The mathematical analysis towards the dependence on the initial data for a discrete thermostatted kinetic framework for biological systems composed of interacting entities

Marco Menale; Bruno Carbonaro; Marco Menale; Bruno Carbonaro

doi:10.3934/biophy.2020016

AIMS Biophysics

2020, Volume 7, Issue 3: 204-218. doi: 10.3934/biophy.2020016

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The mathematical analysis towards the dependence on the initial data for a discrete thermostatted kinetic framework for biological systems composed of interacting entities

Marco Menale ^{1,2
,
,},
Bruno Carbonaro ¹

1.
Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, Viale Lincoln 5, I-81100 Caserta, Italy;
2.
Laboratoire Quartz EA 7393, École Supérieure d'Ingénieurs en Génie Électrique, Productique et Management Industriel, 95092, Cergy Pontoise Cedex, France

Received: 09 April 2020 Accepted: 16 June 2020 Published: 28 June 2020

This paper is devoted to a mathematical proof of the continuous dependence on the initial data for the discrete thermostatted kinetic framework, for all T > 0. This is a versatile model for describing the time-evolution of a biological complex system which is composed by a large number of interacting entities, called active particles, and is subjected to an external force field due to the environment. A thermostat term is introduced in order to keep the 2nd-order moment of the system, corresponding to the physical global activation energy, constant in time. This model is expressed by a system of nonlinear ordinary differential equations with quadratic nonlinearity.
- complex systems,
- Ordinary Differential Equations,
- stability,
- active particle,
- biological systems,
- discrete models,
- kinetic theory,
- thermostat,
- modeling
Citation: Marco Menale, Bruno Carbonaro. The mathematical analysis towards the dependence on the initial data for a discrete thermostatted kinetic framework for biological systems composed of interacting entities[J]. AIMS Biophysics, 2020, 7(3): 204-218. doi: 10.3934/biophy.2020016

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Abstract

This paper is devoted to a mathematical proof of the continuous dependence on the initial data for the discrete thermostatted kinetic framework, for all T > 0. This is a versatile model for describing the time-evolution of a biological complex system which is composed by a large number of interacting entities, called active particles, and is subjected to an external force field due to the environment. A thermostat term is introduced in order to keep the 2nd-order moment of the system, corresponding to the physical global activation energy, constant in time. This model is expressed by a system of nonlinear ordinary differential equations with quadratic nonlinearity.

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