REACTION–DIFFUSION SYSTEM

(Redirected from Reaction-diffusion equation)
'Reaction–diffusion systems' are mathematical models that describe how the concentration of one or more substances distributed in space changes under the influence of two processes: local chemical reactions in which the substances are converted into each other, and diffusion which causes the substances to spread out in space.
As this description implies, reaction–diffusion systems are naturally applied in chemistry. However, the equation can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form
:$$
partial_t oldsymbol{q} = underline{oldsymbol{D}} Delta oldsymbol{q}
+ oldsymbol{R}(oldsymbol{q}),
where each component of the vector 'q'('x',''t'') represents the concentration of one substance, is a diagonal matrix of diffusion coefficients and 'R' accounts for all local reactions. The solutions of reaction-diffusion equations display a wide range of behaviours, including the formation of travelling waves
and wave-like phenomena as well as other self-organized patterns like stripes, hexagons or more intricate structure like dissipative solitons.

Contents

One-component reaction–diffusion equations

Two-component reaction–diffusion equations

Three- and more-component reaction–diffusion equations

Applications and universality

Experiments

See also

References

External links

One-component reaction–diffusion equations

The most simple reaction–diffusion equation concerning the concentration ''u'' of a single substance in one spatial dimension,
:$$
partial_t u = D partial^2_x u + R(u),

is also referred to as the KPP (Kolmogorov-Petrovsky-Piscounov) equation. ^[1] If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is the heat equation. The choice ''R''(''u'') = ''u''(1-''u'') yields Fisher's equation that was originally used to describe the spreading of biological populations, ^[2] the Newell-Whitehead-Segel equation with ''R''(''u'') = ''u''(1-''u''²) to describe Rayleigh-Benard convection, ^[3][4] the more general Zeldovich equation with ''R''(''u'') = ''u''(1-''u'')(''u''-''α'') and 0<''α''<1 that arises in combustion theory, ^[5] and its particular degenerate case with ''R''(''u'') = ''u²''-''u³'' that is
sometimes referred to as Zeldovich equation as well. ^[6]
The dynamics of one-component systems is subject to certain restrictions as the evolution equation can also be written in the variational form
:$$
partial_t u=-rac{deltamathfrak L}{delta u}

and therefore describes a permanent decrease of the "free energy" $mathfrak\; L$ given by the functional
:
D2(partial_xu)^2+V(u)
ight] ext{d}x

with a potential ''V''(''u'') such that ''R''(''u'')=d''V''(''u'')/d''u''.

In systems with more than one stationary homogeneous solution, a typical solution is given by travelling fronts connecting the homogeneous states. These solutions move with constant speed without changing their shape and are of the form ''u''(''x'',''t'') =û(''ξ'') with ''ξ''=''x''-''ct'', where ''c'' is the speed of the travelling wave. Note that
while travelling waves are generically stable structures, all non-monotonous stationary solutions (e.g. localized domains composed of a front-antifront pair) are unstable. For ''c=0'', there is a
simple proof for this statement:^[7] if ''u₀''(''x'') is a stationary solution and ''u''=''u₀''(''x'')+''ũ''(''x'',''t'') is an infinitesimally perturbed solution, linear stability analysis yields the equation
:$$
partial_t ilde{u}=Dpartial_x^2
ilde{u}-U(x) ilde{u},quad U(x) =
-R^{prime}(u)|_{u=u_0(x)}.
With the ansatz ''ũ''=''ψ''(''x'')exp(''-λt'') we
arrive at the eigenvalue problem
:$hat\; Hpsi=lambdapsi,\; qquad$
hat H=-Dpartial_x^2+U(x),

of Schrödinger type where negative eigenvalues result in the instability of the solution. Due to translational invariance ''ψ= ∂_xu₀(x)''
is a neutral eigenfunction with the eigenvalue λ=0, and all other eigenfunctions can be sorted according to an increasing number of knots with the magnitude of the corresponding real eigenvalue increases monotonically with the number of zeros. The eigenfunction ''ψ=∂_''x'' ''u₀''(''x'')'' should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalue ''λ''=0 cannot be the
lowest one, thereby implying instability.
To determine the velocity ''c'' of a moving front, one may go
to a moving coordinate system and look at stationary solutions:
:$$
D partial^2_{xi}hat{u}(xi)+ cpartial_{xi} hat{u}(xi)+R(hat{u}(xi))=0.

This equation has a nice mechanical analogue as the motion of a
mass ''D'' with position ''û'' in the course of the "time" ''ξ'' under
the force ''R'' with the damping coefficient c which allows for a
rather illustrative access to the construction of different
types of solutions and the determination of ''c''.
When going from one to more space dimensions, a number of
statements from one-dimensional systems can still be applied.
Planar or curved wave fronts are typical structures, and a new
effect arises as the local velocity of a curved front becomes
dependent on the local radius of curvature (this can be
seen by going to polar coordinates). This phenomenon leads
to the so-called curvature-driven instability.^[8]

Two-component reaction–diffusion equations

Two-component systems allow for a much larger range of possible
phenomena than their one-component counterparts. An important
idea that was first proposed by Alan Turing is that a state
that is stable in the local system should become unstable in
the presence of diffusion. ^[9] This idea seems
non-intuitional at first glance as diffusion is commonly
associated with a stabilizing effect.
A linear stability analysis however shows that when linearizing
the general two-component system
:
partial_t u\ partial_t v
end{array}
ight) =
left(egin{array}{cc} D_u &0\0&D_v
end{array}
ight)
left( egin{array}{c} Delta u\ Delta v
end{array}
ight) + left(egin{array}{c} F(u,v)\G(u,v)
end{array}
ight)

and perturbing the system against plane waves
:$$
ilde{oldsymbol{q}}_{oldsymbol{k}}(oldsymbol{x},t) =
left(egin{array}{c}
ilde{u}(t)\ ilde{v}(t)end{array}
ight) e^{i
oldsymbol{k} cdot oldsymbol{x}}
close to a stationary homogeneous solution one finds
:$$
left(
egin{array}{c}
partial_t ilde{u}_{oldsymbol{k}}(t)\
partial_t ilde{v}_{oldsymbol{k}}(t)
end{array}
ight) = -k^2left(
egin{array}{c}
D_u ilde{u}_{oldsymbol{k}}(t)\
D_v ilde{v}_{oldsymbol{k}}(t)
end{array}
ight) + oldsymbol{R}^{prime} left(
egin{array}{c}
ilde{u}_{oldsymbol{k}}(t)\
ilde{v}_{oldsymbol{k}}(t)
end{array}
ight).
Turings idea can only be realized in three
equivalence classes of systems characterized
by the signs of the Jacobian
'R'` of the reaction function. In particular, if a finite
wave vector 'k' is supposed to be the most unstable one,
the Jacobian must have the signs
:
ight),
quad left(egin{array}{cc} +&+\-&-end{array}
ight), quad
left(egin{array}{cc} -&+\-&+end{array}
ight), quad
left(egin{array}{cc} -&-\+&+end{array}
ight).
This class of systems is named ''activator-inhibitor system''
after its first representative: close to the ground state, one
component stimulates the production of both components while
the other one inhibits their growth. Its most prominent
representative is the FitzHugh–Nagumo equation
:$$
egin{align}
partial_t u &= d_u^2 Delta u + f(u) - sigma v, \
au partial_t v &= d_v^2 Delta v + u - v
end{align}

with ''f''(''u'')=''λu'' -''u³'' -
''κ'' which describes how an action potential travels
through a nerve. ^[10][11] Here, ''d_u'',
''d_v'', ''τ'', ''σ'' and ''λ'' are
positive constants.
When an activator-inhibitor system undergoes a change of parameters, one may pass
from conditions under which a homogeneous ground state is
stable to conditions under which it is linearly unstable. The
corresponding bifurcation may be either
a Hopf bifurcation to a globally oscillating homogeneous
state with a dominant wave number ''k''=0 or a
''Turing bifurcation'' to a globally patterned state with
a dominant finite wave number. The latter in two
spatial dimensions typically leads to stripe or hexagonal
patterns.

Noisy initial conditions at ''t=0''.

State of the system at ''t=10''.

Almost converged state at ''t=100''.

For the Fitzhugh-Nagumo example, the neutral stability curves marking the
boundary of the linearly stable region for the Turing and Hopf
bifurcation are given by
:$$
egin{array}{rrl}
q_{ ext{n}}^H(k): & rac{1}{ au} + (d_u^2 + rac{1}{ au}
d_v^2)k^2
&=f^{prime}(u_{h}),\
q_{ ext{n}}^T(k): & rac{kappa_3}{1 + d_v^2 k^2}+ d_u^2 k^2
&= f^{prime}(u_{h}).
end{array}

If the bifurcation is subcritical, often localized structures
(dissipative solitons) can be observed in the
hysteretic region where the pattern coexists
with the ground state. Other frequently encountered structures
comprise pulse trains,
spiral waves and target patterns.

Rotating spiral.

Target pattern.

Stationary localized pulse (dissipative soliton).

Three- and more-component reaction–diffusion equations

For a variety of systems, reaction-diffusion equations with
more than two components have been proposed, e.g. as models
for the Belousov-Zhabotinsky reaction,
^[12], for blood clotting^[13] or planar gas discharge systems.
^[14]
While it is known that systems with more components allow for
a variety of phenomena not possible in systems with one or two
components (e.g. stable running pulses in more than one spatial
dimension without global feedback)^[15], up to now a systematic
overview of the possible phenomena in dependence on the properties
of the underlying system is hardly present.

Applications and universality

In recent times, reaction–diffusion systems
have attracted much interest as a prototype model for pattern formation.
The above-mentioned patterns (fronts, spirals, targets, hexagons, stripes
and dissipative solitons) can be found in various types of
reaction-diffusion systems in spite of large discrepancies
e.g. in the local reaction terms. It has also been argued that
reaction-diffusion processes are an essential basis for processes
connected to morphogenesis in biology and may even be related to
animal coats and skin pigmentation.^[16][17] Another reason for the interest
in reaction-diffusion systems is that
although they represent nonlinear partial differential equation,
there are often possibilities for an analytical treatment.^{[18][19][20][21][22]}

Experiments

Well-controllable experiments in chemical reaction-diffusion systems have up to now
been realized in three ways. One the one hand, gel reactors^[23] or filled capillariy tubes^[24] may be used. Second, temperature pulses on catalytic surfaces
have been investigated.^[25][26]
Third, the propagation of running nerve pulses is modelled
using reaction-diffusion systems.^[27]
Aside from these generic examples, it has turned out that under appropriate
circumstances electric transport systems like plasmas^[28] or semiconductors^[29] can be
described in a reaction-diffusion approach. For these systems various experiments
on pattern formation have been carried out.

See also

★ Phase space method

★ Fisher's equation

References

1. A. Kolmogorov
et al, Moscow Univ. Bull. Math. A 1 (1937): 1
2. R. A. Fisher, Ann.
Eug. 7 (1937): 355
3. A. C. Newell and J. A. Whitehead, J. Fluid Mech. 38 (1969): 279
4. L. A. Segel,
J. Fluid Mech. 38 (1969): 203
5. Y. B. Zeldovich and D. A. Frank-Kamenetsky, Acta Physicochim. 9 (1938): 341
6. B. H. Gilding and R. Kersner, Travelling Waves in Nonlinear Diffusion Convection Reaction, Birkhäuser (2004)
7. P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer (1979)
8.
A. S. Mikhailov, Foundations of Synergetics I.
Distributed Active Systems, Springer (1990)
9. A. M. Turing, Phil.
Transact. Royal Soc. B 237 (1952): 37
10. R. FitzHugh, Biophys. J. 1 (1961):
445
11. J. Nagumo et al., Proc. Inst. Radio Engin.
Electr. 50 (1962): 2061
12. V. K. Vanag and I. R. Epstein, Phys. Rev. Lett.
92 (2004): 128301
13. E. S. Lobanova
and F. I. Ataullakhanov, Phys. Rev. Lett.
93 (2004): 098303
14. H.-G. Purwins et al. in: Dissipative Solitons,
Lectures Notes in Physics, Ed. N. Akhmediev and A. Ankiewicz,
Springer (2005)
15. C. P. Schenk et al.,
Phys. Rev. Lett. 78 (1997): 3781
16. H. Meinhardt, Models of
Biological Pattern Formation, Academic Press (1982)
17. J. D. Murray,
Mathematical Biology, Springer (1993)
18.

19.
20.
P. Grindrod,Patterns and Waves: The Theory and Applications of Reaction-Diffusion
Equations, Clarendon Press (1991)
21. J. Smoller,
Shock Waves and Reaction Diffusion Equations, Springer (1994)
22.
B. S. Kerner and V. V. Osipov, Autosolitons. A New Approach to Problems
of Self-Organization and Turbulence, Kluwer Academic Publishers (1994)
23. K.-J. Lee et al.,
Nature 369 (1994): 215
24. C. T. Hamik and O. Steinbock,
New J. Phys. 5 (2003): 58
25. H. H. Rotermund et al.,
Phys. Rev. Lett. 66 (1991): 3083
26. M. D. Graham et al.,
J. Phys. Chem. 97 (1993): 7564
27.
A. L. Hodgkin and A. F. Huxley,
J. Physiol. 117 (1952): 500
28. M. Bode and H.-G. Purwins,
Physica D 86 (1995): 53
29. E. Schöll,
Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors,
Cambridge University Press (2001)

External links

★ Java applet showing a reaction–diffusion simulation

★ Another applet showing RD.