Dynamical spike solutions in a nonlocal model of pattern formation

Abstract

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.

An averaging principle for fast diffusions in domains separated by semi-permeable membranes

“…For example, in the Alt and Lauffenberger's [2] model of leucocytes reacting to a bacterial invasion by moving up a gradient of some chemical attractant produced by the bacteria (see Section 13.4.2 in [47]) a system of three PDEs is reduced to one equation provided bacterial diffusion is much smaller than the diffusion of leukocytes or of chemoattractants (which is typically the case). Similarly, in the early carcinogenesis model of Marcinak-Czochra and Kimmel [16,53,54,55], a system of two ODEs coupled with a single diffusion equation (involving Neumann boundary conditions) is replaced by a socalled shadow system of integro-differential equations with ordinary differentiation, provided diffusion may be assumed fast.…”

Irregular convergence of mild solutions of semilinear equations

“…But the force of Müller conditions is that solutions starting in R never leave R, and so they are in fact solutions for (3.6) with original F i 's. Hence, a more precise statement should read: mild solutions of (3.6) converge to those of (3.7) provided they start in R. This connection between (3.6) and (3.7) has been made in [26]. The latter paper also gives more delicate information on the speed of convergence based on heat semigroup estimates to be found e.g., in [32, p. 25] or [35,Lemma 1.3].…”

Signaling gradients in surface dynamics as basis for planarian regeneration

“…On the other hand these models typically do not generate relevant patterns robustly in situations corresponding to cutting and dissociation experiments. Systems of ordinary differential equations (vanishing diffusivities) coupled to partial differential equations have been studied extensively, yet more recently in regard to their pattern-forming capabilities [37,36,23,38,31,39], finding for instance stable patterns and unbounded solutions developing spikes. In a different direction, the role of hysteresis in diffusion-driven instabilities and de novo formation of stable patterns was investigated in [24,30] Lastly, we point out that our analysis connects with recent efforts to model and understand the role of distinguished surface reactions and bulk-to-surface coupling in morphogenesis [17,15,29,59,60].…”