Influence of physical interactions on spatiotemporal patterns

Spatiotemporal patterns are often modeled using reaction-diffusion equations, which combine complex reactions between constituents with ideal diffusive motion. Such descriptions neglect physical interactions between constituents, which might affect resulting patterns. To overcome this, we study how physical interactions affect cyclic dominant reactions, like the seminal rock-paper-scissors game, which exhibits spiral waves for ideal diffusion. Generalizing diffusion to incorporate physical interactions, we find that weak interactions change the length- and time scales of spiral waves, consistent with a mapping to the complex Ginzburg-Landau equation. In contrast, strong repulsive interactions typically generate oscillating lattices, and strong attraction leads to an interplay of phase separation and chemical oscillations, like droplets co-locating with cores of spiral waves. Our work suggests that physical interactions are relevant for forming spatiotemporal patterns in nature, and it might shed light on how biodiversity is maintained in ecological settings.

Figure 1

Linear stability analysis reveals distinct parameter regions. (a) Schematic of physical interactions and chemical reactions of three species $A, B, C$, and the inert solvent $S$. (b) Stability diagram distinguishing regions of low (L) and high (H) mutation rate $\mu$ as well as strong attraction (A), weak interaction (W), and strong repulsion (R). The critical lines follow from Eq. (4) (black line), Eq. (9a) (blue line), Eq. (9b) (red line), and ${\chi }_{\mathrm{R}}^{\mathrm{eq}}=1/{\varphi }_0$ (white line). The colors represent the length scales $l_{\mathrm{r}}^{\mathrm{m}}$ in regimes AH and AL, $l_{\mathrm{c}}^{\mathrm{m}}$ in RH and RL, and $l_{\mathrm{c}}^{+}$ in WL. (c) Representative dispersion relations $\lambda \left(q\right)$ in the six regimes. Green curves represent real eigenvalues ${\lambda }_{\mathrm{r}}\left(q\right)$ with left root ($q_{\mathrm{r}}^{-}$, green disk), right root ($q_{\mathrm{r}}^{+}$, green circle), and maximum ($q_{\mathrm{r}}^{\mathrm{m}}$, green triangle) marked. Solid orange curves and symbols represent the respective values for the real part of the complex eigenvalues, whereas the dashed orange lines mark the imaginary part $Im\left({\lambda }_{\mathrm{c}}\right)={\omega }_{*}$; see Eq. (5). (d) Typical length scales as a function of $\chi$ at $\mu =0.05>{\mu }_{*}$ (upper panel) and $\mu =0.001<{\mu }_{*}$ (lower panel). The subscript and superscript of the length scale $l$ correspond to those of wave number $q$ in panel (c) using $l=2\pi /q$. (b)–(d) Additional model parameters are $\beta =\sigma =D=1$ and $\zeta =0.6$.

Figure 2

Numerical simulations reveal diverse patterns. (a) Phase diagram with stability lines copied from Fig. 1. Background colors also correspond to Fig. 1, except in region WL, where they mark the length scale $\lambda$ given by Eq. (13). The green dashed lines corresponds to $\lambda \approx 0.7L$, which fits the transition best. Symbols classify different patterns corresponding to examples in panel (b). (b) Snapshots of representative patterns for panel (a). Colors represent the abundance of the three species using RGB triplets: (red, green, blue) = (${\varphi }_A, {\varphi }_B, {\varphi }_C$). We used $L=25.6w, \mathrm{\Delta }x=0.4w$, and $t={10}^5\beta$ for the snapshots in the first row and the one marked with white circle in the second row. Movies of these states are enclosed with the Supplemental Material [46]. (a)–(b) Model parameters are $D=\beta =\alpha =1$ and $\zeta =0.6$. Simulation parameters are $L/w=153.6$ with discretization $\mathrm{\Delta }x/w=0.6$ and we evaluate patterns after time $t={10}^4\beta$.

Figure 3

Length- and time scales of dynamic patterns. (a) Spatial correlation function $g_{AB}\left(r\right)$ as a function of distance $r$ for various physical interactions $\chi$ at $\zeta =0.8$ and $\mu =0.005$. (b) Correlation length scale $l_{\mathrm{corr}}$ determined from first maximum of $g_{AB}\left(r\right)$ as a function of $\chi$ at $\mu =0.005$. The black dashed (dotted) line corresponds to $\frac{1}{3}\lambda$ calculated from Eq. (13) at $\zeta =0.86$ ($\zeta =0.30$). The orange lines are the same as in Fig. 1 at $\mu =0.005$. (c) $l_{\mathrm{corr}}$ as function of $\chi$ for $\zeta =0.6$. The black dashed (dotted) line corresponds to $\frac{1}{3}\lambda$ calculated by Eq. (13) at $\mu =0.03$ ($\mu =0.001$). The green and orange lines are the same as in Fig. 1 at $\mu =0.001$. The brown dotted line is $l_{\mathrm{c}}^{+}$ at $\mu =0.03$. (d) Temporal correlation function ${\tilde{g}}_{AA}\left(t\right)$ as a function of lag time $t$ for various $\chi$ at $\zeta =0.8$ and $\mu =0.005$. (e) Frequency $\omega$ determined from the first maximum of ${\tilde{g}}_{AA}$ as a function of $\chi$ for $\mu =0.005$. The black dashed (dotted) line shows $\omega$ given by Eq. (14) for $\zeta =1.24$ ($\zeta =0.14$). (f) $\omega$ as a function of $\chi$ for $\zeta =0.6$. The black dashed (dotted) line shows $\omega$ given by Eq. (14) at $\mu =0.02$ ($\mu =0.001$). (a)–(f) The vertical dashed blue (red) lines denote the critical interactions ${\chi }_{\mathrm{A}}=-10$ (${\chi }_{\mathrm{R}}^{*}=4$). Additional model parameters are $\beta =\alpha =D=1$.

Figure 4

Details of complex, oscillating patterns. (a) Snapshots showing one temporal period for four physical interactions ($\chi =-10.5,5,6.5,7$; top to bottom) at $\mu =0.05>{\mu }_{*}$ and $\zeta =1$. The corresponding periods are $T\approx 11\beta ,11\beta ,52\beta ,610\beta$. (b) Structure factors $S_{AA}\left(q\right)$ corresponding to data in panel (a). The vertical lines mark the wave number of the most unstable mode determined from linear stability analysis.