The Motion of Weakly Interacting Pulses in Reaction-Diffusion Systems

The interaction of stable pulse solutions on R ¹ is considered when distances between pulses are sufficiently large. We construct an attractive local invariant manifold giving the dynamics of interacting pulses in a mathematically rigorous way. The equations describing the flow on the manifold is also given in an explicit form. By it, we can easily analyze the movement of pulses such as repulsiveness, attractivity and/or the existence of bound states of pulses. Interaction of front solutions are also treated in a similar way.     

The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

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Fish and Plankton Interplay Determines Both Plankton Spatio-Temporal Pattern Formation and Fish School Walks: A Theoretical Study

The fascinating variety of spatio-temporal patterns in aquatic ecosystems and the understanding of the governing mechanisms of its generation and further dynamics requires ongoing experimental and theoretical studies. After introducing a certain hybrid mathematical model, this paper makes an attempt to demonstrate that the predation of a mobile planktivorous fish school on zooplankton can initiate both plankton pattern formation and fish school walks. Nonlinear interactions in the model of a fish-zooplankton-algae trophic chain prevent a simple intuitive understanding of the system dynamics. It is shown that the fish school predation and motion can give rise to plankton spiral waves. In the course of the spiral wave formation, the amplitudes of the spatially averaged plankton density oscillations are decreasing dramatically. Fish school walks are shown to resemble a fractional Brownian motions with a Hurst exponent depending on the fish predation rate.     

A Linearized Spectral-Galerkin Method for Three-Dimensional Riesz-Like Space Fractional Nonlinear Coupled Reaction-Diffusion Equations

In this paper, we establish a novel fractional model arising in the chemical reaction and develop an efficient spectral method for the three-dimensional Riesz-like space fractional nonlinear coupled reaction-diffusion equations. Based on the backward difference method for time stepping and the Legendre-Galerkin spectral method for space discretization, we construct a fully discrete numerical scheme which leads to a linear algebraic system. Then a direct method based on the matrix diagonalization approach is proposed to solve the linear algebraic system, where the cost of the algorithm is of a small multiple of $N^4$ ($N$ is the polynomial degree in each spatial coordinate) flops for each time level. In addition, the stability and convergence analysis are rigorously established. We obtain the optimal error estimate in space, and the results also show that the fully discrete scheme is unconditionally stable and convergent of order one in time. Furthermore, numerical experiments are presented to confirm the theoretical claims. As the applications of the proposed method, the fractional Gray-Scott model is solved to capture the pattern formation with an analysis of the properties of the fractional powers.     

Robust feed-back control of travelling waves in a class of reaction-diffusion distributed biological systems

Reaction-Diffusion (RD) mechanisms can describe many biological phenomena such as neuron firing in the brain, the heartbeat, cellular organization activities or even biological disorders such as fibrillation. The FitzHugh-Nagumo (FHN) model is a particular case of RD systems. It is able to capture the key features of many biological processes and since it is relatively simple it has been widely employed during recent years. Some examples of its predictive capabilities include the representation of the normal behavior of some physiological phenomena, related to a travelling plane wave, as well as biological disorders associated with spiral or irregular fronts. The objective of this work is to design a control law that is able to stabilize complex behaviors (travelling plane wave) in biological systems using the FHN model as a case study. Since, in biological systems there usually exists a lack of detailed information on the system structure, our control law will be designed to be robust, i.e., it must be able to reach the predefined reference regardless the presence of structural uncertainties. To this purpose, we will extend some classical results on the finite-dimensional robust control theory to RD systems by means of order reduction techniques, in particular the Proper Orthogonal Decomposition method.     

Sample Duplication Method for Monte Carlo Simulation of Large Reaction-Diffusion System

The sample duplication method for the Monte Carlo simulation of large reaction-diffusion system is proposed in this paper. It is proved that the sample duplication method will effectively raise the efficiency and statistical precision of the simulation without changing the kinetic behaviour of the reaction-diffusion system and the critical condition for the bifurcation of the steady-states. The method has been applied to the simulation of spatial and time dissipative structure of Brusselator under the Dirichlet boundary condition. The results presented in this paper definitely show that the sample duplication method provides a very efficient way to sol-'e the master equation of large reaction-diffusion system. For the case of two-dimensional system, it is found that the computation time is reduced at least by a factor of two orders of magnitude compared to the algorithm reported in literature.     

Degenerate Four-Virtual-Soliton Resonance for the KP-II

We propose a method for solving the (2+1)-dimensional Kadomtsev-Petviashvili equation with negative dispersion (KP-II) using the second and third members of the disipative version of the AKNS hierarchy. We show that dissipative solitons (dissipatons) of those members yield the planar solitons of the KP-II. From the Hirota bilinear form of the SL(2, ℝ) AKNS flows, we formulate a new bilinear representation for the KP-II, by which we construct one- and two-soliton solutions and study the resonance character of their mutual interactions. Using our bilinear form, for the first time, we create a four-virtual-soliton resonance solution of the KP-II, and we show that it can be obtained as a reduction of a four-soliton solution in the Hirota-Satsuma bilinear form for the KP-II.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 162–170, July, 2005.     

强耦合反应扩散方程组的定性分析

本文研究由四个反应扩散方程组成的强耦合方程组，利用算子半群理论，上、下解方法和能量积分等经典分析方法，论证此类方程组的整体解的存在唯一性和有界性。