Math. Model. Nat. Phenom.
Volume 12, Number 6, 2017Special Issue - Nonlocal and delay equations
|Page(s)||144 - 170|
|Published online||30 December 2017|
Transitions in asymmetric spike patterns with anomalous diffusion
Thompson Rivers University,
British Columbia, Canada
Accepted: 21 September 2017
Complex natural systems at times manifest transitions between disparate diffusive regimes. Efforts to devise measurement techniques capable of identifying the cross-over moments have recently borne fruit, however interpretation of findings remains contentious when the bigger picture is considered. This study uses a phenomenological reaction–diffusion model to obtain a sufficiently general, yet tractable description for the dynamics of a pattern qualitatively redolent of molecular clusters subject to transient anomalous diffusion mechanisms. The resulting system of equations is investigated for substantiation of the reasons difficulties are encountered when attempting to distinguish between various diffusive regimes in experimental settings: a non-monotonic dependence of the pattern’s evolution on parameters defining the diffusion mechanism is a common occurrence, as is a non-injective mapping between a given sequence of diffusion regimes and ensuing drift behaviour. Albeit formally the developed description is limited to piecewise constant anomaly functions, it is useful in computation of approximations to smooth realistic transitions.
Mathematics Subject Classification: 34E10 / 34E20 / 35K57 / 35K67 / 35K92 / 35Q92 / 35R11 / 37M05
Key words: Fractional differential equation / variable order differential equation / matched asymptotic expansions / non-linear pattern formation / reaction–diffusion / Gierer–Meinhardt model / spike solution / transient diffusion regimes / anomalous diffusion
© EDP Sciences, 2017
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