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In the present paper we propose and study a simple model of collagen remodeling occurring in latter stage of tendon healing process. The model is an integro-differential equation describing the possibility of an alignment of collagen fibers in a finite time. We show that the solutions may either exist globally in time or blow-up in a finite time depending on initial data. The latter behavior can be related to the healing of injury without the scar formation in a finite time: a full alignment of collagen fibers. We believe that the present model is an essential ingredient of the full description of collagen remodeling. [ABSTRACT FROM AUTHOR]

We study a class of microscopic models corresponding to the standard macroscopic logistic equation. The models at microscale refer to a number of interacting individuals and are in terms of linear evolution equations related to Markov jump processes. The asymptotic large time behavior for the microscopic models is obtained. Moreover, it is shown that any, even nonfactorized, initial probability density tends in the evolution to a factorized equilibrium probability density. [ABSTRACT FROM AUTHOR]

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 -- [25]. 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 -- [16]. We discuss possible applications of the general theory such as redistribution of individuals -- [10], thermal denaturation of DNA [7], and tendon healing process -- [11]. [ABSTRACT FROM AUTHOR]

We show an alternative-axiomatic approach to the direct sums of countably many Banach spaces, called here B-direct sums (BDS). We develop 'direct summing' of linear operators and the problems of generating the $$C_0$$ -semigroups in such 'Banach direct sums of Banach spaces' by 'the direct sums of the generators'. We study special types of BDS, including M-BDS-the one closely related to Day's construction of a direct sum (M.M. Day, Normed linear spaces, 1973). The main abstract results of the paper concern $$C_0$$ -semigroup generation properties for diagonal operators in M-BDS type direct sums. The paper is motivated by stochastic particle systems and the problem of existing $$C_0$$ -semigroups defined by the corresponding hierarchies for marginal probability densities. We use our abstract M-BDS results to get a solution of the respective differential equation in the $$C_0$$ -semigroup sense. [ABSTRACT FROM AUTHOR]

In this paper we analyze emergent collective phenomena in the evolution of opinions in a society structured into few interacting nodes of a network. The presented mathematical structure combines two dynamics: a first one on each single node and a second one among the nodes, i.e. in the network. The aim of the model is to analyze the effect of a network structure on a society with respect to opinion dynamics and we show some numerical solutions addressed in this direction, i.e. comparing the emergent behaviors of a consensus-dissent dynamic on a single node when the effect of the network is not considered, with respect to the emergent behaviors when the effect of a network structure linking few interacting nodes is considered. We adopt the framework of the Kinetic Theory for Active Particles (KTAP), deriving a general mathematical structure which allows to deal with nonlinear features of the interactions and representing the conceptual framework toward the derivation of speciffic models. A specific model is derived from the general mathematical structure by introducing a consensus-dissent dynamics of interactions and a qualitative analysis is given. [ABSTRACT FROM AUTHOR]

In this paper a mesoscopic approach is proposed to describe the process of breaking of hydrogen bonds during the DNA thermal denaturation, also known as DNA melting. A system of integro-differential equations describing the dynamic of the variable which characterizes the opening of the base pairs is proposed. In the derivation of the model non linear effects arising from the collective behavior, namely the interactions, of base pairs are taken into account. Solutions of the mesoscopic model show significative analogies with the experimental S-shaped curves describing the fraction of broken bonds as a function of temperature at the macroscopic level, althought we instead simulate the variation in time. With this respect a research perspective connecting the theoretical results to the experimental one is proposed. [ABSTRACT FROM AUTHOR]

In the present paper we propose a class of kinetic type equations that describes the replicator dynamics at the mesoscopic level. The equations are highly nonlinear due to the dependence of the transition rates of distribution function. Under suitable assumptions we show the asymptotic (exponential) stability of the solutions to such kinetic equations. [ABSTRACT FROM AUTHOR]

The present paper is a continuation of part one, in which two macroscopic limits of the Boltzmann–Enskog kinetic equation with three small parameters—the Knudsen number, Mach number, and diameter of particles—were considered at the formal level, and conditional theorems were stated. In this paper, the proof of a rigorous result corresponding to one of the two cases considered in part one is given. Under some suitable assumptions on the relations between the small parameters, it is proved that the Boltzmann–Enskog equation results in the Navier–Stokes equation for incompressible fluids together with the Boussinesq relation and the temperature fluctuations equation. The time and velocity averaging is exploited.Dedicated to Professor Janusz Mika on his 70th birthday. [ABSTRACT FROM AUTHOR]