^{1,a)}and Carsten Beta

^{1}

### Abstract

We explore the effect of cross-diffusion on pattern formation in the two-variable Oregonator model of the Belousov-Zhabotinsky reaction. For high negative cross-diffusion of the activator (the activator being attracted towards regions of increased inhibitor concentration) we find, depending on the values of the parameters, Turing patterns, standing waves, oscillatory Turing patterns, and quasi-standing waves. For the inhibitor, we find that positive cross-diffusion (the inhibitor being repelled by increasing concentrations of the activator) can induce Turing patterns, jumping waves and spatially modulated bulk oscillations. We qualitatively explain the formation of these patterns. With one model we can explain Turing patterns, standing waves and jumping waves, which previously was done with three different models.

Many intriguing patterns are observed in the Belousov-Zhabotinsky reaction immersed in a microemulsion, a system for which cross-diffusion terms have been measured. A model that well describes the kinetics of the Belousov-Zhabotinsky reaction is the Oregonator model, and when adding cross-diffusion to this model some of those intriguing patterns observed in experiments can be replicated in simulations.

Financial support by the Alexander von Humboldt foundation is gratefully acknowledged.

I. INTRODUCTION

II. EFFECT OF CROSS-DIFFUSION OF THE ACTIVATOR

III. EFFECT OF CROSS-DIFFUSION OF THE INHIBITOR

IV. DISCUSSION

V. CONCLUSIONS AND OUTLOOK

## Figures

Phase diagram in the f-ξ plane for ε = 0.04 and q = 0.01, with Dx = Dz = 1. BO corresponds to bulk oscillations, (a) to Turing patterns, (b) are quasi-standing waves, (c) is standing waves, (d) is mixture of randomly appearing spots and steady state, (e) is defect-mediated turbulence, and (f) is a mixture of standing waves and defect-mediated turbulence. White zones correspond to a steady state.

Phase diagram in the f-ξ plane for ε = 0.04 and q = 0.01, with Dx = Dz = 1. BO corresponds to bulk oscillations, (a) to Turing patterns, (b) are quasi-standing waves, (c) is standing waves, (d) is mixture of randomly appearing spots and steady state, (e) is defect-mediated turbulence, and (f) is a mixture of standing waves and defect-mediated turbulence. White zones correspond to a steady state.

Space-time plots of size 100 space units (horzontal) × 200 time units (downwards), where light (dark) grey levels represents high (low) concentrations of the activator x. (a) Turing patterns obtained with f = 0.5 and ξ = −9, (b) quasi-standing waves with f = 2.19 and ξ = −10, (c) standing waves, with f = 2.19 and ξ = −5, (d) randomly appearing spots and steady state with f = 2.25 and ξ = −10.5, (e) defect-mediated turbulence with f = 2.19 and ξ = −1, and (f) mixed pattern with f = 2.19 and ξ = −4. Insets in (e) and (f) show detailed space-time plots of the marked area (50 s. u. × 20 t. u.). Other parameters as in figure 1 . Random initial conditions were used.

Space-time plots of size 100 space units (horzontal) × 200 time units (downwards), where light (dark) grey levels represents high (low) concentrations of the activator x. (a) Turing patterns obtained with f = 0.5 and ξ = −9, (b) quasi-standing waves with f = 2.19 and ξ = −10, (c) standing waves, with f = 2.19 and ξ = −5, (d) randomly appearing spots and steady state with f = 2.25 and ξ = −10.5, (e) defect-mediated turbulence with f = 2.19 and ξ = −1, and (f) mixed pattern with f = 2.19 and ξ = −4. Insets in (e) and (f) show detailed space-time plots of the marked area (50 s. u. × 20 t. u.). Other parameters as in figure 1 . Random initial conditions were used.

Phase diagram in the f-ξ plane for ε = 0.04 and q = 0.05, with Dx = Dz = 1. T corresponds to Turing patterns, OT to oscillatory Turing and BO to bulk oscillations. White zones correspond to a steady state.

Phase diagram in the f-ξ plane for ε = 0.04 and q = 0.05, with Dx = Dz = 1. T corresponds to Turing patterns, OT to oscillatory Turing and BO to bulk oscillations. White zones correspond to a steady state.

(a) Space-time plot of oscillatory Turing patterns with ε = 0.04, q = 0.05, f = 1.7 and ξ = −9.5, with size 100 space units (horzontal) × 200 time units (downwards), where light (dark) color represents high (low) concentration of the activator x. (b) Shows a zoom of 50 space units × 25 time units taken from the bottom left of panel (a). Random initial conditions were used.

(a) Space-time plot of oscillatory Turing patterns with ε = 0.04, q = 0.05, f = 1.7 and ξ = −9.5, with size 100 space units (horzontal) × 200 time units (downwards), where light (dark) color represents high (low) concentration of the activator x. (b) Shows a zoom of 50 space units × 25 time units taken from the bottom left of panel (a). Random initial conditions were used.

Two-dimensional Turing patterns obtained with ε = 0.04, q = 0.01 and f = 0.5, with Dx = Dz = 1, (a) labyrinthine pattern obtained with ξ = −9.5, and (b) spot pattern obtained with ξ = −6.5. Snapshots of patterns are taken 150 time units after starting the simulation with random initial conditions. Size of pictures is 50 space units × 50 space units, light (dark) color represents high (low) concentration of the activator x.

Two-dimensional Turing patterns obtained with ε = 0.04, q = 0.01 and f = 0.5, with Dx = Dz = 1, (a) labyrinthine pattern obtained with ξ = −9.5, and (b) spot pattern obtained with ξ = −6.5. Snapshots of patterns are taken 150 time units after starting the simulation with random initial conditions. Size of pictures is 50 space units × 50 space units, light (dark) color represents high (low) concentration of the activator x.

Formation of Turing patterns from standing waves, (a) snapshots taken at 1 time unit intervals 11 time units after starting the simulation with random initial conditions, (b) snapshot taken at 248 t. u. after starting the simulation, and (c) space-time plot 50 s. u. (horizontal) × 250 t. u. (downwards) taken at the gray line in the second panel of (a). Size of pictures is 50 space units × 50 space units. Parameters are ε = 0.04, q = 0.01, f = 2.2, ξ = −7, and Dx = Dz = 1. Light (dark) color represents high (low) concentration of the activator x.

Formation of Turing patterns from standing waves, (a) snapshots taken at 1 time unit intervals 11 time units after starting the simulation with random initial conditions, (b) snapshot taken at 248 t. u. after starting the simulation, and (c) space-time plot 50 s. u. (horizontal) × 250 t. u. (downwards) taken at the gray line in the second panel of (a). Size of pictures is 50 space units × 50 space units. Parameters are ε = 0.04, q = 0.01, f = 2.2, ξ = −7, and Dx = Dz = 1. Light (dark) color represents high (low) concentration of the activator x.

Space-time plots of size 100 space units (vertical) × 120 time units (downwards) with ε = 0.04 q = 0.05, f = 1.4, and Dx = Dz = 1. (a) Turing patterns with ζ = 1, (b) spatially modulated bulk oscillations with ζ = 2 (note that jumping waves are generated from a Turing spot in the lower left corner), and (c) jumping waves that are generated at the right boundary with ζ = 3. Light (dark) color represents high (low) concentration of the activator x. Random initial conditions were used.

Space-time plots of size 100 space units (vertical) × 120 time units (downwards) with ε = 0.04 q = 0.05, f = 1.4, and Dx = Dz = 1. (a) Turing patterns with ζ = 1, (b) spatially modulated bulk oscillations with ζ = 2 (note that jumping waves are generated from a Turing spot in the lower left corner), and (c) jumping waves that are generated at the right boundary with ζ = 3. Light (dark) color represents high (low) concentration of the activator x. Random initial conditions were used.

Phase diagrams in the f-ζ plane for ε = 0.04, with Dx = Dz = 1. In (a), q = 0.01 and in (b), q = 0.05. T corresponds to Turing patterns, MBO to spatially modulated bulk oscillations, JW are jumping waves, MSA is a mixed state with small amplitude, DT is defect mediated turbulence, and M is a mixed state of Turing patterns and localized standing waves.

Phase diagrams in the f-ζ plane for ε = 0.04, with Dx = Dz = 1. In (a), q = 0.01 and in (b), q = 0.05. T corresponds to Turing patterns, MBO to spatially modulated bulk oscillations, JW are jumping waves, MSA is a mixed state with small amplitude, DT is defect mediated turbulence, and M is a mixed state of Turing patterns and localized standing waves.

(a) Space time plot of 50 s. u. (horizontal) × 100 t. u. (downwards) taken at the line in the first panel of b. (b) Sequence of snapshots taken at 0.3 t. u. intervals starting 8.7 t. u. after starting the simulation showing the modulated bulk oscillations. (c) Sequence of snapshots taken at 0.2 t. u. intervals starting 90 t. u. after starting the simulation, showing jumping waves. The size of pictures is 50 s. u. × 50 s. u. Parameters are ε = 0.04, q = 0.01, f = 0.55, ζ = 3, and Dx = Dz = 1. Random initial conditions were used.

(a) Space time plot of 50 s. u. (horizontal) × 100 t. u. (downwards) taken at the line in the first panel of b. (b) Sequence of snapshots taken at 0.3 t. u. intervals starting 8.7 t. u. after starting the simulation showing the modulated bulk oscillations. (c) Sequence of snapshots taken at 0.2 t. u. intervals starting 90 t. u. after starting the simulation, showing jumping waves. The size of pictures is 50 s. u. × 50 s. u. Parameters are ε = 0.04, q = 0.01, f = 0.55, ζ = 3, and Dx = Dz = 1. Random initial conditions were used.

(a–d) Space-time plots of size 100 space units (vertical) × 100 time units (downwards) with ε = 0.04, q = 0.01, f = 2.19, ξ = −5.5, and Dx = Dz = 1. The velocity of the advective field is 0.1 (a), 0.4 (b), 1.6 (c), and 1.7 (d). (e) Phase diagram of patterns obtained. (f) For comparison, phase diagram showing the effect of flow on a systems with wave instability that displays standing waves. SW corresponds to standing waves (as depicted in a), Mixed is a mixed pattern of synchronous standing waves near the inlet and standing waves further down (as depicted in b), SSW corresponds to synchronous standing waves (depicted in c), and C corresponds to a chaotic state (as depicted in d). DSW corresponds to drifting standing waves (g), M corresponds to a mixed pattern of synchronous standing waves and drifting clusters (i), DC corresponds to drifting clusters (j) and S to a stabilized steady state. Synchronous standing waves for the model with waves instability are shown in (h). Figures (f)-(j) are adapted from Ref. 27 .

(a–d) Space-time plots of size 100 space units (vertical) × 100 time units (downwards) with ε = 0.04, q = 0.01, f = 2.19, ξ = −5.5, and Dx = Dz = 1. The velocity of the advective field is 0.1 (a), 0.4 (b), 1.6 (c), and 1.7 (d). (e) Phase diagram of patterns obtained. (f) For comparison, phase diagram showing the effect of flow on a systems with wave instability that displays standing waves. SW corresponds to standing waves (as depicted in a), Mixed is a mixed pattern of synchronous standing waves near the inlet and standing waves further down (as depicted in b), SSW corresponds to synchronous standing waves (depicted in c), and C corresponds to a chaotic state (as depicted in d). DSW corresponds to drifting standing waves (g), M corresponds to a mixed pattern of synchronous standing waves and drifting clusters (i), DC corresponds to drifting clusters (j) and S to a stabilized steady state. Synchronous standing waves for the model with waves instability are shown in (h). Figures (f)-(j) are adapted from Ref. 27 .

Space-time plots for q = 0.01, Dx = 0.1, and Dz = 1 with size 100 s. u. (horizontal) × 200 t. u. (downwards), for f = 2.1 (a), 2.16 (b), 2.17 (c), 2.18 (d), 2.19 (e), and 2.2 (f). In (g), the respective phase diagram is shown. C-SW corresponds to states with spatiotemporal chaos and standing waves, BO to bulk oscillations. Random initial conditions were used.

Space-time plots for q = 0.01, Dx = 0.1, and Dz = 1 with size 100 s. u. (horizontal) × 200 t. u. (downwards), for f = 2.1 (a), 2.16 (b), 2.17 (c), 2.18 (d), 2.19 (e), and 2.2 (f). In (g), the respective phase diagram is shown. C-SW corresponds to states with spatiotemporal chaos and standing waves, BO to bulk oscillations. Random initial conditions were used.

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