The initiation and propagation of travelling waves on membrane interfaces in the Belousov-Zhabotinskii reaction

Travelling reaction-diffusion waves are considered in a simplified model of the Belousov -Zhabotinskii reaction, described mathematically by the two-variable Oregonator. A one-dimensional problem consisting of two regions is considered. Region I (effectively the boundary at x ' = 0) acts as a reservoir with a fixed concentration of the autocatalytic species (hypobromous acid), and provides constant in put of this species in to region II. Region II (the reaction zone 0 < x & < ∞) allows diffusion of the autocatalyst while the catalytic species Ce IV is assumed immobilized on a supporting matrix . The form of the ensuing travelling wavefront and the behaviour in the behind the front as it propagates in to the region of increasing x ', is considered. By examining the large time behaviour it is shown that the propagating front travels with its minimum possible wave speed. Both single travelling waves and periodic wave trains are observed.

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Diffusive mixing of periodic wave trains in reaction–diffusion systems

Journal of Differential Equations ◽

10.1016/j.jde.2011.10.014 ◽

2012 ◽

Vol 252 (5) ◽

pp. 3541-3574 ◽

Cited By ~ 19

Author(s):

Björn Sandstede ◽

Arnd Scheel ◽

Guido Schneider ◽

Hannes Uecker

Keyword(s):

Reaction Diffusion ◽

Periodic Wave ◽

Reaction Diffusion Systems ◽

Diffusion Systems ◽

Wave Trains ◽

Diffusive Mixing

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Coexistence of stably propagating periodic wave trains in intrinsically bistable reaction-diffusion systems

Physics Letters A ◽

10.1016/0375-9601(94)90135-x ◽

1994 ◽

Vol 191 (3-4) ◽

pp. 251-256 ◽

Cited By ~ 4

Author(s):

Shinji Koga

Keyword(s):

Reaction Diffusion ◽

Periodic Wave ◽

Reaction Diffusion Systems ◽

Diffusion Systems ◽

Wave Trains

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Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1994.0133 ◽

1994 ◽

Vol 447 (1929) ◽

pp. 155-174 ◽

Cited By ~ 25

Keyword(s):

Travelling Waves ◽

Reaction Diffusion ◽

Wave Structure ◽

Diffusion Front ◽

Wave Fronts ◽

Chemical Systems ◽

Diffusion Waves ◽

System A ◽

Autocatalytic System ◽

Permanent Form

The propagation of planar reaction-diffusion waves in the isothermal autocatalytic system A + m B → ( m + 1)B, rate kab m , is considered. Attention is paid to the case when the diffusion coefficient of reactant A is much less than that of the autocatalyst; the case when A is completely immobilized is discussed in detail. Permanent-form travelling waves are initiated and their structure is treated. It is found that for sufficiently large values of m this steady wave structure becomes unstable to longitudinal disturbances and an oscillatory wave structure develops. The structure of this propagating oscillatory reaction-diffusion front is also discussed.

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The development of travelling waves in a simple isothermal chemical system. IV. Quadratic autocatalysis with quadratic decay

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1991.0112 ◽

1991 ◽

Vol 434 (1892) ◽

pp. 531-554 ◽

Cited By ~ 25

Keyword(s):

Travelling Waves ◽

Reaction Diffusion ◽

Travelling Wave ◽

Chemical System ◽

Governing Equations ◽

Initial Value ◽

Dimensional Parameter ◽

Diffusion Waves ◽

Permanent Form ◽

Bounds On The Solution

The initiation of travelling reaction-diffusion waves in the chemical system governed by the quadratic autocatalytic or branching reaction A + B → 2B (rate k 1 ab) coupled with the decay or termination step B + B → C (rate k 4 b 2 ) is discussed. The system is described by the non-dimensional parameter K - k 4 / k 1 and parameters representing the local initial input of B. It is shown that a travelling wave of permanent form will develop for all K (and no matter how small the initial input of B). Bounds on the solution of the initial-value problem are obtained as well as numerical integrations of the governing equations. The structure of the permanent form travelling waves that arise is discussed in some detail, as well as the asymptotic limits K → 0 and K → ∞. The behaviour of the solution for this problem is compared with solutions found previously for other related simple autocatalytic systems with autocatalyst decay.

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Two components is too simple: an example of oscillatory Fisher–KPP system with three components

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2019.46 ◽

2019 ◽

Vol 150 (6) ◽

pp. 3097-3120 ◽

Cited By ~ 1

Author(s):

Léo Girardin

Keyword(s):

Steady State ◽

Travelling Waves ◽

Stable Limit Cycle ◽

Periodic Wave ◽

Stable Limit ◽

Predator Prey ◽

Positive Steady State ◽

Long Time ◽

Wave Trains ◽

Globally Attractive

AbstractIn a recent paper by Cantrell et al. [9], two-component KPP systems with competition of Lotka–Volterra type were analyzed and their long-time behaviour largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a predator–prey structure near the steady state, periodic wave trains and point-to-periodic rapid travelling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces.

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Isolated periodic wave trains and local critical wave lengths for a nonlinear reaction-diffusion equation

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2019.03.003 ◽

2019 ◽

Vol 74 ◽

pp. 84-96 ◽

Cited By ~ 1

Author(s):

Wentao Huang ◽

Ting Chen ◽

Jibin Li

Keyword(s):

Diffusion Equation ◽

Reaction Diffusion ◽

Periodic Wave ◽

Reaction Diffusion Equation ◽

Nonlinear Reaction ◽

Nonlinear Reaction Diffusion Equation ◽

Wave Trains

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The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

10.1098/rsta.1991.0001 ◽

1991 ◽

Vol 334 (1633) ◽

pp. 1-24 ◽

Cited By ~ 86

Keyword(s):

Travelling Waves ◽

Chemical Species ◽

Propagation Speed ◽

Reaction Diffusion ◽

Spatial Dimension ◽

Travelling Wave ◽

Diffusion Waves ◽

Long Time ◽

Diffusion Rates ◽

Cubic Autocatalysis

We study the isothermal autocatalytic system , A + n B → ( n + 1)B , where n = 1 or 2 for quadratic or cubic autocatalysis respectively. In addition, we allow the chemical species, A and B, to have unequal diffusion rates. The propagating reaction-diffusion waves that may develop from a local initial input of the autocatalyst, B, are considered in one spatial dimension. We find that travelling wave solutions exist for all propagation speeds v ≥ v * n ,where v * n is a function of the ratio of the diffusion rates of the species A and B and represents the minimum propagation speed. It is also shown that the concentration of the autocatalyst, B, decays exponentially ahead of the wavefront for quadratic autocatalysis. However, for cubic autocatalysis, although the concentration of the autocatalyst decays exponentially ahead of the wavefront for travelling waves which propagate at speed v = v * 2 , this rate of decay is only algebraic for faster travelling waves with v > v * 2 . This difference in decay rate has implications for the selection of the long time wave speed when such travelling waves are generated from an initial-value problem.

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