Homeorhesis in the motion of an individual

Połączone seminarium RTN i Zakładu Biomatematyki i Teorii Gier Homeorhesis is an inherent dynamical feature of any living system. Homeorhesis is a peculiar qualitative and quantitative independence of the exogenous signals acting on the system and varying within a certain, system-relevant range. Nonliving systems do not perform homeorhesis. Mathematically, homeorhesis is the asymptotic convergence (in the infinite-time limit) of certain dynamic equilibria of the dynamical model that describes a living system (see [1], [2]). Preliminary results on the homeorhesis modelling in terms of ordinary differential equations (ODEs) are developed in [3, Appendix], [1], and [2]. In this case, both the actual mode and creode of the system are two dynamic equilibria [2] which correspond to the actual exogenous signal and the most favoured exogenous signal, respectively. In these terms, homeorhesis is the property of a living system that its actual mode in the course of time tends to its creode for any actual exogenous signal (within a certain, system-relevant range). Homeorhesis is a fundamental notion in theoretical biology (and in a more general field, theory of living matter). It is an inherent feature of any living system. Nevertheless, the literature on dynamical modelling in sociology does not include works on homeorhesis. The purpose of the present work is to fill this gap. Report [4] suggests the simplest model for homeorhesis. The present work specifies this model in the case of the motion of a single individual and illustrates the treatment with numerical-simulation results. [1] Mamontov, E., 2007, Modelling homeorhesis by ordinary differential equations, Mathl Comput. Modelling 45(5-6), pp. 694-707. [2] Mamontov, E., 2007, Dynamic-equilibrium solutions of ordinary differential equations and their role in applied problems, Appl. Math. Lett., accepted (paper AML5947). [3] E. Mamontov, K. Psiuk-Maksymowicz and A. Koptioug, Stochastic mechanics in the context of the properties of living systems, Mathl Computer Modelling 44(7-8): 595-607 (2006) [4] E. Mamontov, Homeorhesis and evolutionary properties of living systems: From ordinary differential equations to the active-particle generalized kinetics theory, In: 10th Evolutionary Biology Meeting at Marseilles, September 20-22, 2006 (Association pour l'Etude de l'Evolution Biologique, Centre Regional de Documentation Pedagogique,Marseille, France, 2006), pp. 28-29, abstract; the 13-page PDF file for the full oral presentation can be downloaded from http://www.up.univ-mrs.fr/evol-cgr/home_page/meeting2006.php