Noise-induced temporal dynamics in Turing systems

We examine the ability of intrinsic noise to produce complex temporal dynamics in Turing pattern formation systems, with particular emphasis on the Schnakenberg kinetics. Using power spectral methods, we characterize the behavior of the system using stochastic simulations at a wide range of points in parameter space and compare with analytical approximations. Specifically, we investigate whether polarity switching of stochastic patterns occurs at a defined frequency. We find that it can do so in individual realizations of a stochastic simulation, but that the frequency is not defined consistently across realizations in our samples of parameter space. Further, we examine the effect of noise on deterministically predicted traveling waves and find them increased in amplitude and decreased in speed.

Figure 1

Realization of a stochastic simulation of the Schnakenberg reaction system on a one-dimensional domain, showing repeated polarity switching over time. Only the activator dynamics are shown and the range has been limited for visual emphasis.

Figure 2

Parameter space of the deterministic Schnakenberg system in terms of the dimensionless reaction rates $\alpha$ and $\beta$. Turing instabilities occur to the left of the black line, given by (24). Oscillatory response to perturbations occurs between the red lines, given by (22). Steady states become unstable to the left of the blue line, given by (20). Sampled points are labeled with the corresponding figure number.

Figure 3

Just outside the Turing and oscillatory regimes, (a) the deterministic simulation shows a rapid decay from the initial perturbation. The stochastic simulations show both (b) and (c) static patternlike behavior and (d) and (e) oscillatory patterns in individual realizations. (g) On average, the stochastic behavior is patternlike without a defined frequency of oscillations (40 realizations), as predicted by (f) the analytic spectrum of the Langevin dynamics. The parameters used are ${\kappa }_1=17$, ${\kappa }_2=50$, and the rest are as given in Sec. 2d. The right color bar corresponds to the deterministic results, while the left color bar corresponds to the stochastic results.

Figure 4

Just inside the Turing regime, just outside the oscillatory regime, but (a) without Turing-unstable wave modes, (b) the deterministic simulation shows a rapid decay back to the spatially uniform steady state from the initial perturbation. Shown in (a) are the highest Turing-unstable wave mode (blue solid line) and lowest (green dashed line) as given by (25). The red dash-dotted line shows the domain length used. The stochastic simulations show both (c) and (d) static patternlike behavior and (e) and (f) oscillatory patterns in individual realizations. (h) On average, the stochastic behavior is patternlike without a defined frequency of oscillations (40 realizations), as predicted by (g) the analytic spectrum of the Langevin dynamics. The parameters used are ${\kappa }_2=50$ and the rest are as in Sec. 2d. The right color bar corresponds to the deterministic results, while the left color bar corresponds to the stochastic results.

Figure 5

Just outside the Turing regime, just inside the oscillatory regime, (a) the deterministic simulation shows a rapidly decaying perturbation. Additionally, perturbations of higher magnitude ($1/10$ of the steady state value) did not produce visible oscillations. The stochastic simulations show both (b) and (c) static patternlike behavior and (d) and (e) oscillatory patterns in individual realizations. (g) On average, the stochastic behavior is patternlike without a defined frequency of oscillations (40 realizations), as largely predicted by (f) the analytic spectrum of the Langevin dynamics. The parameters used are ${\kappa }_1=17$ and the rest as in Sec. 2d. The right color bar corresponds to the deterministic results, while the left color bar corresponds to the stochastic results.

Figure 6

Just outside the Turing regime, far inside the oscillatory regime, (a) the deterministic simulation shows an oscillatory decay of the perturbation. Note that the perturbation here is larger than usual; its magnitude is $1/10$ of the steady state value. (b) and (d) The stochastic simulations are difficult to classify, but the power spectra have peaks characteristic of both (c) patterns and (d) background oscillations, if at very low power. (g) On average, the stochastic behavior is patternlike with background oscillations around a defined frequency (40 realizations), as predicted by (f) the analytic spectrum of the Langevin dynamics. The parameters used are ${\kappa }_1=10$, ${\kappa }_2=10$, and the rest as in Sec. 2d. The right color bar corresponds to the deterministic results, while the left color bar corresponds to the stochastic results.

Figure 7

Just inside the Turing regime and just inside the oscillatory regime, but (a) without Turing-unstable wave modes, (b) the deterministic simulation shows a rapidly decaying perturbation. Perturbations of higher magnitude ($1/10$ of the steady state value) did not produce visible oscillations. The stochastic simulations show both (c) and (d) static patternlike behavior and (e) and (f) oscillatory patterns in individual realizations. (h) On average, the stochastic behavior is patternlike without a defined frequency of oscillations (40 realizations), as predicted by (g) the analytic spectrum of the Langevin dynamics. The parameters used are as in Sec. 2d. The right color bar corresponds to the deterministic results, while the left color bar corresponds to the stochastic results. In (a) the highest Turing-unstable wave mode (blue solid line) and lowest (green dashed line) are as given by (25). The red dash-dotted line shows the domain length used.

Figure 8

Just inside the Turing regime, within the oscillatory regime and just outside the unstable regime, but (a) without Turing-unstable wave modes, (b) the deterministic simulation shows an oscillatory decay of the perturbation. Note that the perturbation here is larger than usual; its magnitude is $1/10$ of the steady state value. The stochastic simulations show both (c) and (d) static patternlike behavior and (e) and (f) background oscillations in individual realizations. Most realizations showcase both, as in (h) the average spectrum (29 realizations). (g) The analytic spectrum of the Langevin dynamics predicts the background oscillations as well as the patterning, but the latter not at the right magnitude. The parameters used are ${\kappa }_1=6$, ${\kappa }_2=10$, $L=0.14$, and the rest as in Sec. 2d. The right color bar corresponds to the deterministic results, while the left color bar corresponds to the stochastic results. In (a) the highest Turing-unstable wave mode (blue solid line) and lowest (green dashed line) are as given by (25). The red dash-dotted line shows the domain length used.

Figure 9

Far from the Turing regime, inside the oscillatory and unstable regimes, (a) and (b) the deterministic simulation shows fast-traveling waves. (c) and (e) The stochastic simulations also show waves, but visibly slower. (d) and (f) The sharp wave peaks are constructed of multiples of frequencies, which is (h) consistent across realizations (10 realizations). As the assumptions underlying the Langevin dynamics break down, (g) the analytic spectrum fails to predict the observed shape. The red dashed line shows the dispersion relation. The parameters used are ${\kappa }_1=3$, ${\kappa }_2=10$, $D_u=4\times {10}^{-4}$, $D_v={10}^{-4}$, $L=2$, $K=80$, and the rest as in Sec. 2d. The right color bar corresponds to the deterministic results, while the left color bar corresponds to the stochastic results.