Heterogeneity induces spatiotemporal oscillations in reaction-diffusion systems
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F18%3A00322429" target="_blank" >RIV/68407700:21340/18:00322429 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1103/PhysRevE.97.052206" target="_blank" >http://dx.doi.org/10.1103/PhysRevE.97.052206</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1103/PhysRevE.97.052206" target="_blank" >10.1103/PhysRevE.97.052206</a>
Alternative languages
Result language
angličtina
Original language name
Heterogeneity induces spatiotemporal oscillations in reaction-diffusion systems
Original language description
We report on an instability arising in activator-inhibitor reaction-diffusion (RD) systems with a simple spatial heterogeneity. This instability gives rise to periodic creation, translation, and destruction of spike solutions that are commonly formed due to Turing instabilities. While this behavior is oscillatory in nature, it occurs purely within the Turing space such that no region of the domain would give rise to a Hopf bifurcation for the homogeneous equilibrium. We use the shadow limit of the Gierer-Meinhardt system to show that the speed of spike movement can be predicted from well-known asymptotic theory, but that this theory is unable to explain the emergence of these spatiotemporal oscillations. Instead, we numerically explore this system and show that the oscillatory behavior is caused by the destabilization of a steady spike pattern due to the creation of a new spike arising from endogeneous activator production. We demonstrate that on the edge of this instability, the period of the oscillations goes to infinity, although it does not fit the profile of any well-known bifurcation of a limit cycle. We show that nearby stationary states are either Turing unstable or undergo saddle-node bifurcations near the onset of the oscillatory instability, suggesting that the periodic motion does not emerge from a local equilibrium. We demonstrate the robustness of this spatiotemporal oscillation by exploring small localized heterogeneity and showing that this behavior also occurs in the Schnakenberg RD model. Our results suggest that this phenomenon is ubiquitous in spatially heterogeneous RD systems, but that current tools, such as stability of spike solutions and shadow-limit asymptotics, do not elucidate understanding. This opens several avenues for further mathematical analysis and highlights difficulties in explaining how robust patterning emerges from Turing's mechanism in the presence of even small spatial heterogeneity.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10102 - Applied mathematics
Result continuities
Project
<a href="/en/project/EF17_050%2F0008025" target="_blank" >EF17_050/0008025: International Mobility of Researchers ? MSCA-IF in CTU</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>S - Specificky vyzkum na vysokych skolach
Others
Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
PHYSICAL REVIEW E
ISSN
2470-0045
e-ISSN
2470-0053
Volume of the periodical
97
Issue of the periodical within the volume
5
Country of publishing house
US - UNITED STATES
Number of pages
12
Pages from-to
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UT code for WoS article
000432980600008
EID of the result in the Scopus database
2-s2.0-85047004463