Stable patterns with jump discontinuity in systems with Turing instability and hysteresis

Abstract

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 equilibrium patterns with jump discontinuity. We derive conditions for stability of stationary solutions with jump discontinuity in a suitable topology which allows us to include the discontinuity points and leads to the definition of (ε 0 , A)-stability. Additionally, we provide conditions on stability of patterns in a quasi-stationary model reduction. The analysis is illustrated on the example of three-component model of receptor-ligand binding. The proposed model provides an example of a mechanism of de novo formation of far from the equilibrium patterns in reaction-diffusion-ODE models involving co-existence of DDI and hysteresis.

Signaling gradients in surface dynamics as basis for planarian regeneration

“…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]. Envisioning for instance two species reacting and diffusing with equal diffusion constant on a surface, but one of the species diffusing rapidly into, through, and back out of the bulk, one immediately finds the disparate effective diffusivities necessary for pattern formation in Turing's mechanism, thus providing a biologically realistic mechanism for robust pattern formation.…”

Dynamical spike solutions in a nonlocal model of pattern formation

“…In Ref. [10], conditions for linear stability of such patterns in a topology excluding the discontinuity points have been provided. Then, the emergence and stability of the patterns with jump discontinuity has been proved for a receptor-ligand model with DDI and hysteresis.…”

Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media

“…This is also a receptor-based model for pattern formation. For the biological aspects of the model, see [3] and the references therein. Here, we only remark that u and v stand for the density of free receptors and ligands, respectively.…”

“…Under appropriate conditions on the coefficients, the nullclines f (u, v) = 0 and g(u, v) = 0 intersect (a) at exactly one point (the monostable case; see Figure 2 (a) in Section 6), (b) at exactly three points with two on the middle branch of f = 0 (the DDI case; see Figure 2 (b)) or (c) at exactly three points with one on the middle branch, the other on the right branch of f = 0 (the bistable case; see Figure 2 (c)). In [3], the DDI case was treated, and the existence of steady states with jump discontinuities in u was established and their stability in Weinberger's sense [14] was proved. Also in [4], bifurcation of nonconstant steady states from the constant solution and their instability were proved.…”

“…Then by reflection and periodic extension, we obtain a solution of arbitrary mode due to the translation invariance and the homogeneous Nuemann boundary condition. Moreover, construction of monotone solution is relatively easy if we make use of the solution formula for the equation v + f (v) = 0 (see [3,7]), which is not applicable to the variable coefficient case. In this paper, in order to construct monotone solutions, we develop the method devised by Mimura, Tabata and Hosono [8], which was originally used to treat the constant coefficient case.…”