Nonlinear Reaction-diffusion Systems Research Articles

PREDICTION OF CONFINEMENT EFFECTS ON THE NUMBER OF COVID-19 OUTBREAK IN ALGERIA

We are interested in modeling the interaction of calcium dynamics in a medium including sarcolemma and sarcoplasmic reticulum. The governing equations consist of a nonlinear reaction-diffusion system representing the various calcium fluxes and theirs buffers in the two media. We address the question of existence of weak solutions by using a fixed-point approach. We propose a finite element method for this system, we establish the existence of the discrete solution, and we show that the discrete solution generated by the given scheme converges to the corresponding weak solution for the model studied. Finally, we give some 2D and 3D numerical examples to our model. Mathematics Subject Classification. 92C50, 65M60, 65N12, 92C40. The dates will be set by the publisher.

Controllability of a 4 × 4 quadratic reaction–diffusion system

We consider a 4×4 nonlinear reaction–diffusion system posed on a smooth domain Ω of RN (N≥1) with controls localized in some arbitrary nonempty open subset ω of the domain Ω. This system is a model for the evolution of concentrations in reversible chemical reactions. We prove the local exact controllability to stationary constant solutions of the underlying reaction–diffusion system for every N≥1 in any time T>0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics. The proof is based on a linearization which uses return method and an adequate change of variables that creates cross diffusion which will be used as coupling terms of second order. The controllability properties of the linearized system are deduced from Carleman estimates. A Kakutani's fixed-point argument enables to go back to the nonlinear parabolic system. Then, we prove a global controllability result in large time for 1≤N≤2 thanks to our local controllability result together with a known theorem on the asymptotics of the free nonlinear reaction–diffusion system.

Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point

Abstract In this article, we study complex spatio-temporal solutions in nonlinear time-fractional reaction-diffusion systems. The main attention is paid to nonlinear dynamics near a bifurcation point. Despite the fact that the homogeneous state is stable at the parameters lower than bifurcation ones, a variety of complex solutions can also form in the subcritical domain. As an example, we consider a generalized fractional FitzHugh-Nagumo model. Depending on the given standard bifurcation parameters and the order of fractional derivative, the new types of steady auto-wave solutions in such systems have revealed. By computer simulation, it is shown that fractional reaction-diffusion possess much more complex nonlinear dynamics than their integer counterparts even at a subcritical bifurcation.

Bio-inspired reaction diffusion system applied to image restoration

In this paper, we propose a new model of nonlinear and anisotropic reaction diffusion system applied to image restoration and to contrast enhancement. This model is based on a system of partial differential equations of type Fitzhugh-Nagumo. In the first, we give the comparison with the previous model, then, we show the robustness and the performance of our algorithm through a number of experimental results.

On Exact Multidimensional Solutions of a Nonlinear System of Reaction–Diffusion Equations

We study a nonlinear reaction–diffusion system modeled by a system of two parabolic-type equations with power-law nonlinearities. Such systems describe the processes of nonlinear diffusion in reacting two-component media. We construct multiparameter families of exact solutions and distinguish the cases of blow-up solutions and exact solutions periodic in time and anisotropic in spatial variables that can be represented in elementary functions.

A triangular nonlinear reaction‐fractional diffusion system with a balance law

The aim of this paper is to establish a global existence result for a nonlinear reaction diffusion system with fractional Laplacians of different orders and a balance law. Our method of proof is based on a duality argument and a recent maximal regularity result due to Zhang.

Blow-up phenomena for a nonlinear reaction–diffusion system with time dependent coefficients

We investigate the blow-up phenomena for the solution to a nonlinear reaction–diffusion system with time dependent coefficients subject to null Dirichlet boundary conditions. By virtue of Kaplan’s method, method of subsolutions and supersolutions and modified differential inequality technique, we establish the blow-up criteria for the solution. Moreover, lower and upper bounds for the blow-up time are derived.

Existence of Traveling Waves of General Gray-Scott Models

This work gives a rigorous proof of the existence of propagating traveling waves of a nonlinear reaction–diffusion system which is a general Gray-Scott model of the pre-mixed isothermal autocatalytic chemical reaction of order m ( $$m > 1$$ ) between two chemical species, a reactant A and an auto-catalyst B, $$ A + m B \rightarrow (m+1) B$$ , and a super-linear decay of order $$ n > 1$$ , $$ B \rightarrow C$$ , where $$ 1< n < m$$ . Here C is an inert product. Moreover, we establish that the speed set for existence must lie in a bounded interval for a given initial value $$u_0$$ at $$ - \infty $$ . The explicit bound is also derived in terms of $$u_0$$ and other parameters. The same system also appears in a mathematical model of SIR type in infectious diseases.

Multidimensional exact solutions to the reaction-diffusion system with power-law nonlinear terms

We study a nonlinear reaction-diffusion system that is modeled by a system of parabolic equations with power-law nonlinear terms. The proposed construction of exact solutions enables us to split the process of finding the components depending on time and the spatial coordinates. We construct multiparametric families of exact solutions in elementary functions. The cases are elaborated of blow-up solutions as well as exact solutions time-periodic but spatially anisotropic.

Variational methods for strongly indefinite problems

This paper reviews some recent development in studying variational methods for strongly indefinite problems under the support of National Natural Science Foundation of China. We firstly describe the ideas for establishing the variational setting, then state a deformation theory in locally convex topological linear spaces and several critical point theorems for dealing with strongly indefinite problems based on the deformation. Some applications of the theory are given including mainly: Existence and multiplicity of solutions to non-autonomous stationary Dirac system, in particular, the existence, concentration phenomena and the exponential decay of semi-classical states; existence and multiplicity of global solutions to the nonlinear (non-autonomous, unbounded Hamilton-type) reaction-diffusion system, especially the existence, concentration phenomena and decay of its ground states of the system with singular perturbation; depth study of homoclinic orbits Hamiltonian system, global Schrodinger equation; other initial work, such as bifurcation of Dirac equation on spin manifolds.

Global Existence and Finite Time Blow-up for a Reaction-Diffusion System with Three Components

This paper concerns global existence and finite time blow-up behavior of positive solutions for a nonlinear reaction-diffusion system with different diffusion coefficients. By use of algebraic matrix theory and modern analytical theory, we extend results of Wang (Z. Angew. Math. Phys. 51:160–167, 2000) to a more general system. Furthermore, we give a complete answer to the open problem which was brought forward in Wang (Z. Angew. Math. Phys. 51:160–167, 2000).

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces.

In this paper, we present computational techniques to investigate the effect of surface geometry on biological pattern formation. In particular, we study two-component, nonlinear reaction-diffusion (RD) systems on arbitrary surfaces. We build on standard techniques for linear and nonlinear analysis of RD systems and extend them to operate on large-scale meshes for arbitrary surfaces. In particular, we use spectral techniques for a linear stability analysis to characterise and directly compose patterns emerging from homogeneities. We develop an implementation using surface finite element methods and a numerical eigenanalysis of the Laplace-Beltrami operator on surface meshes. In addition, we describe a technique to explore solutions of the nonlinear RD equations using numerical continuation. Here, we present a multiresolution approach that allows us to trace solution branches of the nonlinear equations efficiently even for large-scale meshes. Finally, we demonstrate the working of our framework for two RD systems with applications in biological pattern formation: a Brusselator model that has been used to model pattern development on growing plant tips, and a chemotactic model for the formation of skin pigmentation patterns. While these models have been used previously on simple geometries, our framework allows us to study the impact of arbitrary geometries on emerging patterns.

Explicit exponential convergence to equilibrium for nonlinear reaction–diffusion systems with detailed balance condition

The convergence to equilibrium of mass action reaction–diffusion systems arising from networks of chemical reactions is studied. The considered reaction networks are assumed to satisfy the detailed balance condition and have no boundary equilibria. We propose a general approach based on the so-called entropy method, which is able to quantify with explicitly computable rates the decay of an entropy functional in terms of an entropy entropy-dissipation inequality based on the totality of the conservation laws of the system.As a consequence follows convergence to the unique detailed balance equilibrium with explicitly computable convergence rates. The general approach is further detailed for two important example systems: a single reversible reaction involving an arbitrary number of chemical substances and a chain of two reversible reactions arising from enzyme reactions.

Finite element approximation of a spatially structured metapopulation PDE model

We present a new fully spatially structured PDE metapopulation model for predator–prey dynamics in d≤3 space dimensions. A nonlinear reaction–diffusion system of Rosenzweig–MacArthur form models predator–prey dynamics in two ‘high’ quality patches embedded in a ‘low’ quality subdomain, where species can diffuse, convect and die. Our model substantially generalizes and improves earlier fully structured metapopulation models. After a nondimensionalization procedure, in order to approximate the metapopulation model we present a fully discrete Galerkin finite element method in two space dimensions, which is a generalization of the finite element method analyzed in a previous single patch predator–prey model. The numerical solutions are illustrated for some test cases using MATLAB. Numerical experiments demonstrate that the initial local extinction of predators in one patch leads to waves of recolonization from another patch. In an appendix we also give an outline for the proof of the well-posedness of the model.

A variational principle for nonpotential perturbations of gradient flows of nonconvex energies

We investigate a variational approach to nonpotential perturbations of gradient flows of nonconvex energies in Hilbert spaces. We prove existence of solutions to elliptic-in-time regularizations of gradient flows by combining the minimization of a parameter-dependent functional over entire trajectories and a fixed-point argument. These regularized solutions converge up to subsequences to solutions of the gradient flow as the regularization parameter goes to zero. Applications of the abstract theory to nonlinear reaction–diffusion systems are presented.

An atlas- and data-driven approach to initializing reaction-diffusion systems in computer cardiac electrophysiology.

The cardiac electrophysiology (EP) problem is governed by a nonlinear anisotropic reaction-diffusion system with a very rapidly varying reaction term associated with the transmembrane cell current. The nonlinearity associated with the cell models requires a stabilization process before any simulation is performed. More importantly, when used in a 3-dimensional (3D) anatomy, it is not sufficient to perform this stabilization on the basis of isolated cells only, since the coupling of the different cells through the tissue greatly modulates the dynamics of the system. Therefore, stabilization of the system must be performed on the entire 3D model. This work develops a novel procedure for the initialization of reaction-diffusion systems for numerical simulations of cardiac EP from steady-state conditions. We exploit surface point correspondence to establish volumetric point correspondence. Upon introduction of a new 3D anatomy with surface point correspondence, a prediction of the cell model steady states is derived from the set of earlier biophysical simulations. We show that the prediction error is typically less than 10% for all model variables, with most variables showing even greater accuracy. When initializing simulations with the predicted model states, it is demonstrated that simulation times can be cut by at least two-thirds and potentially more, which saves hours or days of high-performance computing. Overall, these results increase the clinical applicability of detailed computational EP studies on personalized anatomies.

Homotopy perturbation Laplace transform solution of fractional non-linear reaction diffusion system of Lotka-Volterra type differential equation

This work investigates the fractional non linear reaction diffusion (FNRD) system of Lotka-Volterra type. The system of equations together with the boundary conditions are solved by Homotopy perturbation transform method (HPTM). The series solutions are obtained for the two cases (homogeneous and non-homogeneous) of FNRD system. The effect of fractional parameter on the mass concentration of two species are shown and discussed with the help of 3D graphs.

Numerical methods for a nonlinear reaction–diffusion system modelling a batch culture of biofilm

A biofilm is usually defined as a layer of bacterial cells anchored to a surface. These cells are embedded into a polymer matrix that keeps them attached to each other and to a solid surface. Among a large variety of biofilms, in this paper we consider batch cultures. The mathematical model is formulated in terms of a quasilinear system of diffusion–reaction equations for biomass and nutrients concentrations, which exhibits possible degeneracy and singularities in the nonlinear diffusion coefficient. In the present paper, we propose a set of efficient numerical methods that speeds up the solution of the model. Mainly, Crank–Nicolson finite differences techniques for discretisation are combined with a Newton algorithm for the nonlinearities. Moreover, some numerical examples show the expected behaviour of the biomass and nutrients concentrations and also clearly illustrate some theoretically proved qualitative properties related to exponential decays or convergence to a critical biomass concentration depending on the values of the model parameters.

A Dimensional Splitting of ETD Schemes for Reaction-Diffusion Systems

AbstractNovel dimensional splitting techniques are developed for ETD Schemes which are second-order convergent and highly efficient. By using the ETD-Crank-Nicolson scheme we show that the proposed techniques can reduce the computational time for nonlinear reaction-diffusion systems by up to 70%. Numerical tests are performed to empirically validate the superior performance of the splitting methods.

Analysis of Nonlinear Reaction–Diffusion Systems by the Perturbation Method: Conditions of Application, Construction of Solutions, and Bifurcation Analysis

For nonlinear systems of reaction–diffusion type, we propose a technique for the construction and analysis of solutions based on the method of small parameter. The proposed technique enables us not only to analytically obtain approximate quasiharmonic low-amplitude solutions appearing as a result of bifurcations of spatially homogeneous states of the system but also to determine the type of bifurcation in the system. Examples of application of this approach to the analysis of bifurcations and the construction of solutions of a specific mathematical reaction–diffusion model are given.