Nonlinear oscillations and stability domains in fractional reaction-diffusion systems

We study a fractional reaction-diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing of periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. It is established by computer simulation that there exists a set of stable spatio-temporal tructures of the one-dimensional system under the Neumann and periodic boundary condition. The features of these solutions consist in the transformation of the steady state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.Comment: 15 pages, 5 figure

Analysis of instabilities and pattern formation in time fractional reaction-diffusion systems

We analyzed conditions for Hopf and Turing instabilities to occur in two-component fractional reaction-diffusion systems. We showed that the eigenvalue spectrum and fractional derivative order mainly determine the type of instability and the dynamics of the system. The results of the linear stability analysis are confirmed by computer simulation of the model with cubic nonlinearity for activator variable and linear dependance for the inhibitor one. It is shown that pattern formation conditions of instability and transient dynamics are different than for a standard system. As a result, more complicated pattern formation dynamics takes place in fractional reaction-diffusion systems.Comment: 7 pages, 6 figure

Replicator dynamical systems and their gradient and Hamiltonian properties

We consider the general properties of the replicator dynamical system from the standpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system has been studied. Lyaponuv function for investigation of system evolution has been proposed. The generalization of the replicator dynamics for the case of multi-agent systems has been introduced. We propose a new mathematical model to describe the multi-agent interaction in complex system

Remarks on scaling properties inherent to the systems with hierarchically organized supplying network

We study the emergence of a power law distribution in the systems which can be characterized by a hierarchically organized supplying network. It is shown that conservation laws on the branches of the network can, at some approximation, impose power law properties on the systems. Some simple examples taken from economics, biophysics etc. are considered

Mathematical description of the heat transfer in living tissue (Part II)

In the present monograph we formulate a simple model for heat transfer in living tissue with self - regulation. The initial point of the model is the governing equations describing heat transfer in living tissue at the mesoscopic level, i.e. considering different vessels individually. Then, basing on the well known equivalence of the diffusion type process and random walks, we develop a certain regular procedure that enables us to average these mesoscopic equations practically over all scales of the hierarchical vascular network. The microscopic governing equations obtained in this way describe living tissue in terms of an active medium with continuously distributed self - regulation. One of the interesting results obtained in the present monograph is that there can be the phenomena of ideal self - regulation in large active hierarchical systems. Large hierarchical systems are characterized by such a great information flow that none of its elements can possess whole information required of governing the system behavior. Nevertheless, there exists a cooperative mechanism of regulation which involves individual response of each element to the corresponding hierarchical piece of information and leads to ideal system response due to self - processing of information. The particular results are obtained for bioheat transfer. However, self - regulation in other natural hierarchical systems seems to be organized in a similar way. The characteristics of large hierarchical systems occurring in nature are discussed from the stand point of regulation problems. By way of example, some ecological and economic systems are considered. An cooperative mechanism of self-regulation which enables the system to function ideally is proposed

Quasi-stationary Stefan problem and computer simulation of interface dynamics

The computer simulation of quasistationary Stefan problem has been realized. Different representations of the Laplacian growth model are considered. The main attention has been paid for the interface dynamics represented by integro differential equations. Numerical approach has been realized by use of interpolating polynomials and exact quadrature formulae. As a result system of ordinary differential equations has been simulated.Comment: 13 pages, 9 figure

Analysis of the optimality principles responsible for vascular network architectonics

The equivalence of two optimality principles leading to Murray's law has been discussed. The first approach is based on minimization of biological work needed for maintaining the blood flow through the vessels at required level. The second one is the principle of minimal drag and lumen volume. Characteristic features of these principles are considered. An alternative approach leading to Murray's law has been proposed. For that we model the microcirculatory bed in terms of delivering vascular network with symmetrical bifurcation nodes, embedded uniformly into the cellular tissue. It was shown that Murray's law can be regarded as a direct consequence of the organism capacity for controlling the blood flow redistribution over the microcirculatory beds

The properties of quasispecies dynamics in molecular evolution

We consider the general properties of the quasispecies dynamical system from the standpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system has been studied. Mathematical modelling of the system under consideration has been performed.Comment: 11 pages, 1 figur

Hamiltonian and gradient properties of certain type of dynamical systems

From the sandpoint of neural network dynamics we consider dynamical system of special type pesesses gradient (symmetric) and Hamiltonian (antisymmetric) flows. The conditions when Hamiltonian flow properties are dominant in the system are considered. A simple Hamiltonian has been studied for establishing oscillatory patern conditions in system under consideration

Mathematical modeling of pattern formation in sub- and supperdiffusive reaction-diffusion systems

This paper is concerned with analysis of coupled fractional reaction-diffusion equations. It provides analytical comparison for the fractional and regular reaction-diffusion systems. As an example, the reaction-diffusion model with cubic nonlinearity and Brusselator model are considered. The detailed linear stability analysis of the system with Cubic nonlinearity is provided. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. Computer simulation and analytical methods are used to analyze possible solutions for a linearized system. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. It is shown that the increase of the fractional derivative index leads to the periodic solutions which become stochastic at the index approaching the value of 2. It is established by computer simulation that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary condition. The characteristic features of these solutions consist in the transformation of the steady state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index.Comment: 16 pages, 7 figure