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) 11 mixed-mode oscillations used in reaction-diffusion simulations, all dynamic parameters are shown in Table I. (d) Typical “tea-cup” mode, 14, 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.
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).
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 10 and 11 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).
Parameters in HP reaction-diffusion model.
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