Convective instability and boundary driven oscillations in a reaction-diffusion-advection model

Abstract

In a reaction-diffusion-advection system, with a convectively unstable regime, a perturbation creates a wave train that is advected downstream and eventually leaves the system. We show that the convective instability coexists with a local absolute instability when a fixed boundary condition upstream is imposed. This boundary induced instability acts as a continuous wave source, creating a local periodic excitation near the boundary, which initiates waves travelling both up and downstream. To confirm this, we performed analytical analysis and numerical simulations of a modified Martiel-Goldbeter reaction-diffusion model with the addition of an advection term. We provide a quantitative description of the wave packet appearing in the convectively unstable regime, which we found to be in excellent agreement with the numerical simulations. We characterize this new instability and show that in the limit of high advection speed, it is suppressed. This type of instability can be expected for reaction-diffusion systems that present both a convective instability and an excitable regime. In particular, it can be relevant to understand the signaling mechanism of the social amoeba Dictyostelium discoideum that may experience fluid flows in its natural habitat.

Catalytic Membrane Reactor Model as a Laboratory for Pattern Emergence in Reaction‐diffusion‐advection Media

“…3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 subsets, mass conservation, non-local interactions, higher codimension bifurcations and applications thereof. [87,[92][93][94][95][96][97][98][99][100][101][102][103][104][105][106][107][108] 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56…”

Analysing diffusion and flow-driven instability using semidefinite programming

“…The idea of flow-driven instability was previously presented in [21,[44][45][46], where periodic spatial patterns were observed due to the destabilizing effect of advection. The theoretical prediction was also verified with engineered chemical and biological systems [22,47].…”

Boundary-Driven Oscillations Rescue PdsA-cells

“…At the downstream boundary no-flux (∂ x γ(x = L, t) = 0) boundary conditions were used. For a detailed analysis of the effects of advection in this configuration with WT cells, refer to our previous works (1,24).…”

“…For a careful analysis of BDOs in the CU regime, we refer the reader to Ref. (1). To mimic our experimental setup, we performed numerical simulations in a channel filled with PdsAcells with imposition of a Dirichlet boundary upstream.…”

“…In their natural habitat, D. discoideum cells are exposed to external flows which can significantly influence the wave generation process (17). Previously, we had numerically predicted the existence of a convective instability induced by the influx of cAMP-free buffer through a colony of signaling amoebas (1,18). In such a system it is of utmost importance what kind of chemicals get injected in the system with the advecting flow.…”