Travelling wave fronts in reaction–diffusion systems with spatio-temporal delays

The Turing and Benjamin–Feir instabilities are two of the primary instability mechanisms useful for studying the transition from homogeneous states to heterogeneous spatial or spatio-temporal states in reaction–diffusion systems. We consider the case when the underlying reaction–diffusion system is non-autonomous or has a base state which varies in time, as in this case standard approaches, which rely on temporal eigenvalues, break down. We are able to establish respective criteria for the onset of each instability using comparison principles, obtaining inequalities which involve the in general time-dependent model parameters and their time derivatives. In the autonomous limit where the base state is constant in time, our results exactly recover the respective Turing and Benjamin–Feir conditions known in the literature. Our results make the Turing and Benjamin–Feir analysis amenable for a wide collection of applications, and allow one to better understand instabilities emergent due to a variety of non-autonomous mechanisms, including time-varying diffusion coefficients, time-varying reaction rates, time-dependent transitions between reaction kinetics and base states which change in time (such as heteroclinic connections between unique steady states, or limit cycles), to name a few examples.

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Travelling–wave solutions to monostable reaction–diffusion systems of mixed monotone type

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.1999.0372 ◽

1999 ◽

Vol 455 (1984) ◽

pp. 1561-1598 ◽

Cited By ~ 5

Author(s):

H. Takase ◽

B. D. Sleeman

Keyword(s):

Reaction Diffusion ◽

Travelling Wave ◽

Travelling Wave Solutions ◽

Reaction Diffusion Systems ◽

Wave Solutions ◽

Diffusion Systems ◽

Monotone Type

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Noise-induced suppression of periodic travelling waves in oscillatory reaction–diffusion systems

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2009.0611 ◽

2010 ◽

Vol 466 (2119) ◽

pp. 1903-1917 ◽

Cited By ~ 11

Author(s):

Michael Sieber ◽

Horst Malchow ◽

Sergei V. Petrovskii

Keyword(s):

Travelling Waves ◽

Natural Populations ◽

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Oscillatory Reaction ◽

Stochastic Forcing ◽

Spatial Direction ◽

Reaction Diffusion Systems ◽

Diffusion Systems ◽

Spatio Temporal

Ecological field data suggest that some species show periodic changes in abundance over time and in a specific spatial direction. Periodic travelling waves as solutions to reaction–diffusion equations have helped to identify possible scenarios, by which such spatio-temporal 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 reaction–diffusion systems. Irregular spatio-temporal oscillations, however, appear to be more robust and persist under the same stochastic forcing.

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