1–individual with state un, due to the interaction with individuals of n2,...,nm with states un2,...,unm, respectively, is described by the measurable function A[m] = A[m] M , we consider the stochastic system that is defined by the Markov jump process of N individuals through the following generator Λ acting on densities. The gain term Λ+ is a sum of terms describing the changes from state v of n1–individual into un due to the interaction with n2, ..., nm individuals with states un2, ..., unm, respectively for 2 ≤ m ≤ M and the term (m = 1).
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