Flow-induced arrest of spatiotemporal chaos and transition to a stationary pattern in the Gray-Scott model

Spatiotemporal representation of long-delayed systems: An alternative approach

Dynamical systems with long-delay feedback can exhibit complicated temporal phenomena, which once reorganized in a two-dimensional space are reminiscent of spatiotemporal behavior. In this framework, a normal forms description has been developed to reproduce the dynamics, and the opportunity to treat the corresponding variables as true space and time has since been established. However, recently, an alternative approach has been proposed [F. Marino and G. Giacomelli, Phys. Rev. E 98, 060201(R) (2018)2470-004510.1103/PhysRevE.98.060201] with a different interpretation of the variables involved, which better takes into account their physical character and allows for an easier determination of the normal forms. In this paper, we extend such idea and apply it to a number of paradigmatic examples, paving the way to a rethinking of the concept of spatiotemporal representation of long-delayed systems.

Cross-diffusion-induced subharmonic spatial resonances in a predator-prey system

In this paper we investigate the complex dynamics originated by a cross-diffusion-induced subharmonic destabilization of the fundamental subcritical Turing mode in a predator-prey reaction-diffusion system. The model we consider consists of a two-species Lotka-Volterra system with linear diffusion and a nonlinear cross-diffusion term in the predator equation. The taxis term in the search strategy of the predator is responsible for the onset of complex dynamics. In fact, our model does not exhibit any Hopf or wave instability, and on the basis of the linear analysis one should only expect stationary patterns; nevertheless, the presence of the nonlinear cross-diffusion term is able to induce a secondary instability: due to a subharmonic spatial resonance, the stationary primary branch bifurcates to an out-of-phase oscillating solution. Noticeably, the strong resonance between the harmonic and the subharmonic is able to generate the oscillating pattern albeit the subharmonic is below criticality. We show that, as the control parameter is varied, the oscillating solution (subT mode) can undergo a sequence of secondary instabilities, generating a transition toward chaotic dynamics. Finally, we investigate the emergence of subT-mode solutions on two-dimensional domains: when the fundamental mode describes a square pattern, subharmonic resonance originates oscillating square patterns. In the case of subcritical Turing hexagon solutions, the internal interactions with a subharmonic mode are able to generate the so-called "twinkling-eyes" pattern.

Comment on "Flow-induced arrest of spatiotemporal chaos and transition to a stationary pattern in the Gray-Scott model"

In this Comment, we review the results of pattern formation in a reaction-diffusion-advection system following the kinetics of the Gray-Scott model. A recent paper by Das [Phys. Rev. E 92, 052914 (2015)10.1103/PhysRevE.92.052914] shows that spatiotemporal chaos of the intermittency type can disappear as the advective flow is increased. This study, however, refers to a single point in the space of kinetic parameters of the original Gray-Scott model. Here we show that the wealth of patterns increases substantially as some of these parameters are changed. In addition to spatiotemporal intermittency, defect-mediated turbulence can also be found. In all cases, however, the chaotic behavior is seen to disappear as the advective flow is increased, following a scenario similar to what was reported in our earlier work [I. Berenstein and C. Beta, Phys. Rev. E 86, 056205 (2012)10.1103/PhysRevE.86.056205] as well as by Das. We also point out that a similar phenomenon can be found in other reaction-diffusion-advection models, such as the Oregonator model for the Belousov-Zhabotinsky reaction under flow conditions.

Reply to "Comment on 'Flow-induced arrest of spatiotemporal chaos and transition to a stationary pattern in the Gray-Scott model' "

We reply to the Comment of Berenstein et al. [preceding paper, Phys. Rev. E 94, 046201 (2016)10.1103/PhysRevE.94.046201] on our article [D. Das, Phys. Rev. E 92, 052914 (2015)10.1103/PhysRevE.92.052914] about the effect of streamlined flow on the dynamics of the Gray-Scott model characterized by wave-induced spatiotemporal chaos.