Nonlinear stability of source defects in oscillatory media

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2010

Ecological field data suggests, that some species show periodic changes in abundance over time and in a specific spatial direction. Periodic travelling waves as solutions to reactiondiffusion equations have helped to identify possible scenarios, by which such spatiotemporal patterns may arise. In this paper, such solutions are tested for their robustness against an irregular temporal forcing, since most natural populations can be expected to be subject to erratic fluctuations imposed by the environment. It is found, that small environmental noise is able to suppress periodic travelling waves in stochastic variants of oscillatory reactiondiffusion systems. Irregular spatiotemporal oscillations however appear to be more robust and persist under the same stochastic forcing.

2008

We identify a new 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 spontanous focal sources, leads to establishment of periodic traveling waves. The traveling waves regime is established even if boundary conditions do not favor such solutions. The stable wavelength are within a range bounded both from above and from below, and this range does not coincide with instability bands of the spatially uniform oscillations.

Archive for Rational Mechanics and Analysis, 2012

Extending results of Johnson and Zumbrun showing stability under localized (L 1 ) perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. The main new ingredient is a detailed analysis of linear behavior under modulational data ū′ (x)h0(x), where ū is the background profile and h0 is the initial modulation.

Physical Review E, 2003

Wave propagation in an oscillatory reaction-diffusion, one-dimensional domain of finite size with Dirichlet boundary conditions is analyzed. For sizes below a certain threshold length, the medium cannot sustain wave motion. Above this threshold we find that for a relatively small domain extent, a strong correlation exists between the dynamics of the system and its size. This correlation gradually disappears with increasing domain size. For still larger sizes, we observe an effect of wave pseudo reflection near the boundary. It is shown both numerically and analytically that pseudoreflected waves are periodically generated inside the medium by a fast, self-generated ''source'' near the boundary.

Journal of Mathematical Analysis and Applications, 2021

In this paper, we investigate the asymptotic stability of a semi-trivial wavefront ( u , v ) = ( u 0 , 0 ) in a reaction-diffusion system which including a small parameter e in the reaction term. In the absence of v-component, the dynamics of u-component reduces to a reaction-diffusion system with a quasi-monotone reaction term, hence a family of Fisher type wavefronts u 0 ( ⋅ ; e ) exists. When e = 0 , the partial derivative of the reaction terms for the v-component with respect to v itself is assumed to be upper or lower triangular. Using the spectral analysis and the perturbation method, we find a suitable exponential weighted space in which the semi-trivial wavefront ( u 0 ( ⋅ ; e ) , 0 ) is asymptotically stable for small enough | e | . We apply the stability results to two examples as demonstration.

Journal of Differential Equations, 1983

Physica D: Nonlinear Phenomena, 1995

We study the front dynamics of the bistable reaction-diffusion equations with periodic diffusion and/or convection coefficients in several space dimensions. When traveling wave solutions exist, the solutions of the initial value problem behave as wave fronts propagating with the effective speeds of traveling waves under various initial conditions. Yet due to the bistable nature of the nonlinearity, traveling waves may not always exist when the medium variations from the mean states are large enough. Their existence is closely related to the detailed forms of diffusion and convection coefficients, more so in multidimension than in one. We present a simple sufficient condition for the nonexistence of traveling waves (quenching) using perturbation method. Our two dimensional finite difference numerical computations show a variety of front behaviors, such as: the propagation, quenching and retreat of fronts. We found numerically that quenching occurs in two space dimensions when diffusion is spatially uniform and convection field is a periodic array of rotating vortices if the root mean square of the convection field reaches a critical number.

Discrete and Continuous Dynamical Systems - Series B, 2004

When the steady states at infinity become unstable through a pattern forming bifurcation, a travelling wave may bifurcate into a modulated front which is time-periodic in a moving frame. This scenario has been studied by B. Sandstede and A. Scheel for a class of reaction-diffusion systems on the real line. Under general assumptions, they showed that the modulated fronts exist and are spectrally stable near the bifurcation point. Here we consider a model problem for which we can prove the nonlinear stability of these solutions with respect to small localized perturbations. This result does not follow from the spectral stability, because the linearized operator around the modulated front has essential spectrum up to the imaginary axis. The analysis is illustrated by numerical simulations.

The Journal of Physical Chemistry A, 1999

Various patterns of standing waves are found beyond the onset of the short-wave instability in a model reactiondiffusion system. These include plain and modulated stripes, squares, and rhombi in systems with square and rectangular geometry and patterns with rotational symmetry in systems with circular geometry. We also find standing waves consisting of periodic time sequences of stripes and rhombi, stripes and squares, and stripes, rhombi, and hexagons. The short-wave instability can lead to a much greater variety of spatio-temporal patterns than the aperiodic Turing and the long-wave oscillatory instabilities. For instance, a single oscillatory cycle can display all the basic patterns related to the aperiodic Turing instabilitysstripes, hexagons, and inverted hexagons (honeycomb)sas well as rhombi and modulated stripes.

Journal of Differential Equations, 2012

We consider reaction-diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u0(kx − ωt; k) that are parameterized by the wave number k. We prove stable diffusive mixing of the asymptotic states u0(kx + φ±; k) as x → ±∞ with different phases φ− = φ+ at infinity for solutions that initially converge to these states as x → ±∞. The proof is based on Bloch wave analysis, renormalization theory, and a rigorous decomposition of the perturbations of these wave solutions into a phase mode, which shows diffusive behavior, and an exponentially damped remainder. Depending on the dispersion relation, the asymptotic states mix linearly with a Gaussian profile at lowest order or with a nonsymmetric non-Gaussian profile given by Burgers equation, which is the amplitude equation of the diffusive modes in the case of a nontrivial dispersion relation.