Zero-flux Boundary Conditions Research Articles

Implicit–explicit methods for models for vertical equilibrium multiphase flow

In this paper we provide a derivation of a 1D N-phase flow model in a porous medium under the condition of vertical equilibrium which generalizes the three-phase flow model developed in Guerrero et al. (2013). We identify, from a mathematical point of view, a set of conditions on the capillary pressures that ensure that the linearized, purely parabolic, initial value problem is well-posed.For the numerical simulation of the model, we advocate the use of Implicit–Explicit (IMEX)–Runge–Kutta (RK) schemes for time evolution with a Weighted-Essentially Non-Oscillatory (WENO) spatial discretization of the convective terms. In previous papers (Donat et al., 2013), (Guerrero et al., 2013), we showed that IMEX–RK methods can be useful in the numerical simulation of 1D two-phase and three-phase flows, since the stability restrictions for the time-step of these schemes are less severe than those of fully explicit schemes. On the other hand, their implementation requires, in general, a fairly intensive use of a nonlinear system solver, so that the efficiency and robustness of IMEX schemes for multi-phase flow is directly related to this nonlinear technique.In this paper we describe an efficient nonlinear system solver, based on an appropriate fixed-point iteration technique, in order to find the solution of the nonlinear systems that result from the implicit discretization of the nonlinear diffusive terms in the model. In addition, we implement zero-flux boundary conditions fully consistent with the vertical equilibrium assumptions.A set of numerical examples confirm the efficiency, robustness and reliability of the proposed numerical technique.

Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant

This paper deals with the coupled chemotaxis–haptotaxis model{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,0=Δv+u−v,x∈Ω,t>0,wt=−vw+ηw(1−w−u),x∈Ω,t>0, which was initially proposed by Chaplain and Lolas (2006) [10] to describe the interactions between cancer cells, the matrix degrading enzyme and the host tissue in a process of cancer cell invasion of tissue (extracellular matrix). Here, Ω⊂R2 is a bounded domain with smooth boundary, and χ, ξ, μ and η are positive parameters. As compared to previous mathematical studies, the novelty here consists of allowing for positive values of η, reflecting processes of self-remodeling of the extracellular matrix.Under zero-flux boundary conditions, it is shown that for any sufficiently smooth initial data (u0,w0) satisfying first-order compatibility conditions, the model admits a unique global smooth solution. A crucial ingredient in the proof is an energy-like inequality which, given T>0, yields boundedness of u(⋅,t) in LlogL(Ω). This serves as a starting point for a bootstrap argument used to derive higher regularity estimates sufficient for global extensibility of solutions.

Turing pattern formation in a predator–prey system with cross diffusion

The paper explores the impacts of cross-diffusion on the formation of spatial patterns in a ratio-dependent predator–prey system with zero-flux boundary conditions. Our results show that under certain conditions, cross-diffusion can trigger the emergence of spatial patterns which is however impossible under the same conditions when cross-diffusion is absent. We give a rigorous proof that the model has at least one spatially heterogenous steady state by means of the Leray–Schauder degree theory. In addition, numerical simulations are performed to visualize the complex spatial patterns.

Spatiotemporal dynamics of reaction–diffusion models of interacting populations

The present investigation deals with the necessary conditions for Turing instability with zero-flux boundary conditions that arise in a ratio-dependent predator–prey model involving the influence of logistic population growth in prey and intra-specific competition among predators described by a system of non-linear partial differential equations. The prime objective is to investigate the parametric space for which Turing spatial structure takes place and to perform extensive numerical simulation from both the mathematical and the biological points of view in order to examine the role of diffusion coefficients in Turing instability. Various spatiotemporal distributions of interacting species through Turing instability in two dimensional spatial domain are portrayed and analyzed at length in order to substantiate the applicability of the present model.

Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition

In this paper, we study the spatial-temporal patterns of the solutions to the diffusive non-local Nicholson's blowflies equations with time delay (maturation time) subject to the no flux boundary condition. We establish the existence of both spatially homogeneous periodic solutions and various spatially inhomogeneous periodic solutions by investigating the Hopf bifurcations at the spatially homogeneous steady state. We also compute the normal form on the centre manifold, by which the bifurcation direction and stability of the bifurcated periodic solutions can be determined. The results show that the bifurcated homogeneous periodic solutions are stable, while the bifurcated inhomogeneous periodic solutions can only be stable on the corresponding centre manifold, implying that generically the model can only allow transient oscillatory patterns. Finally, we present some numerical simulations to demonstrate the theoretic results. For these transient patterns, we derive approximation formulas which are confirmed by numerical simulations.

Hopf bifurcation in spatially homogeneous and inhomogeneous autocatalysis models

This paper is concerned with spatially homogeneous and inhomogeneous autocatalysis models with arbitrary order. For the spatially homogeneous model, the existence and stability of Hopf bifurcation surrounding the interior equilibrium are considered. For the model subject to zero-flux boundary conditions, Turing instability of the interior equilibrium is firstly discussed and Turing instability region regarding the diffusion coefficients is established. Then by the center manifold theory and normal form method, the existence of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained, which is much more difficult and complicated than dealing with the spatially homogeneous case. Finally, to verify our analytical results, some examples of numerical simulations are also included.

Spatiotemporal dynamics in a reaction–diffusion toxic-phytoplankton–zooplankton model

In this paper, we investigate the complex dynamics of a spatial toxic-phytoplankton–zooplankton model with Holling type-II functional response. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for local and global stability and for diffusive instability with zero-flux boundary conditions of a positive equilibrium. On the basis of these results, we present the evolutionary processes of pattern formation that involve organism distribution and the interaction of a spatially distributed population with local diffusion. Then, novel numerical evidence of the time evolution of patterns controlled by diffusion and environmental carrying capability in the model are presented, and it is found that the model dynamics exhibits complex pattern replication. Additional studies reveal that pattern formation in the spatially extended model depends on the choice of the initial conditions. These results indicate that the interaction between toxin-producing phytoplankton and zooplankton in marine environments may be partly driven by the forces of diffusion or the environmental carrying capability.

Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients

In this paper we consider the initial boundary value problem for strongly degenerate parabolic equations with discontinuous coefficients. This equation has the both properties of parabolic equation and hyperbolic equation. Moreover, approximate solutions for this equation may not belong to $BV$. These are difficult points for this type of equations.  &nbsp We consider the type of equations under the zero-flux boundary conditions. In particular, we prove the existence and partial uniqueness of weak solutions to such problems. Our proof use the compactness theorem derived by Panov [14] and the estimate of degenerate diffusion term derived by Karlsen-Risebro-Towers [10].

Towards computational modelling of the human foetal electrocardiogram: normal sinus rhythm: AV conduction block and re-entrant tachycardia

Introduction: The 10-100 μV foetal electrocardiogram can be recorded from the maternal abdominal surface from ~12 weeks gestational age (WGA), and RR, PR, QR and QT intervals, and T and P wave dispersions extracted. These non-invasive measures provide quantitative information on sino-atrial node pacemaking rate (via RR), propagation times and velocities (via PR, QT), and ventricular action potential duration and its restitution (QT-RR) during foetal normal sinus rhythm. Arrhythmias are observed in b5% of recordings. Methods: The parameters of biophysically detailed models of adult human cardiac cell electrophysiology, based on voltage clamp, immune-histochemical and mRNA expression data were modified, informed by data from the literature on animal foetal tissue and cardiomyocytes derived from human stem cell lines, to reproduce human foetal RR and QT intervals. One dimensional partial differential equation models of SAN, atrial, AVN and ventricular tissue, with lengths from MRI and intercellular coupling informed by PR and QR intervals, were constructed for hearts from 15–40 WGA. Numerical solutions, with zero-flux boundary conditions, were by the explicit Euler method with time and space steps Δt = 0.005 ms and Δx = 0.1 mm. Results: The 1D model at 15, 30 WGA gave NSR intervals RR: 400, 450; QR: 200, 280 ms. Spatially uniform progressive reduction of {gCa,L and gCa, T} slowed the SAN, and led to AV conduction blocks. Self terminating and persistent re-entry with a period of 340–350 ms within a 1-D ring ventricular tissue model. Conclusion: The preliminary 1-D family of models reproduces the observed timings of fECG intervals from 15–40 WGA, the characteristics of development of foetal AV block, and self terminating ventricular tachycardia.

Entropy formulation of degenerate parabolic equation with zero-flux boundary condition

We consider the general degenerate hyperbolic-parabolic equation: $$u_t + {\rm div} f(u) - \Delta \phi(u) = 0\; {\rm in} Q = (0, T) \times \Omega, \quad T > 0, \quad \Omega \subset \mathbb{R}^N;$$ with initial condition and the zero flux boundary condition. Here \({\phi}\) is a continuous non-decreasing function. Following Burger et al. (J Math Anal Appl 326:108–120, 2007), we assume that f is compactly supported (this is the case in several applications), and we define an appropriate notion of entropy solution. Using vanishing viscosity approximation, we prove existence of entropy solution for any space dimension N ≥ 1 under a partial genuine nonlinearity assumption on f. Uniqueness is shown for the case N = 1, using the idea of Andreianov and Bouhsiss (J Evol Equ 4:273–295, 2004), nonlinear semigroup theory and a specific regularity result for one dimension.

On the Numerical Solutions of One and Two-Stage Model of Carcinogenesis Mutations with Time Delay and Diffusion

In this paper, we focused on numerical solutions of carcinogenesis mutations models that are based on reaction-diffusion systems and Lotka-Volterra food chains. We consider the case with one and two-stages of mutations with appropriate initial conditions and the zero-flux boundary conditions. The main purpose is to construct a stable discretization scheme, which allows much accuracy than those of a standard approach. To this end, we use the spectral method to postprocess numerical solutions for the proposed model by using some classical methods for solving differential equations. The implementation of the algorithm is simple and it does not need to solve the linear or nonlinear system (in case the model is nonlinear). We simulate the one and two-stage carcinogenesis mutations model and compared the results with previously published ones.

Dynamics of a Diffusive Predator-Prey Model with Allee Effect on Predator

The reaction-diffusion Holling-Tanner prey-predator model considering the Allee effect on predator, under zero-flux boundary conditions, is discussed. Some properties of the solutions, such as dissipation and persistence, are obtained. Local and global stability of the positive equilibrium and Turing instability are studied. With the help of the numerical simulations, the rich Turing patterns, including holes, stripes, and spots patterns, are obtained.

Spatiotemporal Complexity of a Leslie-Gower Predator-Prey Model with the Weak Allee Effect

We investigate a diffusive Leslie-Gower predator-prey model with the additive Allee effect on prey subject to the zero-flux boundary conditions. Some results of solutions to this model and its corresponding steady-state problem are shown. More precisely, we give the stability of the positive constant steady-state solution, the refineda prioriestimates of positive solution, and the nonexistence and existence of the positive nonconstant solutions. We carry out the analytical study for two-dimensional system in detail and find out the certain conditions for Turing instability. Furthermore, we perform numerical simulations and show that the model exhibits a transition from stripe-spot mixtures growth to isolated spots and also to stripes. These results show that the impact of the Allee effect essentially increases the model spatiotemporal complexity.

Pattern Formation in a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

The spatiotemporal dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition are investigated analytically and numerically. The asymptotic stability of the positive equilibrium and the existence of Hopf bifurcation around the positive equilibrium are shown; the conditions of Turing instability are obtained. And with the help of numerical simulations, it is found that the model exhibits complex pattern replication: stripes, spots-stripes mixtures, and spots Turing patterns.

Spatiotemporal Pattern in a Self- and Cross-Diffusive Predation Model with the Allee Effect

This paper proposes and analyzes a mathematical model for a predator-prey interaction with the Allee effect on prey species and with self- and cross-diffusion. The effect of diffusion which can drive the model with zero-flux boundary conditions to Turing instability is investigated. We present numerical evidence of time evolution of patterns controlled by self- and cross-diffusion in the model and find that the model dynamics exhibits a cross-diffusion controlled formation growth to spotted and striped-like coexisting and spotted pattern replication. Moreover, we discuss the effect of cross-diffusivity on the stability of the nontrivial equilibrium of the model, which depends upon the magnitudes of the self- and cross-diffusion coefficients. The obtained results show that cross-diffusion plays an important role in the pattern formation of the predator-prey model. It is also useful to apply the reaction-diffusion model to reveal the spatial predation in the real world.

Qualitative Analysis of a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

We investigated the dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition. Some qualitative properties, including the dissipation, persistence, and local and global stability of positive constant solution, are discussed. Moreover, we give the refined a priori estimates of positive solutions and derive some results for the existence and nonexistence of nonconstant positive steady state.

Applications of the kinetic formulation for scalar conservation laws with a zero-flux type boundary condition

The authors are concerned with a zero-flux type initial boundary value problem for scalar conservation laws. Firstly, a kinetic formulation of entropy solutions is established. Secondly, by using the kinetic formulation and kinetic techniques, the uniqueness of entropy solutions is obtained. Finally, the parabolic approximation is studied and an error estimate of order \(\eta ^{\tfrac{1} {3}}\) between the entropy solution and the viscous approximate solutions is established by using kinetic techniques, where η is the size of artificial viscosity.

Computer modelling of haematopoietic stem cells migration

A mathematical model for migration of haematopoietic stem cells towards their niche in the bone marrow has been proposed in the literature. It consists of a chemotaxis system of partial differential equations with nonhomogeneous boundary conditions and an additional ordinary differential equation on a part of the computational boundary. The aim of the current work is to extend appropriately a second order positivity preserving central upwind scheme, originally proposed for a chemotaxis system with zero-flux boundary conditions and to apply it for the numerical solution of the considered problem. This paper introduces a first glance of such modification and outlines open questions in the handling of the nonlinear boundary conditions in a way that preserves the positivity of the solution. The presented numerical tests illustrate the need of the development of new specialized schemes for more complex chemotaxis systems.

Dynamics of a Michaelis–Menten-type predation model incorporating a prey refuge withnoise and external forces

In this paper, we investigate a spatial version of a Michaelis–Menten-type predator–preymodel incorporating a prey refuge, which contains some important factors, such asenvironmental noise, external periodic forces and diffusion processes on both prey andpredator. Based on both mathematical analysis and numerical simulations, we get asufficient condition for diffusion instability with zero-flux boundary conditions. There is anincrease in the equilibrium density of both prey and predator as the prey refuge increasesunder a very restricted set of conditions. It is also proved that the refuge used by prey hasa stabilizing effect, that is, increases the local stability of the interior equilibrium.Moreover, due to the effects of environmental noise, we obtain the transition from regularpatterns to spatiotemporal chaotic dynamics in a corresponding spatially extended model.External periodic forces can induce 1:1 frequency-locking and spatially homogeneousoscillation to appear. Also the interaction between environmental noise and externalperiodic forces in the spatial model incorporating a prey refuge gives rise to a rich dynamicphenomenon.

Complex dynamics of a reaction–diffusion epidemic model

In this paper, we investigate the complex dynamics of a reaction–diffusion S−I model incorporating demographic and epidemiological processes with zero-flux boundary conditions. By the method of Lyapunov function, the global stability of the disease free equilibrium and the epidemic equilibrium was established. In addition, the conditions of Turing instability were obtained and the Turing space in the parameters space were given. Based on these results, we present the evolutionary processes that involves organism distribution and their interaction of spatially distributed population with local diffusion, and find that the model dynamics exhibits a diffusion-controlled formation growth to “holes, holes–stripes, stripes, spots–stripes and spots” pattern replication. Furthermore, we indicate that the diseases’ spread is getting smaller with R0 increasing, and the increasing the diffusion of infectious will increase the speed of diseases spreading. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model.