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Turing and wave instabilities in hyperbolic reaction–diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation

Ritchie, Joshua S.; Krause, Andrew L.; Van Gorder, Robert A.

Turing and wave instabilities in hyperbolic reaction–diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation Thumbnail


Authors

Joshua S. Ritchie

Robert A. Van Gorder



Abstract

Hyperbolic reaction-diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities not present in classical two-species reaction-diffusion systems. We explore the onset of diffusive instabilities and resulting pattern formation for such systems. Starting with a rather general formulation of the problem, we obtain necessary and sufficient conditions for the Turing and wave instabilities in such systems, thereby classifying parameter spaces for which these diffusive instabilities occur. We find that the additional temporal terms do not strongly modify the Turing patterns which form or parameters which admit them, but only their regions of existence. This is in contrast to the case of additional space derivatives, where past work has shown that resulting patterned structures are sensitive to second-order cross-diffusion and first-order advection. We also show that additional temporal terms are necessary for the emergence of spatiotemporal patterns under the wave instability. We find that such wave instabilities exist for parameters which are mutually exclusive to those parameters leading to stationary Turing patterns. This implies that wave instabilities may occur in cases where the activator diffuses faster than the inhibitor, leading to routes to spatial symmetry breaking in reaction-diffusion systems which are distinct from the well studied Turing case.

Citation

Ritchie, J. S., Krause, A. L., & Van Gorder, R. A. (2022). Turing and wave instabilities in hyperbolic reaction–diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation. Annals of Physics, 444, Article 169033. https://doi.org/10.1016/j.aop.2022.169033

Journal Article Type Article
Acceptance Date Jul 12, 2022
Online Publication Date Jul 21, 2022
Publication Date 2022-09
Deposit Date Jul 29, 2022
Publicly Available Date Jul 22, 2023
Journal Annals of Physics
Print ISSN 0003-4916
Publisher Elsevier Masson
Peer Reviewed Peer Reviewed
Volume 444
Article Number 169033
DOI https://doi.org/10.1016/j.aop.2022.169033
Public URL https://durham-repository.worktribe.com/output/1199092

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