98 research outputs found
Instability of Turing patterns in reaction-diffusion-ODE systems
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.Comment: This is a new version of the paper. Presentation of results was
essentially revised according to referee suggestion
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.Comment: 32 pages, 8 picture
Mass concentration in a nonlocal model of clonal selection
Self-renewal is a constitutive property of stem cells. Testing the cancer
stem cell hypothesis requires investigation of the impact of self-renewal on
cancer expansion. To understand better this impact, we propose a mathematical
model describing dynamics of a continuum of cell clones structured by the
self-renewal potential. The model is an extension of the finite
multi-compartment models of interactions between normal and cancer cells in
acute leukemias. It takes a form of a system of integro-differential equations
with a nonlinear and nonlocal coupling, which describes regulatory feedback
loops in cell proliferation and differentiation process. We show that such
coupling leads to mass concentration in points corresponding to maximum of the
self-renewal potential and the model solutions tend asymptotically to a linear
combination of Dirac measures. Furthermore, using a Lyapunov function
constructed for a finite dimensional counterpart of the model, we prove that
the total mass of the solution converges to a globally stable equilibrium.
Additionally, we show stability of the model in space of positive Radon
measures equipped with flat metric. The analytical results are illustrated by
numerical simulations
Shadow limit using Renormalization Group method and Center Manifold method
International audienceMulti-scale analysis and biological applications are two subjects of focus in Willi's research over his professional life. Abstract We study a shadow limit (the infinite diffusion coefficient-limit) of a system of ODEs coupled with a semilinear heat equation in a bounded domain with Neumann boundary conditions. In the literature, it was established formally that in the limit, the original semilinear heat equation reduces to an ODE involving the space averages of the solution to the semilinear heat equation and of the nonlinearity. It is coupled with the original system of ODEs for every space point x .We present derivation of the limit using the renormalization group (RG) and the center manifold approaches. The RG approach provides also further approximating expansion terms. The error estimate in the terms of the inverse of the diffusion coefficient is obtained for the finite time intervals. For the infinite times, the center manifolds for the starting problem and for its shadow limit approximation are compared and it is proved that their distance is of the order of the inverse of the diffusion coefficient
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