Reaction Diffusion System Modeling Research Articles

Minimal reaction schemes for pattern formation.

We link continuum models of reaction-diffusion systems that exhibit diffusion-driven instability to constraints on the particle-scale interactions underpinning this instability. While innumerable biological, chemical and physical patterns have been studied through the lens of Alan Turing's reaction-diffusion pattern-forming mechanism, the connections between models of pattern formation and the nature of the particle interactions generating them have been relatively understudied in comparison with the substantial efforts that have been focused on understanding proposed continuum systems. To derive the necessary reactant combinations for the most parsimonious reaction schemes, we analyse the emergent continuum models in terms of possible generating elementary reaction schemes. This analysis results in the complete list of such schemes containing the fewest reactions; these are the simplest possible hypothetical mass-action models for a pattern-forming system of two interacting species.

Solving Model of Reaction-Diffusion System in Ecological by Symmetry Lie Group Methods

In this paper, we will use analytical methods for solving the fundamental element of a reaction diffusion model in ecology. The model denotes an interaction between two species which describe ecological predations, and mathematically is defined as a system of partial differential equations with initial and boundary conditions. Symmetry Lie group methods are used to transform this model into a system of ordinary differential equations, then this system solved by generalized tanh function method when there exists a small parameter appear in one of the equation.

Analysis of a finite-volume scheme for a single-species biofilm model

An implicit Euler finite-volume scheme for a parabolic reaction-diffusion system modeling biofilm growth is analyzed and implemented. The system consists of a degenerate-singular diffusion equation for the biomass fraction, which is coupled to a diffusion equation for the nutrient concentration, and it is solved in a bounded domain with Dirichlet boundary conditions. By transforming the biomass fraction to an entropy-type variable, it is shown that the numerical scheme preserves the lower and upper bounds of the biomass fraction. The existence and uniqueness of a discrete solution and the convergence of the scheme are proved. Numerical experiments in one and two space dimensions illustrate, respectively, the rate of convergence in space of our scheme and the temporal evolution of the biomass fraction and the nutrient concentration.

Longterm existence of solutions of a reaction diffusion system with non-local terms modeling an immune response—An interpretation-orientated proof

This paper shows the global existence and boundedness of solutions of a reaction diffusion system modeling liver infections. The existence proof is presented step by step and the focus lies on the interpretation of intermediate results in the context of liver infections which is modeled. Non-local effects in the dynamics between the virus and the immune system cells coming from the immune response in the lymphs lead to an integro-partial differential equation. While existence theorems for parabolic partial differential equations are textbook examples in the field, the additional integral term requires new approaches to proving the global existence of a solution. This allows to set up an existence proof with a focus on interpretation leading to more insight in the system and in the modeling perspective at the same time.We show the boundedness of the solution in the L1(Ω)- and the L2(Ω)-norms, and use these results to prove the global existence and boundedness of the solution. A core element of the proof is the handling of oppositely acting mechanisms in the reaction term, which occur in all population dynamics models and which results in reaction terms with opposite monotonicity behavior. In the context of modeling liver infections, the boundedness in the L∞(Ω)-norm has practical relevance: Large immune responses lead to strong inflammations of the liver tissue. Strong inflammations negatively impact the health of an infected person and lead to grave secondary diseases. The gained rough estimates are compared with numerical tests.

Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar

A no-flux initial-boundary value problem for the doubly degenrate parabolic system ut=∇·(uv∇u)+ℓuv,vt=Δv-uv,(⋆)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} u_t = \ abla \\cdot \\big ( uv\ abla u\\big ) + \\ell uv, \\\\ v_t = \\Delta v - uv, \\end{array} \\right. \\qquad \\qquad (\\star ) \\end{aligned}$$\\end{document}is considered in a smoothly bounded convex domain Omega subset mathbb {R}^n, with nge 1 and ell ge 0. The first of the main results asserts that for nonnegative initial data (u_0,v_0)in (L^infty (Omega ))^2 with u_0not equiv 0, v_0not equiv 0 and sqrt{v_0}in W^{1,2}(Omega ), there exists a global weak solution (u, v) which, inter alia, belongs to C^0(overline{Omega }times (0,infty )) times C^{2,1}(overline{Omega }times (0,infty )) and satisfies sup _{t>0} Vert u(cdot ,t)Vert _{L^p(Omega )}<infty for all pin [1,p_0) with p_0:=frac{n}{(n-2)_+}. It is next seen that for each of these solutions one can find u_infty in bigcap _{pin [1,p_0)} L^p(Omega ) such that, within an appropriate topological setting, (u(cdot ,t),v(cdot ,t)) approaches the equilibrium (u_infty ,0) in the large time limit. Finally, in the case nle 5 a result ensuring a certain stability property of any member in the uncountably large family of steady states (u_0,0), with arbitrary and suitably regular u_0:Omega rightarrow [0,infty ), is derived. This provides some rigorous evidence for the appropriateness of (star ) to model the emergence of a strikingly large variety of stable structures observed in experiments on bacterial motion in nutrient-poor environments. Essential parts of the analysis rely on the use of an apparently novel class of functional inequalities to suitably cope with the doubly degenerate diffusion mechanism in (star ).

Inferring the dynamics of “black-box” systems using a learning machine

Given a segment of a time series of a system at a particular set of parameter values, is it possible to infer the dynamic behavior of the system in its parameter space? Here, we show that this goal can be achieved to a certain extent using a self-evolution learning machine. It is found that following an appropriate training strategy that monotonously decreases the cost function, the learning machine in different training stages is just like the system at different parameter sets. Consequently, the dynamic properties of the system are, in turn, usually revealed in the simple-to-complex order. The underlying mechanism can be attributed to the training strategy, which results in the learning machine collapsing to a qualitatively equivalent system of the system behind the time series. Thus, the learning machine enables a novel way of probing the dynamic properties of a “black-box” system without artificially establishing the equations of motion. The given illustrative examples include a representative model of low-dimensional nonlinear dynamical systems and a spatiotemporal model of reaction-diffusion systems.

Space-time histories approach to fast stochastic simulation of bimolecular reactions.

Computational models of reaction-diffusion systems involving low copy numbers or strongly heterogeneous molecular spatial distributions, such as those frequently found in cellular signaling pathways, require approaches that account for the stochastic dynamics of individual particles, as opposed to approaches representing them through their average concentrations. Efforts to remedy the high computational cost associated with particle-based stochastic approaches by taking advantage of Green's functions are hampered by the need to draw random numbers from complicated, and therefore costly, non-standard probability distributions to update particle positions. Here, we introduce an approach that permits the reconstruction of entire molecular trajectories, including bimolecular encounters, in retrospect, after a simulated time step, while avoiding inefficient draws from non-standard distributions. This means that highly accurate stochastic simulations can be performed for system sizes that would be prohibitively costly to simulate with conventional Green's function based methods. The algorithm applies equally well to one, two, and three dimensional systems and can be readily extended to include deterministic forces specified by an interaction potential, such as the Coulomb potential.

Mathematical analysis of a tumor invasion model—global existence and stability

This work studies an outstanding reaction–diffusion system modeling tumor invasion, with interactions among tumor tissue, acid concentration and normal tissue. This model has very different features from the models extensively studied in the mathematics literature. The most challenge issue for mathematical analysis of the present model is the existence of classical solution, since the diffusion of tumor tissue is influenced by the density of normal cells and diffusion degeneracy arises when normal cells are at the carrying capacity. A rigorous proof of global existence and uniqueness of classical solutions is presented. Moreover, we study global dynamics of the solution, and show asymptotic stability of the four possible constant equilibria under various scenarios.

Dynamic behaviors of a predator–prey model with weak additive Allee effect on prey

This paper deals with a reaction–diffusion system modeling predator–prey interaction with additive Allee effect. We first analyze nonnegative constant equilibrium solutions and study the stability of the unique positive solution. Then we investigate the steady state bifurcation and Hopf bifurcation from the unique positive constant solution, respectively. The main tools used here include the bifurcation theory, the space decomposition and the implicit function theorem. Moreover, we describe the global structure from a simple eigenvalue. Finally, for the case d1=0, we prove the existence of far-from-equilibrium steady states with jump discontinuities. Moreover, by applying the singular perturbation method, we give a proof of the existence of large amplitude solutions when d1 is sufficiently small.

Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers.

The known three-component reaction–diffusion system modeling competition and co-existence of different language speakers is under study. A modification of this system is proposed, which is examined by the Lie symmetry method; furthermore, exact solutions in the form of traveling fronts are constructed and their properties are identified. Plots of the traveling fronts are presented and the relevant interpretation describing the language shift that has occurred in Ukraine during the Soviet times is suggested.

Global dynamics of a reaction-diffusion system with intraguild predation and internal storage

This paper presents a reaction-diffusion system modeling interactions of the intraguild predator and prey in an unstirred chemostat, in which the predator can also compete with its prey for one single nutrient resource that can be stored within individuals. Under suitable conditions, we first show that there are at least three steady-state solutions for the full system, a trivial steady-state solution with neither species present, and two semitrivial steady-state solutions with just one of the species. Then we establish that coexistence of the intraguild predator and prey can occur if both of the semitrivial steady-state solutions are invasible by the missing species. Comparing with the system without predation, our numerical simulations show that the introduction of predation in an ecosystem can enhance the coexistence of species. Our mathematical arguments also work for the linear food chain model (top-down predation), in which the top-down predator only feeds on the prey but does not compete for nutrient resource with the prey. In our numerical studies, we also do a comparison of intraguild predation and top-down predation.

Spiral waves with interfacial oscillatory chemical reactions emerge in a model of reaction-diffusion systems

We propose a model for describing oscillatory chemical reactions involving a variety of immiscible solvents. This model describes the interaction between chemical oscillations on both sides of the interface of different solvents and the effect of the interface on chemical oscillations. Based on this model, we study the spiral chemical wave in a chemical oscillation with interfaces of different solvents, and report a new type of spiral wave. The interface does not hinder the propagation of chemical waves, which can continue to propagate through the interface. The frequency of the chemical wave is increased after the chemical wave passed through the interface. The work of this paper enriched the categories of the dynamics of pattern formation. We believe that the results of this study can provide direction and guidance for chemical experimental research about interfacial oscillatory chemical reactions.

Turing patterns in a reaction–diffusion system modeling hunting cooperation

A reaction–diffusion system governing the prey–predator interaction with hunting cooperation is investigated. Definitive boundedness of solutions is proved via the existence of positive invariants and attractive sets. Linear stability of the coexistence equilibria is performed and conditions guaranteeing the occurrence of Turing instability are found. Numerical simulations on the obtained results are provided.

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.

Asynchronous Instabilities of Crime Hotspots for a 1-D Reaction-Diffusion Model of Urban Crime with Focused Police Patrol

We analyze the existence and linear stability of steady-state localized hotspot patterns for a 1-D three-component singularly perturbed reaction-diffusion (RD) system modeling urban crime in the pr...

An efficient hybrid method for stochastic reaction-diffusion biochemical systems with delay

Many chemical reactions, such as gene transcription and translation in living cells, need a certain time to finish once they are initiated. Simulating stochastic models of reaction-diffusion systems with delay can be computationally expensive. In the present paper, a novel hybrid algorithm is proposed to accelerate the stochastic simulation of delayed reaction-diffusion systems. The delayed reactions may be of consuming or non-consuming delay type. The algorithm is designed for moderately stiff systems in which the events can be partitioned into slow and fast subsets according to their propensities. The proposed algorithm is applied to three benchmark problems and the results are compared with those of the delayed Inhomogeneous Stochastic Simulation Algorithm. The numerical results show that the new hybrid algorithm achieves considerable speed-up in the run time and very good accuracy.

A reaction diffusion system modeling virus dynamics and CTLs response with chemotaxis

In this paper, we study the effect of chemotactic movement of CTLs on HIV-1 infection dynamics by a reaction diffusion system with chemotaxis. Choosing a typical chemosensitive function, we find that chemoattractive movement of CTLs due to HIV infection does not change stability of a positive steady state of the model, meaning that the stability/instability of the model remains the same as in the model without spatial effect. However, chemorepulsion movement of CTLs can destabilize the positive steady state as the strength of the chemotactic sensitivity increases. In this case, Turing instability occurs, which may result in Hopf bifurcation or steady state bifurcation, and spatial inhomogeneous pattern forms.

Approximate Solutions for A Model of Reaction-Diffusion System with Slow Reaction and Fast Diffusion

In this paper, perturbation and finite difference methods are used to solve a reaction diffusion system. This system is modeled for describing the interaction between species in ecology. The interaction is interpreted as traveling wave solutions for both species under three types of initial conditions which describe some ecological cases. Types of traveling wave solutions are found and studied using numerical and approximate methods when there exists a small parameter appear in one of the equation. The solutions of the two methods are compared and show a good agreement.

Competitive exclusion and coexistence of a delayed reaction–diffusion system modeling two predators competing for one prey

In this work, we study a time delayed reaction–diffusion system with homogeneous Neumann boundary conditions. This system describes two predators competing for the same prey. By the method of upper and lower solutions, we obtain sufficient conditions for the competitive exclusion principle to hold and sufficient conditions of the global asymptotic stability of positive constant solution. By taking time delay as the bifurcation parameter, spatially homogeneous and inhomogeneous Hopf bifurcation at the positive constant solution are proved to occur for a sequence of critical values of the delay parameter. It is shown that there are three coexistence forms for the three species: steady states, spatial homogeneous and inhomogeneous periodic oscillations.

Explicitly Solvable Nonlocal Eigenvalue Problems and the Stability of Localized Stripes in Reaction‐Diffusion Systems

The transverse stability of localized stripe patterns for certain singularly perturbed two‐component reaction‐diffusion (RD) systems in the asymptotic limit of a large diffusivity ratio is analyzed. In this semi‐strong interaction regime, the cross‐sectional profile of the stripe is well‐approximated by a homoclinic pulse solution of the corresponding 1‐D problem. The linear instability of such homoclinic stripes to transverse perturbations is well known from numerical simulations to be a key mechanism for the creation of localized spot patterns. However, in general, owing to the difficulty in analyzing the associated nonlocal and nonself‐adjoint spectral problem governing stripe stability for these systems, it has not previously been possible to provide an explicit analytical characterization of these instabilities, including determining the growth rate and the most unstable mode within the band of unstable transverse wave numbers. Our focus is to show that such an explicit characterization of the transverse instability of a homoclinic stripe is possible for a subclass of RD system for which the analysis of the underlying spectral problem reduces to the study of a rather simple algebraic equation in the eigenvalue parameter. Although our simplified theory for stripe stability can be applied to a class of RD system, it is illustrated only for homoclinic stripe and ring solutions for a subclass of the Gierer–Meinhardt model and for a three‐component RD system modeling patterns of criminal activity in urban crime.