The aim of this paper is to contribute to the understanding of the pattern formation phenomenon in reaction-diffusion equations coupled with ordinary differential equations. Such systems of equations arise, for example, from modeling of interactions between cellular processes such as cell growth, differentiation or transformation and diffusing signaling factors. We focus on stability analysis of solutions of a prototype model consisting of a single reaction-diffusion equation coupled to an ordinary differential equation. We show that such systems are very different from classical reaction-diffusion models. They exhibit diffusion-driven instability (Turing instability) under a condition of autocatalysis of non-diffusing component. However, the same mechanism which destabilizes constant solutions of such models, destabilizes also all continuous spatially heterogeneous stationary solutions, and consequently, there exist no stable Turing patterns in such reaction-diffusion-ODE systems. We provide a rigorous result on the nonlinear instability, which involves the analysis of a continuous spectrum of a linear operator induced by the lack of diffusion in the destabilizing equation. These results are extended to discontinuous patterns for a class of nonlinearities.
We explore a mechanism of pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions between cellular processes and diffusing growth factors. We focused on the model of early carcinogenesis proposed by Marciniak-Czochra and Kimmel, which is an example of a wider class of pattern formation models with an autocatalytic non-diffusing component. We present a numerical study showing emergence of periodic and irregular spike patterns due to diffusi…
We consider the classical Turing instability in a reaction-diffusion system as the secend part of our study on pattern formation. We prove that nonlinear dynamics of a general perturbation of the Turing instability is determined by the finite number of linear growing modes over a time scale of $ln(1/δ)$, where &δ$ is the strength of the initial perturbation.
In this paper, we study a strongly coupled two-prey one-predator system. We first prove the unique positive equilibrium solution is globally asymptotically stable for the corresponding kinetic system (the system without diffusion) and remains locally linearly stable for the reaction-diffusion system without cross-diffusion, hence it does not belong to the classical Turing instability scheme. Moreover we prove that the positive equilibrium solution is globally asymptotically stable for the reaction-diffusion system without cross-diffusion. But it becomes linear unstable only when cross-diffusio…
The modelling of linear and nonlinear reaction-subdiffusion processes is more subtle than normal diffusion and causes different phenomena. The resulting equations feature a spatial Laplacian with a temporal memory term through a time fractional derivative. It is known that the precise form depends on the interaction of dispersal and reaction, and leads to qualitative differences. We refine these results by defining generalised spectra through dispersion relations, which allows us to examine the onset of instability and in particular inspect Turing type instabilities. These results are numerica…
Biological, physical, medical, and numerical applications involving membrane problems on different scales are numerous. We propose an extension of the standard Turing theory to the case of two domains separated by a permeable membrane. To this aim, we study a reaction-diffusion system with zero-flux boundary conditions on the external boundary and Kedem-Katchalsky membrane conditions on the inner membrane. We use the same approach as in the classical Turing analysis but applied to membrane operators. The introduction of a diagonalization theory for compact and self-adjoint membrane operators i…
Classical models of pattern formation are based on diffusion-driven instability (DDI) of constant stationary solutions of reaction-diffusion equations, which leads to emergence of stable, regular Turing patterns formed around that equilibrium. In this paper we show that coupling reaction-diffusion equations with ordinary differential equations (ODE) may lead to a novel pattern formation phenomenon in which DDI causes destabilization of both constant solutions and Turing patterns. Bistability and hysteresis effects in the null sets of model nonlinearities yield formation of far from the equilib…
A general system of several ordinary differential equations coupled with a reaction-diffusion equation in a bounded domain with zero-flux boundary condition is studied in the context of pattern formation. These initial-boundary value problems may have regular (i.e. sufficiently smooth) stationary solutions. This class of {\it close-to-equilibrium} patterns includes stationary solutions that emerge due to the Turing instability of a spatially constant stationary solution. The main result of this work is instability of all regular patterns. It suggests that stable stationary solutions arising in…
In this paper we provide an example of a class of two reaction-diffusion-ODE equations with homogeneous Neumann boundary conditions, in which Turing-type instability not only destabilizes constant steady states but also induces blow-up of nonnegative spatially heterogeneous solutions. Solutions of this problem preserve nonnegativity and uniform boundedness of the total mass. Moreover, for the corresponding system with two non-zero diffusion coefficients, all nonnegative solutions are global in time. We prove that a removal of diffusion in one of the equations leads to a finite-time blow-up of …
Turing patterns in reaction-diffusion (RD) systems have classically been studied only in RD systems which do not explicitly depend on independent variables such as space. In practise, many systems for which Turing patterning is important are not homogeneous with ideal boundary conditions. In heterogeneous systems with stable steady states, the steady states are also necessarily heterogeneous which is problematic for applying the classical analysis. Whilst there has been some work done to extend Turing analysis to some heterogeneous systems, for many systems it is still difficult to determine i…
Coupling a reaction-diffusion equation with ordinary differential equations (ODE) may lead to diffusion-driven instability (DDI) which, in contrast to the classical reaction-diffusion models, causes destabilization of both, constant solutions and Turing patterns. Using a shadow-type limit of a reaction-diffusion-ODE model, we show that in such cases the instability driven by nonlocal terms (a counterpart of DDI) may lead to formation of unbounded spike patterns.
Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary…
We present a rigorous analysis of the slow passage through a Turing bifurcation in the Swift-Hohenberg equation using a novel approach based on geometric blow-up. We show that the formally derived multiple scales ansatz which is known from classical modulation theory can be adapted for use in the fast-slow setting, by reformulating it as a blow-up transformation. This leads to dynamically simpler modulation equations posed in the blown-up space, via a formal procedure which directly extends the established approach to the time-dependent setting. The modulation equations take the form of non-au…
Motivated by numerical simulations showing the emergence of either periodic or irregular patterns, we explore a mechanism of pattern formation arising in the processes described by a system of a single reaction-diffusion equation coupled with ordinary differential equations. We focus on a basic model of early cancerogenesis proposed by Marciniak-Czochra and Kimmel [Comput. Math. Methods Med. {\bf 7} (2006), 189--213], [Math. Models Methods Appl. Sci. {\bf 17} (2007), suppl., 1693--1719], but the theory we develop applies to a wider class of pattern formation models with an autocatalytic non-di…
Mode-Wise Spectral Criteria for Coupled Mass Transport in Hybrid PDE--ODE Tumor Microenvironments
College of Engineering, Boston University（波士顿大学工程学院）；Department of Integrated Systems Engineering, Ohio State University（俄亥俄州立大学综合工程系）；Department of Mathematics, University of Tennessee（田纳西大学数学系）；Fujian Cancer Hospital（福建省肿瘤医院）
We study a p-adic reaction-diffusion system and the associated Turing patterns. We establish an instability criteria and show that the Turing patterns are not classical patterns consisting of alternating domains. Instead of this, a Turing pattern consists of several domains (clusters), each of them supporting a different pattern but with the same parameter values. This type of patterns are typically produced by reaction-diffusion equations on large networks.
It is well known that for reaction-diffusion systems with differential isotropic diffusions, a Turing instability yields striped solutions. In this paper we study the impact of weak anisotropy by directional advection on such solutions, and the role of quadratic terms. We focus on the generic form of planar reaction-diffusion systems with two components near such a bifurcation. Using Lyapunov-Schmidt reduction and Floquet-Bloch decomposition we derive a rigorous parameter expansion for existence and stability against large wavelength perturbations. This provides detailed formulae for the loci …
Striped patterns are known to bifurcate in reaction-diffusion systems with differential isotropic diffusions at a supercritical Turing instability. In this paper we study the impact of weak anisotropy by directional advection on the stability of stripes with respect to various lattice modes, and the role of quadratic terms therein. We focus on the generic form of planar reaction-diffusion systems with two components near such a bifurcation. Using centre manifold reduction we derive a rigorous parameter expansion for the critical eigenvalues for lattice mode perturbations, specifically nearly s…
We prove that the famous diffusive Brusselator model can support more complicated spatial-temporal wave structure than the usual temporal-oscillation from a standard Hopf bifurcation. In our investigation, we discover that the diffusion term in the model is neither a usual parabolic stabilizer nor a destabilizer as in the Turing instability of uniform state, but rather plays the role of maintaining an equivariant Hopf bifurcation spectral mechanism. At the same time, we show that such a mechanism can occur around any nonzero wave number and this finding is also different from the former works …
This paper explores the classification of parameter spaces for reaction-diffusion systems of two chemical species on stationary domains. The dynamics of the system are explored both in the absence and presence of diffusion. The parameter space is fully classified in terms of the types and stability of the uniform steady state. In the absence of diffusion the results on the classification of parameter space are supported by simulations of the corresponding vector-field and some trajectories around the uniform steady state. In the presence of diffusion, the main findings are the quantitative ana…
A reaction-diffusion system exhibiting Turing's diffusion driven instability is considered. The equation for an activator is supplemented by unilateral terms of the type $s_{-}(x)u^{-}$, $s_{+}(x)u^{+}$ describing sources and sinks active only if the concentration decreases below and increases above, respectively, the value of the basic spatially constant solution which is shifted to zero. We show that the domain of diffusion parameters in which spatially non-homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral term…
Collective motion of chemotactic bacteria as E. Coli relies, at the individual level, on a continuous reorientation by runs and tumbles. It has been established that the length of run is decided by a stiff response to a temporal sensingof chemical cues along the pathway.We describe a novel mechanism for pattern formation stemming from the stiffness of chemotactic response relying on a kinetic chemotaxis model which includes a recently discovered formalism for the bacterial chemotaxis. We prove instability both for amicroscopic description in the space-velocity space and for the macroscopic equ…
