Reaction Cross-diffusion Systems Research Articles

Global smooth solutions for triangular reaction-cross diffusion systems

For a class of reaction cross-diffusion systems of two equations with a cross-diffusion term in the first equation and with self-diffusion terms, we prove that the unique local smooth solution given by Amann theorem is actually global. This class of systems arises in Population dynamics, and extends the triangular Shigesada-Kawasaki-Teramoto system when general power-laws growth are considered in the reaction and diffusion rates.

Exact propagating wave solutions in reaction cross-diffusion system

Reaction-diffusion systems with cross-diffusion terms in addition to, or instead of, the usual self-diffusion demonstrate interesting features which motivate their further study. The present work is aimed at designing a toy reaction-cross-diffusion model with exact solutions in the form of propagating fronts. We propose a minimal model of this kind which involves two species linked by cross-diffusion, one of which governed by a linear equation and the other having a polynomial kinetic term. We classify the resulting exact propagating front solutions. Some of them have some features of the Fisher-KPP fronts and some features of the ZFK-Nagumo fronts.

Weak–strong uniqueness of renormalized solutions to reaction–cross-diffusion systems

The weak–strong uniqueness for renormalized solutions to reaction–cross-diffusion systems in a bounded domain with no-flux boundary conditions is proved. The system generalizes the Shigesada–Kawasaki–Teramoto population model to an arbitrary number of species. The diffusion matrix is neither symmetric nor positive definite, but the system possesses a formal gradient-flow or entropy structure. No growth conditions on the source terms are imposed. It is shown that any renormalized solution coincides with a strong solution with the same initial data, as long as the strong solution exists. The proof is based on the evolution of the relative entropy modified by suitable cutoff functions.

Trend to equilibrium of renormalized solutions to reaction–cross-diffusion systems

The convergence to equilibrium of renormalized solutions to reaction–cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, while the diffusion matrix is of cross-diffusion type and hence nondiagonal and neither symmetric nor positive semi-definite, but the system admits a formal gradient-flow or entropy structure. The diffusion term generalizes the population model of Shigesada, Kawasaki and Teramoto to an arbitrary number of species. By showing that any renormalized solution satisfies the conservation of masses and a weak entropy–entropyproduction inequality, it can be proved under the assumption of no boundary equilibria that all renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.

About reaction–diffusion systems involving the Holling-type II and the Beddington–DeAngelis functional responses for predator–prey models

We consider in this paper a microscopic model (that is, a system of three reaction–diffusion equations) incorporating the dynamics of handling and searching predators, and show that its solutions converge when a small parameter tends to 0 towards the solutions of a reaction–cross diffusion system of predator–prey type involving a Holling-type II or Beddington–DeAngelis functional response. We also provide a study of the Turing instability domain of the obtained equations and (in the case of the Beddington–DeAngelis functional response) compare it to the same instability domain when the cross diffusion is replaced by a standard diffusion.

The full Keller–Segel model is well-posed on nonsmooth domains**This work is dedicated to Prof Willi Jäger.

In this paper we prove that the full Keller–Segel system, a quasilinear strongly coupled reaction-crossdiffusion system of four parabolic equations, is well-posed in the sense that it always admits an unique local-in-time solution in an adequate function space, provided that the initial values are suitably regular. The proof is done via an abstract solution theorem for nonlocal quasilinear equations by Amann and is carried out for general source terms. It is fundamentally based on recent nontrivial elliptic and parabolic regularity results which hold true even on rather general nonsmooth spatial domains. For space dimensions 2 and 3, this enables us to work in a nonsmooth setting which is not available in classical parabolic systems theory. Apparently, there exists no comparable existence result for the full Keller–Segel system up to now. Due to the large class of possibly nonsmooth domains admitted, we also obtain new results for the ‘standard’ Keller–Segel system consisting of only two equations as a special case.

Splitting Schemes and Segregation in Reaction Cross-Diffusion Systems

One of the most fascinating phenomenon observed in reaction diffusion systems is the emergence of segregated solutions, i.e., population densities with disjoint supports. We analyze such a reaction...

Lumped finite elements for reaction–cross-diffusion systems on stationary surfaces

We consider a lumped surface finite element method (LSFEM) for the spatial approximation of reaction–diffusion equations on closed compact surfaces in R3 in the presence of cross-diffusion. We provide a fully-discrete scheme by applying the Implicit–Explicit (IMEX) Euler method. We provide sufficient conditions for the existence of polytopal invariant regions for the numerical solution after spatial and full discretisations. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. To support our theoretical findings, we provide two numerical tests. The first test confirms that in the absence of lumping numerical solutions violate the invariant region leading to blow-up due to the nature of the kinetics. The second experiment is an example of Turing pattern formation in the presence of cross-diffusion on the sphere.

Bifurcation of Reaction Cross-Diffusion Systems

This paper is devoted to a reaction cross-diffusion system under Neumann boundary conditions. Firstly, the existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are investigated by means of Lyapunov–Schmidt reduction. Next, the linear stability and Hopf bifurcations of homogeneous steady-state solutions are described in detail. In particular, the Hopf bifurcation direction and the stability of bifurcating time-periodic solutions are determined by using center manifold reduction and normal form theory. Finally, some of the main results are illustrated by an application to a predator–prey model with Allee effect and one-dimensional spatial domain [Formula: see text].

On triangular reaction cross-diffusion systems with possible self-diffusion

We present new results of existence of global solutions for a class of reaction cross-diffusion systems of two equations presenting a cross-diffusion term in the first equation, and possibly presenting a self-diffusion term in any (or both) of the two equations. This class of systems arises in Population dynamics, and notably includes the triangular SKT system. In particular, we recover and extend existing results for the triangular SKT system. Our proof relies on entropy and duality methods.

New results for triangular reaction cross diffusion system

We present an approach based on entropy and duality methods for “triangular” reaction cross diffusion systems of two equations, in which cross diffusion terms appear only in one of the equations. Thanks to this approach, we recover and extend many existing results on the classical “triangular” Shigesada–Kawasaki–Teramoto model.

Analysis of a finite volume method for a bone growth system in vivo

This paper is devoted to the convergence analysis of a finite volume method and numerical simulations of a reaction–cross diffusion system arising from a bone growth model. This model describes the evolution of mesenchymal stem cells, osteoblasts, bone matrix and osteogenic growth factor. We propose a numerical scheme based on an implicit finite volume method constructed on an orthogonal mesh. Lack of the regularity of the approximate system is overcome by stability results which allow to obtain estimates on the translates, apply the Kolmogorov theorem in order to get compactness and show the convergence of the proposed scheme. The efficiency and robustness of the scheme are shown in simulating a situation in bone growth: the healing of a skull fracture in rats.