Spontaneous traveling waves in oscillatory systems with cross diffusion

We identify a type of pattern formation in spatially distributed active systems. We simulate one-dimensional two-component systems with predator-prey local interaction and pursuit-evasion taxis between the components. In a sufficiently large domain, spatially uniform oscillations in such systems are unstable with respect to small perturbations. This instability, through a transient regime appearing as spontaneous focal sources, leads to establishment of periodic traveling waves. The traveling wave regime is established even if boundary conditions do not favor such solutions. The stable wavelength is within a range bounded both from above and from below, and this range does not coincide with instability bands of the spatially uniform oscillations.

Figure 1

Different regimes resulting from random perturbation of uniform oscillations. Shown are density plots: space $x$ is horizontal, time $t$ is vertical increasing upward; $u=1$ corresponds to black and $u=0$ corresponds to white. (a) TB model, $L=15$, taxis $\left(h_{-}=1,h_{+}=0,D_u=D_v=0\right)$, and periodic boundary conditions, $t∊\left[0,2500\right]$. (b) Same as (a), except boundary conditions are no-flux, $t∊\left[0,1200\right]\cup \left[43800,45000\right]$. (c) Same as (a) except $D_u=D_v=0.05$, $h_{+}=h_{-}=0$. (d) Same as (a) except $w=0.07$, $L=10$, $t∊\left[0,1200\right]\cup \left[36300,37500\right]$. (e) Same as (a) except $h_{+}=0.1$, $L=50$, $t∊\left[0,1200\right]\cup \left[5000,6200\right]$. (f) Same as (a) except $h_{+}=0.1$, $D_u=D_v=0.02$, $L=50$, $t∊\left[0,400\right]\cup \left[1900,3900\right]$. (g) RM model, $h_{-}=1$, $h_{+}=D_u=D_v=0$, $L=25$, $t∊\left[0,2500\right]$.

Figure 2

(Color online) Examples of profiles of spontaneously established periodic waves. Shown are dependencies of $u$ and $v$ on $x$ at fixed $t$ and direction of propagation by arrows. Parameters are the same as in Fig. 1 except interval length $L$, specifically, (a) as in Figs. 1a, 1b, as in Figs. 1d, 1c, as in Figs. 1e, 1d, as in Figs. 1f, 1e, and as in Fig. 1g.

Figure 3

(Color online) Variability of the wave trains at changing $L$. (a) Number of waves in the interval $\left[0,L\right]$ as $L$ gradually increases (by 0.2 every 2000 time units, solid red line) and decreases (by 0.2 every 1000 time units, dashed blue line). Oblique dashed lines: $n=L/2.5$ and $n=L/8$ to guide the eyes. (b) Wave periods measured at a point as a function of $L$ as it decreases. (c) Same as $L$ increases. (d) Density plots of two episodes of simulation of 1200 time unit duration each, with $L$ increasing by 0.2 every 1000 time units. Lower episode: soft transition from steady one-wave solution to modulated one-wave solution $\left(L:4.8\to 5.6\right)$. Upper episode: subsequent sudden transition from modulated one-wave solution to a steady two-wave solution $\left(L:7.2\to 7.6\right)$.

Figure 4

(Color online) Emergence of spontaneous periodic waves through instabilities. (a) Spectrum $\lambda \left(k\right)$ of harmonic perturbations of the spatially uniform oscillations (black lines with points) and the histograms of the wave numbers of spontaneous wave trains in the simulations shown in Fig. 3 for increasing $L$ (red solid line) and decreasing $L$ (blue dashed line). (b) Same, for the RM model, parameters as in Fig. 1f, histograms obtained by increasing $L$ by 0.2 every 2000 time units from 5 to 50 and decreasing it back to 2 by 0.2 every 1000 time units. (c) Emergence of standing periodic waves via an instability of the spatially uniform oscillations. Parameters as in Figs. 1a, 3, $L=12.26$, $\Delta x=L/63$, $\Delta t=5\times {10}^{-5}$, and $t∊\left[0,1000\right]$. (d) Emergence of spontaneous periodic wave trains via an instability of periodic standing waves. Continuation of (c), $t∊\left[44460,45460\right]$.