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

1991 ◽  
Vol 334 (1633) ◽  
pp. 1-24 ◽  
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.

The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. II. An initial-value problem with an immobilized or nearly immobilized autocatalyst

1991 ◽  
Vol 336 (1644) ◽  
pp. 497-539 ◽  
Keyword(s):  
Travelling Waves ◽  
Chemical Species ◽  
Travelling Wave ◽  
Small Time ◽  
Initial Value ◽  
Slab Geometry ◽  
Diffusion Rates ◽  
Large Spike ◽  
Reaction Schemes ◽  
Cubic Autocatalysis

We study the isothermal autocatalytic reaction schemes, A + B -> 2B (quadratic autocatalysis), and A + 2B → 3B (cubic autocatalysis), where A is a reactant and B is an autocatalyst. We consider the situation when a quantity of B is introduced locally into a uniform expanse of A, in one-dimensional slab geometry. In addition, we allow the chemical species A and B to have unequal diffusion rates D A and D B respectively, and study the two closely related cases, ( D B / D A ) = 0 and 0 < ( D B / D A ) < 1. When (D b /D a ) = 0 a spike forms in the concentration of B, which grows indefinitely, and we can obtain both large and small time asymptotic solutions. For 0 < ( D B / D A ) < 1 there is a long induction period during which a large spike forms in the concentration of B, before a minimum speed travelling wave is generated. We can relate the results for case ( D B / D A ) = 0 to the solution when 0 < ( D B / D A ) < 1 to obtain detailed information about its behaviour.

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

1991 ◽  
Vol 434 (1892) ◽  
pp. 531-554 ◽  
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.

The development of travelling waves in a simple isothermal chemical system III. Cubic and mixed autocatalysis

1990 ◽  
Vol 430 (1880) ◽  
pp. 509-524 ◽  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Chemical System ◽  
Mixed System ◽  
Dimensional Parameter ◽  
Diffusion Wave ◽  
Symmetrical Form ◽  
Asymptotic Speed ◽  
Mixed Systems ◽  
Cubic Autocatalysis

Autocatalytic chemical reactions can support isothermal travelling waves of constant speed and form. This paper extends previous studies to cubic autocatalysis and to mixed systems where quadratic and cubic autocatalyses occur concurrently. A + B → 2B, rate = k q ab , (1) A + 2B → 3B, rate = k c ab 2 . (2) For pure cubic autocatalysis the wave has, at large times, a constant asymptotic speed v 0 (where v 0 = 1/√2 in the appropriate dimensionless units). This result is confirmed by numerical investigation of the initial-value problem. Perturbations to this stable wave-speed decay at long times as t -3/2 e -1/8 t . The mixed system is governed by a non-dimensional parameter μ = k q / k c a 0 which measures the relative rates of transformation by quadratic and cubic modes. In the mixed case ( μ ≠ 0) the reaction-diffusion wave has a form appropriate to a purely cubic autocatalysis so long as μ lies between ½ and 0. When μ exceeds ½, the reaction wave loses its symmetrical form, and all its properties steadily approach those of quadratic autocatalysis. The value μ = ½ is the value at which rates of conversion by the two paths are equal.

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. II. Abruptly vanishing diffusivity

1995 ◽  
Vol 350 (1694) ◽  
pp. 361-378 ◽  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Wave Form ◽  
Initial Development ◽  
Finite Concentration ◽  
Distinct Process ◽  
And Diffusion

In this paper we continue our study of some of the qualitative features of chemical polymerization processes by considering a reaction-diffusion equation for the chemical concentration in which the diffusivity vanishes abruptly at a finite concentration. The effect of this diffusivity cut-off is to create two distinct process zones; in one there is both reaction and diffusion and in the other there is only reaction. These zones are separated by an interface across which there is a jump in concentration gradient. Our analysis is focused on both the initial development of this interface and the large time evolution of the system into a travelling wave form. Some distinct differences from our previous analysis of smoothly vanishing diffusivity are found.

Travelling waves for a reaction–diffusion system in population dynamics and epidemiology

2005 ◽  
Vol 135 (4) ◽  
pp. 663-675 ◽  
Author(s):  
Shangbing Ai ◽  
Wenzhang Huang
Keyword(s):  
Population Dynamics ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Diffusion Constant ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Dimensional System ◽  
Travelling Wave Solutions ◽  
Wave Solutions

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

Travelling waves for delayed reaction–diffusion equations with global response

2005 ◽  
Vol 462 (2065) ◽  
pp. 229-261 ◽  
Author(s):  
Teresa Faria ◽  
Wenzhang Huang ◽  
Jianhong Wu
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Index Formula ◽  
Delayed Response ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Non Local

We develop a new approach to obtain the existence of travelling wave solutions for reaction–diffusion equations with delayed non-local response. The approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain Banach space, coupled with an index formula of the associated Fredholm operator and some careful estimation of the nonlinear perturbation. The general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation, and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non-local and delayed response for the matured population.

The development of travelling waves in a simple isothermal chemical system II. Cubic autocatalysis with quadratic and linear decay

1990 ◽  
Vol 430 (1879) ◽  
pp. 315-345 ◽  
Keyword(s):  
Numerical Solutions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Chemical System ◽  
Necessary Condition ◽  
Travelling Wave Solution ◽  
Diffusion Front ◽  
Initial Value ◽  
Linear Decay

The possibility of travelling reaction-diffusion waves developing in the isothermal chemical system governed by the cubic autocatalytic reaction A + 2B → 3B (rate k 3 ab 2 ) coupled with either the linear decay step B → C (rate k 2 b ) or the quadratic decay step B + B → C (rate k 4 b 2 ) is examined. Two simple solutions are obtained,namely the well-stirred analogue of the spatially inhomogeneous problem and the solution for small input of the autocatalyst B. Both of these suggest that, for the quadratic decay case, a wave will develop only if the non-dimensional parameter k ═ k 4 / k 3 a 0 < 1 (where a 0 is the initial concentration of the reactant A), with there being no restriction on the initial input of the autocatalyst B. However, for the linear decay case the initiation of a travelling wave depends on the parameter v ═ k 2 / k 3 a 2 0 and that, in addition, there is an input threshold on B before the formation of a wave will occur. The equations governing the fully developed travelling waves are then considered and it is shown that for the quadratic decay case the situation is similar to previous work in quadratic autocatalysis with linear decay, with a necessary condition for the existence of a travelling-wave solution being that K < 1. However, the case of linear decay is quite different, with a necessary condition for the existence of a travelling wave solution now found to be v < 1/4 Numerical solutions of the equations governing this case reveal further that a solution exists only for v < v c , with v c ≈ 0.0465, and that there are two branches of solution for 0 < v < v c . The behaviour of these lower branch solutions as v → 0 is discussed. The initial-value problem is then considered. For the quadratic decay case it is shown that the uniform state a ═ a 0 , b ═ 0 is globally asymptotically stable (i. e. a → a 0 , b → 0 uniformly for large times) for all k > 1. For the linear decay case it is shown that the development of a travelling wave requires β 0 > v (where β 0 is a measure of the initial input of B) for v < v c . These theoretical results are then complemented by numerical solutions of the initial-value problem for both cases, which confirm the various predictions of the theory. The behaviour of the solution of the equations governing the travelling waves is then discussed in the limits K → 0, v → 0 and K → 1. In the first case the solution approaches the solution for K ═ 0 (or v =0) on the length scale of the reaction-diffusion front, with there being a long tail region of length scale O ( K -1 ) (or O ( v -1 )) in which the autocatalyst B decays to zero. In the latter case we find that the concentration of reactant A is 1 + O [(1 - k )] and autocatalyst B is O[(1 - k 2 ] with the thickness of the reaction-diffusion front becoming large, of thickness O [(1- k ) -3/2 ].

Stability of Turing-Type Patterns in a Reaction–Diffusion System with an External Gradient

2017 ◽  
Vol 27 (01) ◽  
pp. 1750003 ◽  
Author(s):  
Tilmann Glimm ◽  
Jianying Zhang ◽  
Yun-Qiu Shen
Keyword(s):  
Diffusion Coefficients ◽  
Chemical Species ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Reaction Diffusion Equations ◽  
Generic Model ◽  
Bifurcation Diagrams ◽  
Analytic Approximations ◽  
Small Parameters ◽  
The Stability

We investigate the stability of Turing-type patterns in one spatial dimension in a system of reaction–diffusion equations with a term depending linearly on the spatial position. The system is a generic model of two interacting chemical species where production rates are dependent on a linear external gradient. This is motivated by mathematical models in developmental biology. In a previous paper, we found analytic approximations of Turing-like steady state patterns. In the present article, we derive conditions for the stability of these patterns and show bifurcation diagrams in two small parameters related to the slope of the external gradient and the ratio of the diffusion coefficients.

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