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### Abstract

Systems with the same local dynamics but different types of diffusive instabilities may show the same type of patterns. In this paper, we show that under the influence of advective flow the scenario of patterns that is formed at different velocities change; therefore, we propose the use of advective flow as a tool to uncover the underlying instabilities of a reaction-diffusion system.

One of the challenges in pattern formation in reaction-diffusion systems is identifying the instability or instabilities behind a certain pattern, since different instabilities or combination of them may result in the same kind of pattern. Here, it is shown that the effect of advective flow is different for similar patterns with different underlying instabilities, which becomes a kind of proof of principle to construct a technique that allows distinguishing similar patterns arising from different instabilities.

Financial support by the Alexander von Humboldt foundation and Carsten Beta for his hospitality is gratefully acknowledged.

### Key Topics

- Flow instabilities
- 40.0
- Bifurcations
- 7.0
- Diffusion
- 6.0
- Phase diagrams
- 4.0
- Boundary value problems
- 3.0

## Figures

Space-time plots with size 80 space units (horizontal) × 200 time units (downwards). All systems are started from random initial conditions, and no-flux boundary conditions are used. The diffusion coefficients are *D _{x} * =

*D*=

_{y}*D*= 0.1 (a);

_{z}*D*= 0.1 and

_{x}*D*=

_{y}*D*= 2 (b); and

_{z}*D*=

_{x}*D*= 0.1 and

_{z}*D*= 2 (c). For other parameters see text. Light (dark) color represents high (low) concentration of the activator

_{y}*x*.

Space-time plots with size 80 space units (horizontal) × 200 time units (downwards). All systems are started from random initial conditions, and no-flux boundary conditions are used. The diffusion coefficients are *D _{x} * =

*D*=

_{y}*D*= 0.1 (a);

_{z}*D*= 0.1 and

_{x}*D*=

_{y}*D*= 2 (b); and

_{z}*D*=

_{x}*D*= 0.1 and

_{z}*D*= 2 (c). For other parameters see text. Light (dark) color represents high (low) concentration of the activator

_{y}*x*.

Effect of advective flow for a system with *D _{x} * =

*D*=

_{y}*D*= 0.1. Space-time plots with size 80 space units (horizontal) and 400 time units (downwards). For the initial 100 time units, the velocity

_{z}*v*is 0 and then is turned to 0.2 (a), 0.5 (b), and 1.5 (c). The square in (a), with 80 space units × 20 time units is expanded at the right. Light (dark) color represents high (low) concentration of the activator

*x*. (d) Phase diagram for patterns at different velocities of the advective flow.

Effect of advective flow for a system with *D _{x} * =

*D*=

_{y}*D*= 0.1. Space-time plots with size 80 space units (horizontal) and 400 time units (downwards). For the initial 100 time units, the velocity

_{z}*v*is 0 and then is turned to 0.2 (a), 0.5 (b), and 1.5 (c). The square in (a), with 80 space units × 20 time units is expanded at the right. Light (dark) color represents high (low) concentration of the activator

*x*. (d) Phase diagram for patterns at different velocities of the advective flow.

Effect of advective flow for a system with *D _{x} * = 0.1 and

*D*=

_{y}*D*= 2. Space-time plots with size 80 space units (horizontal) and 400 time units (downwards). For the initial 100 time units, the velocity

_{z}*v*is 0 and then is turned to 1.5 (a) and 3.0 (b). The square in (b), with 80 space units × 20 time units is expanded at the right. Light (dark) color represents high (low) concentration of the activator

*x*. (c) Phase diagram for patterns at different velocities of the advective flow.

Effect of advective flow for a system with *D _{x} * = 0.1 and

*D*=

_{y}*D*= 2. Space-time plots with size 80 space units (horizontal) and 400 time units (downwards). For the initial 100 time units, the velocity

_{z}*v*is 0 and then is turned to 1.5 (a) and 3.0 (b). The square in (b), with 80 space units × 20 time units is expanded at the right. Light (dark) color represents high (low) concentration of the activator

*x*. (c) Phase diagram for patterns at different velocities of the advective flow.

Effect of advective flow for a system with *D _{x} * =

*D*= 0.1 and

_{z}*D*= 2. Space-time plots with size 80 space units (horizontal) and 400 time units (downwards). For the initial 100 time units, the velocity

_{y}*v*is 0 and then is turned to 0.3 (a), 0.7 (b), 1.5 (c), and 2.5 (d). Light (dark) color represents high (low) concentration of the activator

*x*. (e) Phase diagram for patterns at different velocities of the advective flow.

Effect of advective flow for a system with *D _{x} * =

*D*= 0.1 and

_{z}*D*= 2. Space-time plots with size 80 space units (horizontal) and 400 time units (downwards). For the initial 100 time units, the velocity

_{y}*v*is 0 and then is turned to 0.3 (a), 0.7 (b), 1.5 (c), and 2.5 (d). Light (dark) color represents high (low) concentration of the activator

*x*. (e) Phase diagram for patterns at different velocities of the advective flow.

Effect of advective flow for a system with *D _{x} * = 0.1 and

*D*=

_{y}*D*= 2 and

_{z}*f*

_{y}=

*f*

_{z}= 1. Space-time plots with size 80 space units (horizontal) and 300 time units (downwards). For the initial 100 time units, the velocity

*v*is 0 and then is turned to 0.1 (a) and 2.5 (b). The square in (b), with 30 space units × 20 time units is expanded at the right. Light (dark) color represents high (low) concentration of the activator

*x*. (c) Phase diagram for patterns at different velocities of the advective flow.

Effect of advective flow for a system with *D _{x} * = 0.1 and

*D*=

_{y}*D*= 2 and

_{z}*f*

_{y}=

*f*

_{z}= 1. Space-time plots with size 80 space units (horizontal) and 300 time units (downwards). For the initial 100 time units, the velocity

*v*is 0 and then is turned to 0.1 (a) and 2.5 (b). The square in (b), with 30 space units × 20 time units is expanded at the right. Light (dark) color represents high (low) concentration of the activator

*x*. (c) Phase diagram for patterns at different velocities of the advective flow.

Effect of advective flow on a system with *D _{x} * =

*D*= 0.1 and

_{y}*D*= 2. Space-time plots with size 80 space units (horizontal) and 300 time units (downwards). For the initial 100 time units, the velocity

_{z}*v*is 0 and then is turned to 0.2 (a), 0.4 (b), and 1.0 (c). Light (dark) color represents high (low) concentration of the activator

*x*. (d) Phase diagram for patterns at different velocities of the advective flow.

Effect of advective flow on a system with *D _{x} * =

*D*= 0.1 and

_{y}*D*= 2. Space-time plots with size 80 space units (horizontal) and 300 time units (downwards). For the initial 100 time units, the velocity

_{z}*v*is 0 and then is turned to 0.2 (a), 0.4 (b), and 1.0 (c). Light (dark) color represents high (low) concentration of the activator

*x*. (d) Phase diagram for patterns at different velocities of the advective flow.

Detail of space-time plots with size 100 space units (horizontal) and 40 time units (downwards) when changing *v* from 0 to 1.0 ((a) and (c)) or to 3.0 ((b) and (d)) on systems with *D _{u} * = 0.1,

*D*= 2, and ξ = 0 ((a) and (b)) and for

_{w}*D*=

_{u}*D*= 0.1 and ξ = −10 ((c) and (d)). Phase diagram for the effect of advective flow on systems with

_{w}*D*= 0.1,

_{u}*D*= 2, and ξ = 0 (e) and for

_{w}*D*=

_{u}*D*= 0.1 and ξ = −10 (f).

_{w}Detail of space-time plots with size 100 space units (horizontal) and 40 time units (downwards) when changing *v* from 0 to 1.0 ((a) and (c)) or to 3.0 ((b) and (d)) on systems with *D _{u} * = 0.1,

*D*= 2, and ξ = 0 ((a) and (b)) and for

_{w}*D*=

_{u}*D*= 0.1 and ξ = −10 ((c) and (d)). Phase diagram for the effect of advective flow on systems with

_{w}*D*= 0.1,

_{u}*D*= 2, and ξ = 0 (e) and for

_{w}*D*=

_{u}*D*= 0.1 and ξ = −10 (f).

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