Nonlinear Reaction-diffusion Systems Research Articles

Uniform Null Controllability for a Degenerating Reaction-Diffusion System Approximating a Simplified Cardiac Model

This paper is devoted to analyze the null controllability of a nonlinear reaction-diffusion system approximating a parabolic-elliptic system modeling electrical activity in the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by means of a single control is shown. The proof is based on the combination of Carlemans estimates and weighted energy inequalities.

The local integral equation method for pattern formation simulations in reaction–diffusion systems

A meshless local integral equation (LIE) method is proposed for numerical simulation of 2D pattern formation in nonlinear reaction diffusion systems. The method uses weak formulation of the differential governing equations on local sub-domains with using the Green function of the Laplace operator as the test function. The moving least square (MLS) approximation is employed for spatial variations of field variables while the time evolution is discretized by using suitable finite difference approximations. The effect of parameters and conditions are studied by considering the well known Schnakenberg model.

On Uniform Decay of the Entropy for Reaction–Diffusion Systems

This work provides entropy decay estimates for classes of nonlinear reaction–diffusion systems modeling reversible chemical reactions under the detailed-balance condition. We obtain explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the Log-Sobolev estimate and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: (i) vanishing diffusion constants in some chemical components and (ii) usage of different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.

The scaling and shift of morphogen gene expression boundary in a nonlinear reaction diffusion system.

The scaling and shift of the gene expression boundary in a developing embryo are two key problems with regard to morphogen gradient formation in developmental biology. In this study, a bigradient model was applied to a nonlinear reaction diffusion system (NRDS) to investigate the location of morphogen gene expression boundary. In contrast to the traditional synthesis-diffusion-degradation model, the introduction of NRDS in this study contributes to the precise gene expression boundary at arbitrary location along the anterior-posterior axis other than simply midembryo even when the linear characteristic lengths of two morphogens are equal. The scaling location depends on the ratio of two morphogen influxes (w) and concentrations (r) as well as the nonlinear reaction diffusion parameters (a, n). We also formulate a direct relationship between the shift in the gene expression boundary and the influx of morphogen and find that enhancing the morphogen influx is helpful to build up a robust gene expression boundary. By analyzing the robustness of the morphogen gene expression boundary and comparing with the relevant results in linear reaction diffusion system, we determine the precise range of the ratio of the two morphogen influxes with a lower shift in the morphogen gene expression boundary and increased system robustness.

On the performance of anisotropic mesh adaptation for scroll wave turbulence dynamics in reaction–diffusion systems

Nonlinear reaction–diffusion systems are widely employed to study the spatio-temporal chaotic behavior that occurs in excitable media such as cardiac tissue where sufficiently strong perturbations can excite nonlinear propagating waves which can form spiral waves in two dimensions or scroll waves in three dimensions. The numerical simulation of these waves calls for grids that are extremely fine over the whole computational domain to accurately predict the trajectories and multiplication of wave fronts and therefore leads to huge computational challenges. Mesh adaptation methods can reduce the number of degrees of freedom required for a given accuracy but they also have a cost and it is not clear if they are competitive with respect to very fine uniform meshes. Previous mesh adaptation techniques applied to spatio-temporal chaotic behavior have been mostly limited to the two-dimensional case. The purpose of this paper is to explore the efficiency of a three-dimensional anisotropic finite element mesh adaptivity for simulating scroll wave turbulence. The computational efficiency of the proposed method is assessed using reference solutions obtained on a uniform refined mesh with more than 44 millions degrees of freedom. The proposed method reduces significantly the number of elements leading to huge saving in memory as well as in computational time. Examples of the dynamics of ventricular fibrillation in cardiac tissue will be presented illustrating the performance of the overall methodology.

Cross-diffusion induced stationary patterns in a prey–predator system with parental care for predators

This paper is concerned about a strongly-coupled nonlinear reaction–diffusion system, which describes a prey–predator model with parental care for predators. The long time behaviors of the solution are discussed, stability and instability of the positive constant equilibrium are studied. Our results show that even though the unique positive constant steady-state is stable for the kinetic system and for the self-diffusion reaction system, cross-diffusion can generate the stationary patterns (nonconstant positive steady states). The numerical simulation is also given to illustrate the theoretical result.

The effect of exclusion on nonlinear reaction–diffusion system in inhomogeneous media

We study a minimal model for the formation of clusters on surfaces in the presence of surface defects. A reaction–diffusion model is considered in which atoms undergo reactions at the defect centers to form clusters. Volume exclusion between particles is introduced through a drift term in the reaction–diffusion equation. This drift arises due the repulsive force field produced by the clustering atoms on the adatoms and vice versa. Approximate linearized equations are obtained which give useful insight into the process. The Green’s function method is used to obtain solutions of the coupled linearized equations. We have found that the clusterization process slows down due to exclusion. As a result the size of the clusters reduces.

Non-linear instability and exact solutions to some delay reaction–diffusion systems

We deal with coupled delay non-linear reaction–diffusion systems of the formut=k1uxx+F(u,u¯,w,w¯),wt=k2wxx+G(u,u¯,w,w¯),where u=u(x,t), w=w(x,t), u¯=u(x,t−τ), and w¯=w(x,t−τ), and τ is the delay time. For a wide class of the kinetic functions F and G, we determine global instability conditions; once these conditions hold, any solution of the system is unstable. The solution instability is proved with an exact approach without making any assumptions or approximations (this approach can be useful for analyzing other non-linear delay models, including biological, biochemical, biophysical, etc.). We discuss some ill-posed Cauchy-type and initial-boundary-value problems connected with the global instability. We present multiparameter exact solutions involving an arbitrary number of free parameters and give an exact solution that represents a non-linear superposition of a traveling wave and a periodic standing wave. All of the systems considered contain two arbitrary functions of two or three arguments. We also study other non-linear systems of delay PDEs including reaction–diffusion system with two different time-varying delays, multicomponent systems of reaction–diffusion equations and more complex, higher-order non-linear systems with delay. The results may be used in solving certain problems and testing approximate analytical and numerical methods for certain classes of similar and more complex non-linear delay systems.

Global existence for fully nonlinear reaction–diffusion systems describing multicomponent reactive flows

We consider combustion problems in the presence of complex chemistry and nonlinear diffusion laws leading to fully nonlinear multispecies reaction–diffusion equations. We establish results of existence of solution and maximum principle, i.e. positivity of the mass fractions, which rely on specific properties of the models. The nonlinear diffusion coefficients are obtained by resolution of the so-called Stefan–Maxwell equations.

Automatic simplification of systems of reaction–diffusion equations by a posteriori analysis

Many mathematical models in biology and physiology are represented by systems of nonlinear differential equations. In recent years these models have become increasingly complex in order to explain the enormous volume of data now available. A key role of modellers is to determine which components of the model have the greatest effect on a given observed behaviour. An approach for automatically fulfilling this role, based on a posteriori analysis, has recently been developed for nonlinear initial value ordinary differential equations [J.P. Whiteley, Model reduction using a posteriori analysis, Math. Biosci. 225 (2010) 44–52]. In this paper we extend this model reduction technique for application to both steady-state and time-dependent nonlinear reaction–diffusion systems. Exemplar problems drawn from biology are used to demonstrate the applicability of the technique.

Direct discontinuous Galerkin method for nonlinear reaction–diffusion systems in pattern formation

Nonlinear reaction–diffusion systems are often employed in mathematical modeling for pattern formation. Most of the work to date has been concerned within one-dimensional or rectangular domains. However, it is recognised that in most applications multidimensional complex geometrical domains are typically more important. In this paper we solve reaction–diffusion systems by combining direct discontinuous Galerkin (DDG) finite element methods with implicit integration factor (IIF) time integration method, on triangular meshes. This allows us solve the nonlinear algebraic systems on an element-by-element bases with significant gains in computational time. Numerical solutions of two reaction–diffusion systems, the well-studied Schnakenberg model and chloride–iodide–malonic acid (CIMA) reactive model, are presented to demonstrate effects of various domain geometries on the resulting biological patterns. Our numerical results are in good agreement with other numerical and analytical results, and with experimental results.

Reaction-Diffusion Patterns in Plant Tip Morphogenesis: Bifurcations on Spherical Caps

We study a chemical reaction-diffusion model (the Brusselator) for pattern formation on developing plant tips. A family of spherical cap domains is used to represent tip flattening during development. Applied to conifer embryos, we model the chemical prepatterning underlying cotyledon ("seed leaf") formation, and demonstrate the dependence of patterns on tip flatness, radius, and precursor concentrations. Parameters for the Brusselator in spherical cap domains can be chosen to give supercritical pitchfork bifurcations of patterned solutions of the nonlinear reaction-diffusion system that correspond to the cotyledon patterns that appear on the flattening tips of conifer embryos.

Lie and conditional symmetries of the three-component diffusive Lotka–Volterra system

Lie and Q-conditional symmetries of the classical three-component diffusive Lotka–Volterra system in the case of one space variable are studied. The group-classification problems for finding Lie symmetries and Q-conditional symmetries of the first type are completely solved. Notably, non-Lie symmetries (Q-conditional symmetry operators) for a multi-component nonlinear reaction–diffusion system are constructed for the first time. The results are compared with those derived for the two-component diffusive Lotka–Volterra system. The conditional symmetry obtained for the non-Lie reduction of the three-component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found. Particularly, the exact solution describing different scenarios of competition between three species is constructed.

Exponential attractors for random dynamical systems and applications

The paper is devoted to constructing a random exponential attractor for some classes of stochastic PDE's. We first prove the existence of an exponential attractor for abstract random dynamical systems and study its dependence on a parameter and then apply these results to a nonlinear reaction-diffusion system with a random perturbation. We show, in particular, that the attractors can be constructed in such a way that the symmetric distance between the attractors for stochastic and deterministic problems goes to zero with the amplitude of the random perturbation.

2-D coupled computational model of biological cell proliferation and nutrient delivery in a perfusion bioreactor

Tissue engineering aims to regenerate, repair or replace organs or tissues which have become defective due to trauma, disease or age related degeneration. This engineering may take place within the patient’s body or tissue can be regenerated in a bioreactor for later implantation into the patient. Regeneration of soft tissue is one of the most demanding applications of tissue engineering. Producing proper nutrient supply, uniform cell distribution and high cell density are the important challenges. Many experimental models exist for tissue growth in a bioreactor. It is important to put experiments into a theoretical framework. Mathematical modelling in terms of physical and biochemical mechanisms is the best tool to understand experimental results.In this work a mathematical model of convective and diffusive transport of nutrients and cell evolution in a perfusion bioreactor is developed. A cell-seeded porous scaffold is placed in a perfusion bioreactor and fluid delivers the nutrients to the cells for their growth. The model describes the key features of the tissue engineering processes which includes the interaction between the cell growth, variation of material permeability due to cell proliferation, flow of fluid through the material and delivery of nutrients to the cells. The fluid flow through the porous scaffold is modelled by Darcy’s law, and the delivery of nutrients to the cells is modelled by the advection–diffusion equation. A non-linear reaction diffusion system is used to model the cell growth. The growth of cells is modelled by logistic growth. COMSOL (a commercial finite element solver) is used to numerically solve the model. The results show that the distribution of cells and total cell number in the scaffold does not depend on the initial cell density but depend on the material permeability.

Self-Similarity and Attraction in Stochastic Nonlinear Reaction-Diffusion Systems

Self-similarity solutions play an important role in many fields of science. We explore self-similarity in some stochastic partial differential equations (spdes). Important issues are not only the existence of stochastic self-similarity but also whether a self-similar solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of spdes in a form to which stochastic center manifold theory may be applied, we resolve these issues in this class. For definiteness, a first example of self-similarity in solutions of Burgers equation driven by some stochastic force is studied. Under suitable assumptions a stationary solution is constructed which yields the existence of a stochastically self-similar solution for the stochastic Burgers equation. Further, the asymptotic convergence to the self-similar solution is proved. Second, in more general stochastic reaction-diffusion systems, stochastic center manifold theory provides a framework for constructing s...

A critical exponent of Fujita type for a nonlinear reaction–diffusion system on a Riemannian manifold

In this paper, we study the global existence and nonexistence of positive solutions to the following nonlinear reaction–diffusion system {ut−Δu=W(x)vp+S(x)in Mn×(0,∞),vt−Δv=F(x)ud+G(x)in Mn×(0,∞),u(x,0)=u0(x)in Mn,v(x,0)=v0(x)in Mn, where Mn(n≥3) is a non-compact complete Riemannian manifold, Δ is the Laplace–Beltrami operator, and S(x),G(x) are non-negative Lloc1 functions. We assume that both u0(x) and v0(x) are non-negative, smooth and bounded functions, and constants p,d>1. When p=d, there is an exponent p∗ which is critical in the following sense. When p∈(1,p∗], the above problem has no global positive solution for any non-negative constants S(x),G(x) not identically zero. When p∈[p∗,∞), the problem has a global positive solution for some S(x),G(x)>0 and u0(x),v0(x)≥0.

Geometry sensing by self-organized protein patterns

In the living cell, proteins are able to organize space much larger than their dimensions. In return, changes of intracellular space can influence biochemical reactions, allowing cells to sense their size and shape. Despite the possibility to reconstitute protein self-organization with only a few purified components, we still lack knowledge of how geometrical boundaries affect spatiotemporal protein patterns. Following a minimal systems approach, we used purified proteins and photolithographically patterned membranes to study the influence of spatial confinement on the self-organization of the Min system, a spatial regulator of bacterial cytokinesis, in vitro. We found that the emerging protein pattern responds even to the lateral, two-dimensional geometry of the membrane such that, as in the three-dimensional cell, Min protein waves travel along the longest axis of the membrane patch. This shows that for spatial sensing the Min system does not need to be enclosed in a three-dimensional compartment. Using a computational model we quantitatively analyzed our experimental findings and identified persistent binding of MinE to the membrane as requirement for the Min system to sense geometry. Our results give insight into the interplay between geometrical confinement and biochemical patterns emerging from a nonlinear reaction-diffusion system.

Null controllability of a degenerated reaction–diffusion system in cardiac electro-physiology

This Note is devoted to the analysis of the null controllability of a nonlinear reaction–diffusion system, approximating a parabolic–elliptic system, modeling electrical activity in the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by a single control force acting on a subdomain is shown. The proof needs a precise estimate with respect to the degenerating parameter and it is done combining Carleman estimates and energy inequalities.

A PDE model of clonal plant competition with nonlinear diffusion

We present a spatial competition model for clonal plant growth that combines two different mechanisms of competition. The first one is described by the standard underlying Kolmogorov Model for two interacting populations. A second competition mechanism, more specific to clonal plant growth, expresses the motility of each species and their capacity to resist to competitor's space intrusion. This model leads to a degenerated nonlinear reaction–diffusion system in which the diffusion coefficient of each species vanishes wherever the other species is beyond a certain biomass value.Indeed, we show that pattern forming instability cannot occur in general competition systems if the exclusion ability of both species is small even if the diffusions are degenerated. The effect of diffusion degeneracy on patterns formation is carried out. Numerical simulations of this two-level competition model are performed when the reaction terms are given by the competitive Lotka–Volterra equations. We, finally, discuss the potential of such nonlinear reaction–diffusion systems to be a surrogate model for phalanx–guerilla species competition.