^{1}, Qingyu Gao

^{1,a)}, Shirui Gong

^{1}, Yuemin Zhao

^{1}and Irving R. Epstein

^{2,a)}

### Abstract

Diverse spatiotemporal patterns are generated in a three-variable reaction-diffusion model that supports 1^{1} mixed-mode oscillations. Diffusion-induced instability results in spatiotemporal patterns such as amplitude-modulated overtargets (circular super-waves superimposed on spiral waves) and superspirals. The types of superstructure waves are determined by the ratio of diffusion coefficients, which controls the interaction and competition between two local oscillatory modes, one of which is the original homogeneous 1^{1} mixed-mode oscillation, resulting in periodic amplitude modulation in space. Variation of the control parameter can reverse the chirality and radial propagation direction (outward or inward rotation) of a superspiral pattern. These amplitude-modulated patterns may provide insight into mechanisms of pattern development in some living systems.

This work was supported in part by grants (Nos. 21073232 and 50921002) from the National Natural Science Foundation of China (NNSFC), PAPD and Grant No. CHE-1012428 from the U.S. National Science Foundation (NSF).

I. INTRODUCTION

II. MODEL AND MIXED-MODE OSCILLATIONS

III. AMPLITUDE MODULATION AND SUPERSTRUCTURE WAVES RESULTING FROM DIFFUSION-INDUCED INSTABILITY

IV. STRUCTURE ANALYSIS OF AMPLITUDE-MODULATED WAVES

V. LOCAL OSCILLATORY DYNAMICS CORRESPONDING TO SPATIOTEMPORAL PATTERNS

VI. BIFURCATION DIAGRAM OF AMPLITUDE-MODULATED WAVES

VII. CONCLUSION

### Key Topics

- Diffusion
- 29.0
- Bifurcations
- 10.0
- Spiral arms
- 9.0
- Time series analysis
- 9.0
- Chiral symmetries
- 5.0

## Figures

Attractors of distinct dynamical modes in the Hastings-Powell model without diffusion. (a) High frequency mode of the subsystem between prey and predator (*dw*/*dt* = 0 *w* _{ s } = 1.75); (b) low frequency mode due to predator-superpredator interaction, dynamic parameters are shown in Table I, except *b* _{2} = 1.30; (c) 1^{1} mixed-mode oscillations used in reaction-diffusion simulations, all dynamic parameters are shown in Table I. (d) Typical “tea-cup” mode, 1^{4}, mixing high and low frequency modes, *b* _{2} = 2.0 and other parameters are as Table I.

Attractors of distinct dynamical modes in the Hastings-Powell model without diffusion. (a) High frequency mode of the subsystem between prey and predator (*dw*/*dt* = 0 *w* _{ s } = 1.75); (b) low frequency mode due to predator-superpredator interaction, dynamic parameters are shown in Table I, except *b* _{2} = 1.30; (c) 1^{1} mixed-mode oscillations used in reaction-diffusion simulations, all dynamic parameters are shown in Table I. (d) Typical “tea-cup” mode, 1^{4}, mixing high and low frequency modes, *b* _{2} = 2.0 and other parameters are as Table I.

Three typical spiral wave patterns in variable *v* with different diffusion coefficients. Kinetic parameters are given in Table I. Right hand column shows amplitude curves of wave arms from the spiral core to the boundary along the propagating spiral. (a) Twin-armed spiral waves for *Dv* = 0, *Dw* = 0.1; (b) simple spiral waves for *Dv* = 0.1, *Dw* = 0; (c) spiral waves for *Dv* = 0.1, *Dw* = 0.1, which have periodically amplitude-modulated wave arms.

Three typical spiral wave patterns in variable *v* with different diffusion coefficients. Kinetic parameters are given in Table I. Right hand column shows amplitude curves of wave arms from the spiral core to the boundary along the propagating spiral. (a) Twin-armed spiral waves for *Dv* = 0, *Dw* = 0.1; (b) simple spiral waves for *Dv* = 0.1, *Dw* = 0; (c) spiral waves for *Dv* = 0.1, *Dw* = 0.1, which have periodically amplitude-modulated wave arms.

Four different superstructure waves superimposed on the basic spirals. Each plot shows snapshots of *v* on a 2000 × 2000 grid. The plots on the left are 3D plots with *v* shown on z-axis. Vertical views are shown on the right. Parameters are given in Table I and *Dw* is fixed at 0.1. (a) Overtargeted spiral waves for *Dv* = 0.0009; (b) superspiral waves for *Dv* = 0.025, with a clockwise superstructure; (c) half-plane modulated spiral waves for *Dv* = 0.04; (d) superspiral waves for *Dv* = 0.1, with an anticlockwise superstructure. Each plot in this figure is enhanced online. [URL: http://dx.doi.org/10.1063/1.4768895.1] [URL: http://dx.doi.org/10.1063/1.4768895.2] [URL: http://dx.doi.org/10.1063/1.4768895.3] [URL: http://dx.doi.org/10.1063/1.4768895.4]10.1063/1.4768895.110.1063/1.4768895.210.1063/1.4768895.310.1063/1.4768895.4

Four different superstructure waves superimposed on the basic spirals. Each plot shows snapshots of *v* on a 2000 × 2000 grid. The plots on the left are 3D plots with *v* shown on z-axis. Vertical views are shown on the right. Parameters are given in Table I and *Dw* is fixed at 0.1. (a) Overtargeted spiral waves for *Dv* = 0.0009; (b) superspiral waves for *Dv* = 0.025, with a clockwise superstructure; (c) half-plane modulated spiral waves for *Dv* = 0.04; (d) superspiral waves for *Dv* = 0.1, with an anticlockwise superstructure. Each plot in this figure is enhanced online. [URL: http://dx.doi.org/10.1063/1.4768895.1] [URL: http://dx.doi.org/10.1063/1.4768895.2] [URL: http://dx.doi.org/10.1063/1.4768895.3] [URL: http://dx.doi.org/10.1063/1.4768895.4]10.1063/1.4768895.110.1063/1.4768895.210.1063/1.4768895.310.1063/1.4768895.4

Spatial power spectra (left) and space-time plots (right) for spiral waves with the superstructures shown in Fig. 3. Time runs from bottom to top (0–5000). Space in space-time plots runs from the spiral core (1000, 1000) to the boundary (1800, 1000) on the 2000 × 2000 lattices. Black arrows in space-time plots show the propagation direction of the wave numbers that correspond to the circles indicated in the spatial power spectra. The corresponding spiral patterns of these plots are (a) overtargeted spiral waves; (b) clockwise superspiral waves (inward rotation); (c) half-plane amplitude-modulating patterns; (d) anticlockwise superspiral waves (outward rotation).

Spatial power spectra (left) and space-time plots (right) for spiral waves with the superstructures shown in Fig. 3. Time runs from bottom to top (0–5000). Space in space-time plots runs from the spiral core (1000, 1000) to the boundary (1800, 1000) on the 2000 × 2000 lattices. Black arrows in space-time plots show the propagation direction of the wave numbers that correspond to the circles indicated in the spatial power spectra. The corresponding spiral patterns of these plots are (a) overtargeted spiral waves; (b) clockwise superspiral waves (inward rotation); (c) half-plane amplitude-modulating patterns; (d) anticlockwise superspiral waves (outward rotation).

Time series of the point (1200,1200) on the 2000 × 2000 lattices of spiral waves. Peaks of the oscillations are connected by a red line. (a) *Dv* = 0, *Dw* = 0.1, twin-armed spiral waves; (b) *Dv* = 0.0009, *Dw* = 0.1, overtargeted spiral waves; (c) *Dv* = 0.1, *Dw* = 0.1, superspiral waves; (d) *Dv* = 0, *Dw* = 0.1, simple spiral wave with single arm.

Time series of the point (1200,1200) on the 2000 × 2000 lattices of spiral waves. Peaks of the oscillations are connected by a red line. (a) *Dv* = 0, *Dw* = 0.1, twin-armed spiral waves; (b) *Dv* = 0.0009, *Dw* = 0.1, overtargeted spiral waves; (c) *Dv* = 0.1, *Dw* = 0.1, superspiral waves; (d) *Dv* = 0, *Dw* = 0.1, simple spiral wave with single arm.

Analysis of local dynamics and spatial correlation. (a) Times series at the point (1200, 1200) in superspiral pattern with *Dv* = *Dw* = 0.1; red line shows the periodic amplitude modulation; *T* _{ s } is the period of the oscillatory packets; *T* _{ b } is the period of a single oscillation. (b) Attractors of the same point in (*u*, *v*, *w*)-space. Black and red trajectories indicate the periods of 1^{0} and 1^{1} oscillations, respectively. Inset shows the Poincaré map for the green section through the attractor. (c) Fourier spectra of the time series containing two fundamental frequencies: the frequency of basic oscillations (*f* _{ b }) and the frequency of oscillatory packets (*f* _{ s }); (d) spatial correlation coefficients of the superspiral patterns for ten periods of basic oscillations (*T* _{ b }), with time step equal to *T* _{ b }/10.

Analysis of local dynamics and spatial correlation. (a) Times series at the point (1200, 1200) in superspiral pattern with *Dv* = *Dw* = 0.1; red line shows the periodic amplitude modulation; *T* _{ s } is the period of the oscillatory packets; *T* _{ b } is the period of a single oscillation. (b) Attractors of the same point in (*u*, *v*, *w*)-space. Black and red trajectories indicate the periods of 1^{0} and 1^{1} oscillations, respectively. Inset shows the Poincaré map for the green section through the attractor. (c) Fourier spectra of the time series containing two fundamental frequencies: the frequency of basic oscillations (*f* _{ b }) and the frequency of oscillatory packets (*f* _{ s }); (d) spatial correlation coefficients of the superspiral patterns for ten periods of basic oscillations (*T* _{ b }), with time step equal to *T* _{ b }/10.

Bifurcation diagram for amplitude-modulated waves. Abscissa is the diffusion coefficient ratio (*δ*) of components *v* and *w*, and ordinate is the frequency of oscillatory packets generated by diffusion-induced instability. Inset shows the detailed evolution for small *δ*. Five different spiral waves mentioned above are indicated in the figure, where the half-plane behavior as shown in Fig. 3(c) corresponds to the transition point (*δ* = 0.4).

Bifurcation diagram for amplitude-modulated waves. Abscissa is the diffusion coefficient ratio (*δ*) of components *v* and *w*, and ordinate is the frequency of oscillatory packets generated by diffusion-induced instability. Inset shows the detailed evolution for small *δ*. Five different spiral waves mentioned above are indicated in the figure, where the half-plane behavior as shown in Fig. 3(c) corresponds to the transition point (*δ* = 0.4).

## Tables

Parameters in HP reaction-diffusion model.

Parameters in HP reaction-diffusion model.

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