System Of Reaction-diffusion Equations Research Articles

Analysis of a nutrient-phytoplankton model in the presence of viral infection

In this paper, a system of reaction-diffusion equations arising in a nutrient-phytoplankton populations is investigated. The equations model a situation in which phytoplankton population is divided into two groups, namely susceptible phytoplankton and infected phytoplankton. A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given. If the diffusion coefficient of the infected phytoplankton is treated as bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.

Flat bones and sutures formation in the human cranial vault during prenatal development and infancy: A computational model

The processes of flat bones growth, sutures formation and interdigitation in the human calvaria are controlled by a complex interaction between genetic, biochemical and environmental factors that regulate bone formation and resorption during prenatal development and infancy. Despite previous experimental evidence accounting for the role of the main biochemical factors acting on these processes, the underlying mechanisms controlling them are still unknown. Therefore, we propose a mathematical model of the processes of flat bone and suture formation, taking into account several biological events. First, we model the growth of the flat bones and the formation of sutures and fontanels as a reaction diffusion system between two proteins: TGF-β2 and TGF-β3. The former is expressed by osteoblasts and allows adjacent mesenchymal cells differentiation on the bone fronts of each flat bone. The latter is expressed by mesenchymal cells at the sutures and inhibits their differentiation into osteoblasts at the bone fronts. Suture interdigitation is modelled using a system of reaction diffusion equations that develops spatio-temporal patterns of bone formation and resorption by means of two molecules (Wnt and Sclerostin) which control mesenchymal cells differentiation into osteoblasts at these sites. The results of the computer simulations predict flat bone growth from ossification centers, sutures and fontanels formation as well as bone formation and resorption events along the sutures, giving rise to interdigitated patterns. These stages were modelled and solved by the finite elements method. The simulation results agree with the morphological characteristics of calvarial bones and sutures throughout human prenatal development and infancy.

Invasion Generates Periodic Traveling Waves (Wavetrains) in Predator-Prey Models with Nonlocal Dispersal

Periodic Traveling waves (wavetrains) have been studied extensively in systems of reaction-diffusion equations. An important motivation for this work is the identification of periodic Traveling waves of abundance in ecological data sets. However, for many natural populations diffusion is a poor representation of movement, and spatial convolution with a dispersal kernel is more realistic because of its ability to reflect rare long-distance dispersal events. In marked contrast to the literature on reaction-diffusion systems, there has been almost no previous work on periodic Traveling waves in models with nonlocal dispersal. In this paper the author considers the generation of such waves by the invasion of the unstable coexistence state in cyclic predator-prey systems with nonlocal dispersal for which the dispersal kernel is thin-tailed (exponentially bounded). The main result is formulae for the wave period and amplitude when the parameters of the local population dynamics are close to a Hopf bifurcation p...

Using genetic data to estimate diffusion rates in heterogeneous landscapes.

Having a precise knowledge of the dispersal ability of a population in a heterogeneous environment is of critical importance in agroecology and conservation biology as it can provide management tools to limit the effects of pests or to increase the survival of endangered species. In this paper, we propose a mechanistic-statistical method to estimate space-dependent diffusion parameters of spatially-explicit models based on stochastic differential equations, using genetic data. Dividing the total population into subpopulations corresponding to different habitat patches with known allele frequencies, the expected proportions of individuals from each subpopulation at each position is computed by solving a system of reaction-diffusion equations. Modelling the capture and genotyping of the individuals with a statistical approach, we derive a numerically tractable formula for the likelihood function associated with the diffusion parameters. In a simulated environment made of three types of regions, each associated with a different diffusion coefficient, we successfully estimate the diffusion parameters with a maximum-likelihood approach. Although higher genetic differentiation among subpopulations leads to more accurate estimations, once a certain level of differentiation has been reached, the finite size of the genotyped population becomes the limiting factor for accurate estimation.

Competitive exclusion and coexistence in a two-strain pathogen model with diffusion.

We consider a two-strain pathogen model described by a system of reaction-diffusion equations. We define a basic reproduction number R0 and show that when the model parameters are constant (spatially homogeneous), if R0 >1 then one strain will outcompete the other strain and drive it to extinction, but if R0 ≤ 1 then the disease-free equilibrium is globally attractive. When we assume that the diffusion rates are equal while the transmission and recovery rates are heterogeneous, then there are two possible outcomes under the condition R0 < 1: 1) Competitive exclusion where one strain dies out. 2) Coexistence between the two strains. Thus, spatial heterogeneity promotes coexistence.

Retinal oxygen distribution and the role of neuroglobin.

The retina is the tissue layer at the back of the eye that is responsible for light detection. Whilst equipped with a rich supply of oxygen, it has one of the highest oxygen demands of any tissue in the body and, as such, supply and demand are finely balanced. It has been suggested that the protein neuroglobin (Ngb), which is found in high concentrations within the retina, may help to maintain an adequate supply of oxygen via the processes of transport and storage. We construct mathematical models, formulated as systems of reaction-diffusion equations in one-dimension, to test this hypothesis. Numerical simulations show that Ngb may play an important role in oxygen transport, but not in storage. Our models predict that the retina is most susceptible to hypoxia in the regions of the photoreceptor inner segment and inner plexiform layers, where Ngb has the potential to prevent hypoxia and increase oxygen uptake by 30-40%. Analysis of a simplified model confirms the utility of Ngb in transport and shows that its oxygen affinity ([Formula: see text] value) is near optimal for this process. Lastly, asymptotic analysis enables us to identify conditions under which the piecewise linear and quadratic approximations to the retinal oxygen profile, used in the literature, are valid.

Pattern formation in a diffusion-ODE model with hysteresis

Coupling diffusion process of signaling molecules with nonlinear interactions of intracellular processes and cellular growth/transformation leads to a system of reaction-diffusion equations coupled with ordinary differential equations (diffusion-ODE models), which differ from the usual reaction-diffusion systems. One of the mechanisms of pattern formation in such systems is based on the existence of multiple steady states and hysteresis in the ODE subsystem. Diffusion tries to average different states and is the cause of spatio-temporal patterns. In this paper, we provide a systematic description of stationary solutions of such systems having the form of transition or boundary layers. The solutions are discontinuous in the case of non-diffusing variables whose quasi-stationary dynamics exhibit hysteresis. The considered model is motivated by biological applications and elucidates a possible mechanism of formation of patterns with sharp transitions.

Pattern Formation in a Coupled Brusselator Model

The Brusselator model is widely used to illustrate and study basic features of models of chemical reactions involving trimolecular steps. A single layer model is a system of reaction diffusion equations of the form ...

Leader in a diffusion competition model

A one-dimensional Cauchy problem is considered for a system of reaction-diffusion equations that, in the point version, generalizes the Volterra competition model. It is proved that the number of the leader in the propagation velocity of nonvanishing solution values at the periphery is independent of nonnegative finite initial distributions.

One Dimensional Finite Element Model to Study the Effect of Diffusion of Free Ca2+ in Hepatocytes

Abstract The main functional cell of the liver is hepatocyte. The frequently working of liver depends on the particular level of cytosolic Ca 2+ in hepatocyte. Cytosolic calcium is a second messenger which responsible for reproduction. Calcium controls many biological processes like movement of cell, secretion, muscular contraction, cell differentiation and buffering processes etc. The calcium diffusion in hepatocyte can be described by the system of reaction-diffusion equations. The main objective of this paper is to study the effect of diffusion of cytosolic calcium concentration in hepatocyte in presence of buffers. A one Dimensional mathematical model is developed in the form of diffusion equation. This model has studied the effect of the important parameters like immobile buffer, diffusion coefficient and influx over [Ca 2+ ] profile on hepatocyte. It is found that diffusion has significant effect in the presence of low affinity buffer compared to high affinity buffer.

Instability of multi-spot patterns in shadow systems of reaction-diffusion equations

Our aim in this paper is to prove the instability of multi-spot patterns in a shadow system, which is obtained as a limiting system of a reaction-diffusion model as one of the diffusion coefficients goes to infinity. Instead of investigating each eigenfunction for a linearized operator, we characterize the eigenspace spanned by unstable eigenfunctions.

Abstract 138: A role for Wnt signaling in regulation of Warburg metabolism in colon cancer

Abstract For most aerobic organisms, oxidative phosphorylation (OXPHOS) is the predominant metabolic pathway for production of cellular energy. Compared to anaerobic metabolism, OXPHOS is much more efficient at producing ATP. However, despite an abundance of environmental oxygen and the induction of angiogenesis by a colon cancer tumor, the predominant metabolic pathway utilized is glycolysis. This paradox is also known as Warburg metabolism. In mouse xenograft tumors grown from SW620 colon cancer cells, partial inhibition of beta-catenin dependent Wnt signaling shifts metabolism of the tumor, where glycolysis markers decrease. We also determined that transcription of the pyruvate dehydrogenase kinase-1 gene (PDK1), is regulated by Wnt signaling. PDK1 protein normally inactivates mitochondrial pyruvate dehydrogenase (PDH) through phosphorylation in order to increase the conversion of pyruvate to lactate in the cytosol. We concluded that Wnt signaling directs Warburg metabolism in colon cancer via regulation of a key regulator of glycolysis. When staining for phosphorylated PDH in the xenograft tumors as a measure of active PDK, we discovered a unique spotted pattern of discrete regions of increased phospho-PDH at regular intervals throughout the xenograft tumor. The pattern was accentuated when Wnt signaling was reduced and it was completely abolished with PDK1 expression. This regular spotted pattern was also seen in beta-catenin expression, suggesting that Wnt signaling may be responsible for establishing the spotted pattern. We hypothesized that this patterning could be modeled mathematically as a Turing pattern. Beta-catenin dependent Wnt signaling and its associated inhibitors have been previously characterized to form Turing patterns, or reaction-diffusion systems, in multiple developmental biology systems. Our collaborative group has developed a system of reaction-diffusion equations that describes the formation of these spots in relation to varying concentrations of Wnt signaling, Wnt inhibitors, and nutrients. Included in the model are equations for glycolytic cells, oxidative cells, Wnt signaling activity, Wnt inhibitors, PDK activity, lactate, HIF, and a general nutrient term. Wnt activity and Wnt inhibitor equations are based on the Gierer-Meinhardt activator-inhibitor model. The tumor cells switch metabolic regimes based on PDK activity level (high activity implies a tendency towards glycolysis, and low activity tends toward oxidative phosphorylation). Using novel imaging techniques and mathematical modeling, we have demonstrated that beta-catenin dependent Wnt signaling regulates expression of PDK1 to drive glycolysis in xenograft tumors. This increased glycolysis exists in a regular Turing pattern throughout the tumor. Our mathematical models will allow us to predict changes to tumor metabolism and behavior in response to modulation of Wnt signaling or external stimuli. Citation Format: George T. Chen, Mary Lee, Kira Pate, Kehui Wang, Robert A. Edwards, John S. Lowengrub, Marian L. Waterman. A role for Wnt signaling in regulation of Warburg metabolism in colon cancer. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 138. doi:10.1158/1538-7445.AM2014-138

Improved Duality Estimates and Applications to Reaction-Diffusion Equations

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.

Pattern Formation in a Reaction-Diffusion System with a Singularity

Systems of reaction-diffusion equations are widely used to model morphogenesis (organisation of forms and patterns in living organisms) in different biological contexts. A particular application describes the formation of animal coat patterns and distribution of structures on the epidermis. The patterns of interest are either an arrangement of local maxima or a stripe-like distribution of chemical concentrations over the domain of interest that persist over time. In mathematical terms, a pattern is a spatially inhomogeneous solution of the reaction-diffusion system which is asymptotically stable. The common feature of systems displaying pattern formation is the presence of a steady state that is asymptotically stable to spatially-homogeneous perturbations but asymptotically unstable to spatially-inhomogeneous perturbations.We study whether it is possible to relax this assumption with an analysis of a reaction-diffusion system with a singularity that is a slight modification of a model for hair follicle spacing . We prove existence of global solutions for the reaction-diffusion system and examine stability of spatially inhomogeneous stationary solutions.

About the Quasi Steady State Approximation for a Reaction Diffusion System Describing a Chain of Irreversible Chemical Reactions

We present here a rigorous proof of the quasi steady state approximation in the context of a system of reaction-diffusion equations coming out of a typical chain of irreversible chemical reactions.

A Class of Mathematical Models Describing Processes in Spatially Heterogeneous Biofilm Communities

We present a class of deterministic continuum models for spatially heterogeneous biofilm communities. The prototype is a single-species biofilm growth model, which is formulated as a highly non-linear system of reaction-diffusion equations for the biomass density and the concentration of the growth controlling substrate. While the substrate concentration satisfies a standard semi-linear reaction-diffusion equation the equation for the biomass density comprises two non-linear diffusion effects: a porous medium-type degeneracy and super diffusion. When further biofilm processes are taken into account equations for several substrates and multiple biomass components have to be included in the model. The structure of these multi-component extensions is essentially different from the mono-species case, since the diffusion operator forthe biomass componentsdepends on the total biomass in the system and the equations are strongly coupled. We present the prototype biofilm growth model and give an overview of its multi-component extensions. Moreover, we summarize analytical results that were obtained for these models.

Traveling wave solutions of competitive models with free boundaries

We study two systems of reaction diffusion equations with monostable or bistable type of nonlinearitiesand with free boundaries. These systems are used as multi-species competitive model.For two-species models, we prove the existence of traveling wave solutions, each of which consistsof two semi-waves intersecting at the free boundary.For three-species models, we also obtain some traveling wave solutions. In this case, however, everytraveling wave solution consists oftwo semi-waves and one compactly supported wave in between, each intersecting with its neighbors at the free boundaries.

Systems of reaction-diffusion equations with spatially distributed hysteresis

We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis on the right-hand side. The input of the hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the assumed discontinuity of hysteresis. These conditions are formulated in terms of geometry of the manifolds defining the hysteresis thresholds and the spatial profile of the initial data.

An Efficient, Nonlinear Stability Analysis for Detecting Pattern Formation in Reaction Diffusion Systems

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, nonlinear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and nonlinear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have nonlinear patterning regimes not previously detected. The real power of this technique, however, is its simplicity and applicability to larger complex systems where other nonlinear methods become intractable. This is demonstrated through analysis of a chemotaxis regulatory network comprised of interacting proteins and phospholipids. In each case, predictions of this method are verified against results of numerical simulation, linear stability, asymptotic, and/or full PDE bifurcation analyses.

Mathematical Modeling of Atherosclerotic Plaque Formation Coupled with a Non-Newtonian Model of Blood Flow

We deal with a mathematical model of atherosclerosis plaque formation, which describes the early formation of atherosclerotic lesions. The model assumes that the inflammatory process starts with the penetration of low-density lipoproteins cholesterol in the intima, and that penetration will occur in the area of lower shear stress. Using a system of reaction-diffusion equations, we first provide a one-dimensional model of lesion growth. Then we perform numerical simulations on an idealized two-dimensional geometry of the carotid artery bifurcation before and after the formation of the atherosclerotic plaque. For that purpose, we consider the blood as an incompressible non-Newtonian fluid with shear-thinning viscosity. We also present a study of the wall shear stress and blood velocity behavior in a geometry with one plaque and also with two plaques in different positions.