# Linear Stability of Fractional Reaction - Diffusion Systems

Mathematical Modelling of Natural Phenomena (2010)

- Volume: 2, Issue: 2, page 77-105
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topNec, Y., and Nepomnyashchy, A. A.. "Linear Stability of Fractional Reaction - Diffusion Systems." Mathematical Modelling of Natural Phenomena 2.2 (2010): 77-105. <http://eudml.org/doc/222256>.

@article{Nec2010,

abstract = {
Theoretical framework for linear stability of an anomalous sub-diffusive
activator-inhibitor system is set. Generalized Turing instability conditions are found to depend on
anomaly exponents of various species. In addition to monotonous instability, known from
normal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly
exponents for both species the type of unstable modes is determined by the ratio of the reactants'
diffusion coefficients. When the ratio exceeds its normal critical value, the monotonous modes
become stable, whereas oscillatory instability persists until the anomalous critical value is
also exceeded. An exact formula for the anomalous critical value is obtained. It is shown
that in the short wave limit the growth rate is a power law of the wave number. When
the anomaly exponents differ, disturbance modes are governed by power laws of the distinct
exponents. If the difference between the diffusion anomaly exponents is small, the splitting
of the power law exponents is absent at the leading order and emerges only as a next-order
effect. In the short wave limit the onset of instability is governed by the anomaly exponents,
whereas the ratio of diffusion coefficients influences the complex growth rates. When the
exponent of the inhibitor is greater than that of the activator, the system is always unstable
due to the inhibitor enhanced diffusion relatively to the activator. If the exponent of the
activator is greater, the system is always stable. Existence of oscillatory unstable modes is
also possible for waves of moderate length.
},

author = {Nec, Y., Nepomnyashchy, A. A.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {reaction-diffusion; anomaly exponent; monotonous instablity; oscillatory instability;
critical diffusion coefficients' ratio},

language = {eng},

month = {3},

number = {2},

pages = {77-105},

publisher = {EDP Sciences},

title = {Linear Stability of Fractional Reaction - Diffusion Systems},

url = {http://eudml.org/doc/222256},

volume = {2},

year = {2010},

}

TY - JOUR

AU - Nec, Y.

AU - Nepomnyashchy, A. A.

TI - Linear Stability of Fractional Reaction - Diffusion Systems

JO - Mathematical Modelling of Natural Phenomena

DA - 2010/3//

PB - EDP Sciences

VL - 2

IS - 2

SP - 77

EP - 105

AB -
Theoretical framework for linear stability of an anomalous sub-diffusive
activator-inhibitor system is set. Generalized Turing instability conditions are found to depend on
anomaly exponents of various species. In addition to monotonous instability, known from
normal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly
exponents for both species the type of unstable modes is determined by the ratio of the reactants'
diffusion coefficients. When the ratio exceeds its normal critical value, the monotonous modes
become stable, whereas oscillatory instability persists until the anomalous critical value is
also exceeded. An exact formula for the anomalous critical value is obtained. It is shown
that in the short wave limit the growth rate is a power law of the wave number. When
the anomaly exponents differ, disturbance modes are governed by power laws of the distinct
exponents. If the difference between the diffusion anomaly exponents is small, the splitting
of the power law exponents is absent at the leading order and emerges only as a next-order
effect. In the short wave limit the onset of instability is governed by the anomaly exponents,
whereas the ratio of diffusion coefficients influences the complex growth rates. When the
exponent of the inhibitor is greater than that of the activator, the system is always unstable
due to the inhibitor enhanced diffusion relatively to the activator. If the exponent of the
activator is greater, the system is always stable. Existence of oscillatory unstable modes is
also possible for waves of moderate length.

LA - eng

KW - reaction-diffusion; anomaly exponent; monotonous instablity; oscillatory instability;
critical diffusion coefficients' ratio

UR - http://eudml.org/doc/222256

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.