System Of Reaction-diffusion Equations Research Articles

A Variational Approach to Morphogenesis : Recovering Spatial Phenotypic Features from Epigenetic Landscapes.

We model the process of cell fate determination of the flower Arabidopsis-thaliana employing a system of reaction-diffusion equations governed by a potential field. This potential field mimics the flower's epigenetic landscape as defined by Waddington. It is derived from the underlying genetic regulatory network (GRN), which is based on detailed experimental data obtained during cell fate determination in the early stages of development of the flower. The system of equations has a variational structure, and we use minimax techniques (in particular the Mountain Pass Lemma) to show that the minimal energy solution of our functional is, in fact, the one that traverses the epigenetic landscape (the potential field) in the spatial order that corresponds to the correct architecture of the flower, that is, following the observed geometrical features of the meristem. This approach can generally be applied to systems with similar structures to establish a genotype to phenotype correspondence. From a broader perspective, this problem is related to phase transition models with a multiwell vector potential, and the results and methods presented here can potentially be applied in this case.

Strong convergence of attractors of reaction-diffusion system with rapidly oscillating terms in an orthotropic porous medium

A system of reaction-diffusion equations in a perforated domain with rapidly oscillating terms in the equations and in the boundary conditions is considered. It is not assumed that the uniqueness theorem conditions are satisfied for the corresponding initial-boundary value problem. We have proved the strong convergence of the trajectory attractors of this system to the trajectory attractors of the homogenized reaction-diffusion system with a ‘strange term’ (potential).

Analysis of a model of the Calvin cycle with diffusion of ATP

<p style='text-indent:20px;'>The dynamics of a mathematical model of the Calvin cycle, which is part of photosynthesis, is analysed. Since diffusion of ATP is included in the model a system of reaction-diffusion equations is obtained. It is proved that for a suitable choice of parameters there exist spatially inhomogeneous positive steady states, in fact infinitely many of them. It is also shown that all positive steady states, homogeneous and inhomogeneous, are nonlinearly unstable. The only smooth steady state which could be stable is a trivial one, where all concentrations except that of ATP are zero. It is found that in the spatially homogeneous case there are steady states with the property that the linearization about that state has eigenvalues which are not real, indicating the presence of oscillations. Numerical simulations exhibit solutions for which the concentrations are not monotone functions of time.</p>

Wave blocking in a bistable system by local introduction of a population: application to sterile insect techniques on mosquito populations

The Sterile Insect Technique (SIT) is a classic vector control method that has been successfully applied to fight against diverse insect plagues since the 1950s. In recent years, this strategy has been used to control mosquito populations, in order to limit the spread of the diseases they transmit. In this paper, we consider a system of reaction-diffusion equations to model the mosquito population and study the effect of the release of sterile mosquito males. Then, we propose to analyze the release in a limited area inside a wider area containing a natural mosquito population. We are interested in protecting a mosquito free region from invasion by mosquitoes from an exterior domain by controlling the population in a release band at the border between the two regions: we construct a barrier blocking the invasion of mosquitoes from the exterior. We adapt the geometric method of Lewis and Keener (see Lewis and Keener [SIAM J. Appl. Math. 61 (2000) 293-316]) in this framework and extend their main result to find relations on the size of the release region and the density of the released sterile males that allow us to block the invasion. Numerical simulations are also performed to illustrate our results.

Blow-up Properties of a Coupled System of Reaction-Diffusion Equations

This paper is concerned with a Coupled Reaction-diffusion system defined in a ball with homogeneous Dirichlet boundary conditions. Firstly, we studied the blow-up set showing that, under some conditions, the blow-up in this problem occurs only at a single point. Secondly, under some restricted assumptions on the reaction terms, we established the upper (lower) blow-up rate estimates. Finally, we considered the Ignition system in general dimensional space as an application to our results.

Finite-time boundary stabilization of the reaction-diffusion system with switching time-delay input

The paper is devoted to the study of boundary finite-time control for a reaction-diffusion (RD) system with switching time-delayed input. The RD system with switching time-delay input is converted to a switching system of RD equation cascaded with a transport equation with non-delay boundary input. Next, a novel switching controller is designed for the cascaded RD-transport system based on the backstepping technique, and this causes the closed-loop system to be convergence in a finite-time. Simulation results are provided to exhibit the effectiveness of the proposed method.

Virus-Host Interactions Shape Viral Dispersal Giving Rise to Distinct Classes of Traveling Waves in Spatial Expansions

Reaction-diffusion waves have long been used to describe the growth and spread of populations undergoing a spatial range expansion. Such waves are generally classed as either pulled, where the dynamics are driven by the very tip of the front and stochastic fluctuations are high, or pushed, where cooperation in growth or dispersal results in a bulk-driven wave in which fluctuations are suppressed. These concepts have been well studied experimentally in populations where the cooperation leads to a density-dependent growth rate. By contrast, relatively little is known about experimental populations that exhibit density-dependent dispersal. Using bacteriophage T7 as a test organism, we present novel experimental measurements that demonstrate that the diffusion of phage T7, in a lawn of host E. coli, is hindered by steric interactions with host bacteria cells. The coupling between host density, phage dispersal and cell lysis caused by viral infection results in an effective density-dependent diffusion coefficient akin to cooperative behavior. Using a system of reaction-diffusion equations, we show that this effect can result in a transition from a pulled to pushed expansion. Moreover, we find that a second, independent density-dependent effect on phage dispersal spontaneously emerges as a result of the viral incubation period, during which phage is trapped inside the host unable to disperse. Additional stochastic agent-based simulations reveal that lysis time dramatically affects the rate of diversity loss in viral expansions. Taken together, our results indicate both that bacteriophage can be used as a controllable laboratory population to investigate the impact of density-dependent dispersal on evolution, and that the genetic diversity and adaptability of expanding viral populations could be much greater than is currently assumed.

Signaling gradients in surface dynamics as basis for planarian regeneration.

Based on experimental data, we introduce and analyze a system of reaction-diffusion equations for the regeneration of planarian flatworms. We model dynamics of head and tail cells expressing positional control genes that translate into localized signals which in turn guide stem cell differentiation. Tissue orientation and positional information are encoded in a long range wnt-related signaling gradient. Our system correctly reproduces typical cut and graft experiments, and improves on previous models by preserving polarity in regeneration over orders of magnitude in body size during growth phases. Key to polarity preservation in our model flatworm is the sensitivity of cell differentiation to gradients of wnt-related signals relative to the tissue surface. This process is particularly relevant in small tissue layers close to cuts during their healing, and modeled in a robust fashion through dynamic boundary conditions.

Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications

Abstract We analyze Hopf bifurcation and its properties of a class of system of reaction-diffusion equations involving two discrete time delays. First, we discuss the existence of periodic solutions of this class under Neumann boundary conditions, and determine the required conditions on parameters of the system at which Hopf bifurcation arises near equilibrium point. Bifurcation analysis is carried out by choosing one of the delay parameter as a bifurcation parameter and fixing the other in its stability interval. Second, some properties of periodic solutions such as direction of Hopf bifurcation and stability of bifurcating periodic solution are studied through the normal form theory and the center manifold reduction for functional partial differential equations. Moreover, an algorithm is developed in order to determine the existence of Hopf bifurcation (and its properties) of variety of system of reaction-diffusion equations that lie in the same class. The benefit of this algorithm is that it puts a very complex and long computations of existence of Hopf bifurcation for each equation in that class into a systematic schema. In other words, this algorithm consists of the conditions and formulae that are useful for completing the existence analysis of Hopf bifurcation by only using coefficients in the characteristic equation of the linearized system. Similarly, it is also useful for determining the direction analysis of the Hopf bifurcation merely by using the coefficients of the second degree Taylor polynomials of functions in the right hand side of the system. Finally, the existence of Hopf bifurcation for three different problems whose governing equations stay in that class is given by utilizing the algorithm derived, and thus the feasibility of the algorithm is presented.

Cardiac memory phenomenon, time-fractional order nonlinear system and bidomain-torso type model in electrocardiology

Cardiac memory, also known as the Chatterjee phenomenon, refers to the persistent but reversible T-wave changes on ECG caused by an abnormal electrical activation pattern. After a period of abnormal ventricular activation in which the myocardial repolarization is altered and delayed (such as with artificial pacemakers, tachyarrhythmias with wide QRS complexes or ventricular pre-excitation), the heart remembers and mirrors its repolarization in the direction of the vector of "abnormally" activated QRS complexes. This phenomenon alters patterns of gap junction distribution and generates changes in repolarization seen at the level of ionic-channel, ionic concentrations, ionic-current gating and action potential. In this work, we propose a mathematical model of cardiac electrophysiology which takes into account cardiac memory phenomena. The electrical activity in heart through torso, which is dependent on the prior history of accrued heartbeats, is mathematically modeled by a modified bidomain system with time fractional-order dynamics (which are used to describe processes that exhibit memory). This new bidomain system, that I name "<i>it memory bidomain system</i>", is a degenerate nonlinear coupled system of reaction-diffusion equations in shape of a fractional-order differential equation coupled with a set of time fractional-order partial differential equations. Cardiac memory is represented via fractional-order capacitor (associate to capacitive current) and fractional-order cellular membrane dynamics. First, mathematical model is introduced. Afterward, results on generalized Gronwall inequality within the framework of coupled fractional differential equations are developed. Next, the existence and uniqueness of solution of state system are proved as well as stability result. Further, some preliminary numerical applications are performed to show that memory reproduced by fractional-order derivatives can play a significant role on key dependent electrical properties including APD, action potential morphology and spontaneous activity.

Existence and stability traveling wave solutions for a system of social outbursts

We study the existence and stability of planar traveling wave solutions to the system of reaction-diffusion equations:{ut=∂2u∂x2+Γu(1−u)1+e−β(v−1)−u,x∈Rn,t>0,vt=∂2v∂x2−v(1+u)p+1,x∈Rn,t>0, which was introduced in [8] as a model for the spatiotemporal dynamics of rioting activity and social tension. The “wave-like” spread of rioting activity observed in the 2005 French Riots [12] motivates the study of these type of solutions, which were numerically analyzed in [46]. Of particular interest is the qualitative difference in traveling wave solutions between the p<0 and p≥0 case. We present results for the existence and stability of these type of solutions for p≥0 and provide the asymptotic approach rates of the traveling wave solutions to the steady states as |x|→∞. For the case of p<0 we study the stability of such solutions using Evans Functions and a combination of analytical and numerical methods to show irrefutable evidence of spectral stability.

A moving grid finite element method applied to a mechanobiochemical model for 3D cell migration

This work presents the development, analysis and numerical simulations of a biophysical model for 3D cell deformation and movement, which couples biochemical reactions and biomechanical forces. We propose a mechanobiochemical model which considers the actin filament network as a viscoelastic and contractile gel. The mechanical properties are modelled by a force balancing equation for the displacements, the pressure and contractile forces are driven by actin and myosin dynamics, and these are in turn modelled by a system of reaction-diffusion equations on a moving cell domain. The biophysical model consists of highly non-linear partial differential equations whose analytical solutions are intractable. To obtain approximate solutions to the model system, we employ the moving grid finite element method. The numerical results are supported by linear stability theoretical results close to bifurcation points during the early stages of cell migration. Numerical simulations exhibited show both simple and complex cell deformations in 3-dimensions that include cell expansion, cell protrusion and cell contraction. The computational framework presented here sets a strong foundation that allows to study more complex and experimentally driven reaction-kinetics involving actin, myosin and other molecular species that play an important role in cell movement and deformation.

Mathematical modeling of plant cell fate transitions controlled by hormonal signals.

Coordination of fate transition and cell division is crucial to maintain the plant architecture and to achieve efficient production of plant organs. In this paper, we analysed the stem cell dynamics at the shoot apical meristem (SAM) that is one of the plant stem cells locations. We designed a mathematical model to elucidate the impact of hormonal signaling on the fate transition rates between different zones corresponding to slowly dividing stem cells and fast dividing transit amplifying cells. The model is based on a simplified two-dimensional disc geometry of the SAM and accounts for a continuous displacement towards the periphery of cells produced in the central zone. Coupling growth and hormonal signaling results in a nonlinear system of reaction-diffusion equations on a growing domain with the growth rate depending on the model components. The model is tested by simulating perturbations in the level of key transcription factors that maintain SAM homeostasis. The model provides new insights on how the transcription factor HECATE is integrated in the regulatory network that governs stem cell differentiation.

Mathematical model of signal propagation in excitable media

<p style='text-indent:20px;'>This article deals with a model of signal propagation in excitable media based on a system of reaction-diffusion equations. Such media have the ability to exhibit a large response in reaction to a small deviation from the rest state. An example of such media is the nerve tissue or the heart tissue. The first part of the article briefly describes the origin and the propagation of the cardiac action potential in the heart. In the second part, the mathematical properties of the model are discussed. Next, the numerical algorithm based on the finite difference method is used to obtain computational studies in both a homogeneous and heterogeneous medium with an emphasis on interactions of the propagating signals with obstacles in the medium.

Steady-state solutions of one-dimensional competition models in an unstirred chemostat via the fixed point index theory

AbstractThe existence and nonexistence of semi-trivial or coexistence steady-state solutions of one-dimensional competition models in an unstirred chemostat are studied by establishing new results on systems of Hammerstein integral equations via the classical fixed point index theory. We provide three ranges for the two parameters involved in the competition models under which the models have no semi-trivial and coexistence steady-state solutions or have semi-trivial steady-state solutions but no coexistence steady-state solutions or have semi-trivial or coexistence steady-state solutions. It remains open to find the largest range for the two parameters under which the models have only coexistence steady-state solutions. We apply the new results on systems of Hammerstein integral equations to obtain results on steady-state solutions of systems of reaction-diffusion equations with general separated boundary conditions. Such type of results have not been studied in the literature. However, these results are very useful for studying the competition models in an unstirred chemostat. Our results on Hammerstein integral equations and differential equations generalize and improve some previous results.

Superconvergence analysis of two-grid methods for bacteria equations

In this paper, two-grid methods (TGMs) are developed for a system of reaction-diffusion equations of bacterial infection with initial and boundary conditions. The backward Euler (B-E) and Crank–Nicolson (C-N) fully discrete schemes are established, and the existence and uniqueness of the solutions of these schemes are proved. Moreover, based on the combination technique of the interpolation and Ritz projection and derivative transfer trick which are important ingredients in the TGMs, the superclose estimates of order O(h2 + H4 + τ) and O(h2 + H4 + τ2) in H1-norm are deduced for the above schemes, respectively, where h is fine mesh size, H is coarse mesh size, and τ is time step size. Then, by the interpolated postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some other popular finite elements are discussed and numerical results are provided to verify the theoretical analysis, which show that the computing cost of TGMs are only half of Galerkin finite element methods (FEMs) for the test problem.

Prescribed-time tracking for triangular systems of reaction-diffusion PDEs

Approximate controllability of systems of coupled parabolic partial differential equations has been of interest for a few decades, where the existence of open-loop control laws performing approximate state transitions within a finite time is studied. In this work, we specialize to systems of reaction-diffusion equations where the connectivity structure is triangular in the reaction parameters and the controls appear at the boundary. We first generate controllers by combining a decoupling backstepping approach with differential flatness that allow us to generate admissible trajectories for system outputs from a given initial condition. As a byproduct of our approach, we achieve approximate state transitioning for the system within a finite terminal time. We enhance our control law by introducing time-varying error feedback controllers which reject variations in initial conditions within the terminal time. The resulting control law not only performs the approximate control task but also output trajectory tracking, all within the terminal time which can be prescribed independently of initial conditions.

Stability of Traveling Waves for Systems of Reaction-Diffusion Equations with Multiplicative Noise

We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable traveling wave solutions to the deterministic system ret...

Viruses competition in the genotype space

This paper is devoted to the study of persistence and evolution of two viruses taking into account virus mutation, reproduction, and genotype dependent mortality, either natural or determined by an antiviral treatment. The model describes the virus density distribution u(x; t) for the first virus and v(y; t) for the second one as functions of genotypes x and y considered as continuous variables and of time t. The model consists of a system of reaction-diffusion equations with integral terms characterizing virus competition for host cells. The analysis of the model shows the conditions of the existence of virus strains.

Dynamical stabilization and traveling waves in integrodifference equations

Integrodifference equations are discrete-time analogues of reaction-diffusion equations and can be used to model the spatial spread and invasion of non-native species. They support solutions in the form of traveling waves, and the speed of these waves gives important insights about the speed of biological invasions. Typically, a traveling wave leaves in its wake a stable state of the system. Dynamical stabilization is the phenomenon that an unstable state arises in the wake of such a wave and appears stable for potentially long periods of time, before it is replaced with a stable state via another transition wave. While dynamical stabilization has been studied in systems of reaction-diffusion equations, we here present the first such study for integrodifference equations. We use linear stability analysis of traveling-wave profiles to determine necessary conditions for the emergence of dynamical stabilization and relate it to the theory of stacked fronts. We find that the phenomenon is the norm rather than the exception when the non-spatial dynamics exhibit a stable two-cycle.