Delay-induced wave instabilities in single-species reaction-diffusion systems

The Turing (wave) instability is only possible in reaction-diffusion systems with more than one (two) components. Motivated by the fact that a time delay increases the dimension of a system, we investigate the presence of diffusion-driven instabilities in single-species reaction-diffusion systems with delay. The stability of arbitrary one-component systems with a single discrete delay, with distributed delay, or with a variable delay is systematically analyzed. We show that a wave instability can appear from an equilibrium of single-species reaction-diffusion systems with fluctuating or distributed delay, which is not possible in similar systems with constant discrete delay or without delay. More precisely, we show by basic analytic arguments and by numerical simulations that fast asymmetric delay fluctuations or asymmetrically distributed delays can lead to wave instabilities in these systems. Examples, for the resulting traveling waves are shown for a Fisher-KPP equation with distributed delay in the reaction term. In addition, we have studied diffusion-induced instabilities from homogeneous periodic orbits in the same systems with variable delay, where the homogeneous periodic orbits are attracting resonant periodic solutions of the system without diffusion, i.e., periodic orbits of the Hutchinson equation with time-varying delay. If diffusion is introduced, standing waves can emerge whose temporal period is equal to the period of the variable delay.

Figure 1

The analytical results for the MSE $\lambda$ of the equilibrium $u^{*}=1$ of Eq. (15) coincide very well with our numerical results ($a=1$, ${\tau }_0=1$).

Figure 2

D-curves of Eq. (18) for ${\tau }_0=1$. The (red) arrows indicate the direction of the D-curve with increasing $\omega$. No Turing or wave instability is possible (see text).

Figure 3

D-curves of Eq. (19) with an asymmetric delay distribution $\rho \left(\tau \right)=0.7\delta (\tau -0.48)+0.3\delta (\tau -1.52)$. Panel (b) is an enlarged figure of panel (a) and shows the parameter region, where a wave instability is possible. The red arrows indicate the direction of the D-curves with increasing $\omega$.

Figure 4

MSE $\lambda \left(k\right)$ for different periods $T_p$ of a rectangular delay fluctuation Eq. (22). Two limiting cases for slowly and fast time-varying delay can be identified, i.e., large and small periods $T_p$, respectively.

Figure 5

Convergence of the MSE $\lambda \left(k\right)$ for the two limiting cases of a (a) slow and (b) fast delay fluctuation. The same parameters as in Fig. 4 are used.

Figure 6

1D-Turing pattern for the system Eq. (15) with the distributed delay $\rho \left(\tau \right)=0.7\delta (\tau -0.48)+0.3\delta (\tau -1.52)$ and periodic boundary condition ($a=6.1$, $D=0.1$).

Figure 7

Snapshots of 2D-Turing pattern for the system Eq. (15) with the distributed delay $\rho \left(\tau \right)=0.7\delta (\tau -0.48)+0.3\delta (\tau -1.52)$ and periodic boundary conditions ($a=6.1$, $D=0.1$). The snapshots are taken at (a) $t=1000$, (b) $t=2000$, (c) $t=3000$, and (d) $t=5000$. The size of space is $18\times 18$. An initial condition near the equilibrium $u^{*}=1$ with an arbitrary small perturbation was used.

Figure 8

MLE $\mathrm{\Lambda }$ of Eq. (15) for $D=0$ (black) and $D=0.05$ (red dashed) dependent on the parameter $a$. A sine-shaped delay with ${\tau }_m=1.5$, ${\tau }_A=0.5$, and $T_p=2{\pi }^2$ is used. Periodic boundary conditions and random initial conditions near the equilibrium $u^{*}=1$ are considered. The space size is 10.

Figure 9

(a) Periodic orbit for Hutchinson equation with a sine-shaped fluctuating delay with ${\tau }_m=1.5$, ${\tau }_A=0.5$, $T_p=2{\pi }^2$, and $a=1.45$. (b) The MSE $\lambda \left(k\right)$ for a spatial homogeneous state with the periodic orbit shown in (a).

Figure 10

Asymptotic state of Eq. (15) in case of a diffusion-driven instability for parameters as in Fig. 9 and (a) $D=0.01$, (b) $D=0.1$. Space size is 32, and periodic boundary conditions are considered. The initial condition is chosen close to the homogeneous periodic orbit in Fig. 9.

Figure 11

Spatial spectrum for structures in Fig. 10 with (a) $D=0.01$ and (b) $D=0.1$. A larger diffusion coefficient leads to a larger wavelength of the structures. The dominant peaks are located at wave numbers where the MSE $\lambda \left(k\right)$ of the homogeneous periodic solution is positive; cf. Fig. 9.