2104:
719:
1790:
2940:
2099:{\displaystyle {\begin{pmatrix}\partial _{t}{\tilde {u}}_{\boldsymbol {k}}(t)\\\partial _{t}{\tilde {v}}_{\boldsymbol {k}}(t)\end{pmatrix}}=-k^{2}{\begin{pmatrix}D_{u}{\tilde {u}}_{\boldsymbol {k}}(t)\\D_{v}{\tilde {v}}_{\boldsymbol {k}}(t)\end{pmatrix}}+{\boldsymbol {R}}^{\prime }{\begin{pmatrix}{\tilde {u}}_{\boldsymbol {k}}(t)\\{\tilde {v}}_{\boldsymbol {k}}(t)\end{pmatrix}}.}
1628:
2310:
33:
2640:
3034:. 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
1779:
1406:
3021:
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). An introduction and systematic overview of the possible phenomena in dependence on
2493:
170:
3038:
in biology and may even be related to animal coats and skin pigmentation. Other applications of reaction–diffusion equations include ecological invasions, spread of epidemics, tumour growth, dynamics of fission waves, wound healing and visual hallucinations. Another reason for the interest in
2139:
2935:{\displaystyle {\begin{aligned}q_{\text{n}}^{H}(k):&{}\quad {\frac {1}{\tau }}+\left(d_{u}^{2}+{\frac {1}{\tau }}d_{v}^{2}\right)k^{2}&=f^{\prime }(u_{h}),\\q_{\text{n}}^{T}(k):&{}\quad {\frac {\kappa }{1+d_{v}^{2}k^{2}}}+d_{u}^{2}k^{2}&=f^{\prime }(u_{h}).\end{aligned}}}
1000:
3058:
Aside from these generic examples, it has turned out that under appropriate circumstances electric transport systems like plasmas or semiconductors can be described in a reaction–diffusion approach. For these systems various experiments on pattern formation have been carried out.
1372:
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
1643:
1342:
1623:{\displaystyle {\begin{pmatrix}\partial _{t}u\\\partial _{t}v\end{pmatrix}}={\begin{pmatrix}D_{u}&0\\0&D_{v}\end{pmatrix}}{\begin{pmatrix}\partial _{xx}u\\\partial _{xx}v\end{pmatrix}}+{\begin{pmatrix}F(u,v)\\G(u,v)\end{pmatrix}}}
726:
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
1124:
2329:
87:
3246:
Kolmogorov, A., Petrovskii, I. and
Piskunov, N. (1937) Study of a Diffusion Equation That Is Related to the Growth of a Quality of Matter and Its Application to a Biological Problem. Moscow University Mathematics Bulletin, 1,
768:
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
2305:{\displaystyle {\begin{pmatrix}+&-\\+&-\end{pmatrix}},\quad {\begin{pmatrix}+&+\\-&-\end{pmatrix}},\quad {\begin{pmatrix}-&+\\-&+\end{pmatrix}},\quad {\begin{pmatrix}-&-\\+&+\end{pmatrix}}.}
662:
511:
2558:
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
1171:, and all other eigenfunctions can be sorted according to an increasing number of nodes with the magnitude of the corresponding real eigenvalue increases monotonically with the number of zeros. The eigenfunction
2991:
826:
2957:), spiral waves and target patterns. These three solution types are also generic features of two- (or more-) component reaction–diffusion equations in which the local dynamics have a stable limit cycle
2645:
2319:
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
3828:
Gupta, Ankur; Chakraborty, Saikat (January 2009). "Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors".
324:
2334:
2620:
2588:
1774:{\displaystyle {\tilde {\boldsymbol {q}}}_{\boldsymbol {k}}({\boldsymbol {x}},t)={\begin{pmatrix}{\tilde {u}}(t)\\{\tilde {v}}(t)\end{pmatrix}}e^{i{\boldsymbol {k}}\cdot {\boldsymbol {x}}}}
2604:
46:
are mathematical models that correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local
3047:
Well-controllable experiments in chemical reaction–diffusion systems have up to now been realized in three ways. First, gel reactors or filled capillary tubes may be used. Second,
1215:
538:
2582:
Subcritical Turing bifurcation: formation of a hexagonal pattern from noisy initial conditions in the above two-component reaction–diffusion system of
Fitzhugh–Nagumo type.
4805:
4311:
Lee, Kyoung-Jin; McCormick, William D.; Pearson, John E.; Swinney, Harry L. (1994). "Experimental observation of self-replicating spots in a reaction–diffusion system".
2967:
2979:
330:
3760:
Schenk, C. P.; Or-Guil, M.; Bode, M.; Purwins, H.-G. (May 12, 1997). "Interacting Pulses in Three-Component
Reaction-Diffusion Systems on Two-Dimensional Domains".
2634:
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
1030:
4686:
2990:
2488:{\displaystyle {\begin{aligned}\partial _{t}u&=d_{u}^{2}\,\nabla ^{2}u+f(u)-\sigma v,\\\tau \partial _{t}v&=d_{v}^{2}\,\nabla ^{2}v+u-v\end{aligned}}}
1389:
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
165:{\displaystyle \partial _{t}{\boldsymbol {q}}={\underline {\underline {\boldsymbol {D}}}}\,\nabla ^{2}{\boldsymbol {q}}+{\boldsymbol {R}}({\boldsymbol {q}}),}
3965:
Holmes, E. E.; Lewis, M. A.; Banks, J. E.; Veit, R. R. (1994). "Partial
Differential Equations in Ecology: Spatial Interactions and Population Dynamics".
3039:
reaction–diffusion systems is that although they are nonlinear partial differential equations, there are often possibilities for an analytical treatment.
1365:
with the damping coefficient c which allows for a rather illustrative access to the construction of different types of solutions and the determination of
3108:
3075:
numerical solution methods are proposed. To highest degree of detail reaction-diffusion systems are described with particle based simulation tools like
2578:
to a globally patterned state with a dominant finite wave number. The latter in two spatial dimensions typically leads to stripe or hexagonal patterns.
4762:
Fröhner, Christoph, and Frank Noé. "Reversible interacting-particle reaction dynamics." The
Journal of Physical Chemistry B 122.49 (2018): 11240-11250.
3165:
546:
78:
456:
4640:
Isaacson, Samuel A.; Peskin, Charles S. (2006). "Incorporating
Diffusion in Complex Geometries into Stochastic Chemical Kinetics Simulations".
450:
The dynamics of one-component systems is subject to certain restrictions as the evolution equation can also be written in the variational form
220:
accounts for all local reactions. The solutions of reaction–diffusion equations display a wide range of behaviours, including the formation of
4301:
B. S. Kerner and V. V. Osipov, Autosolitons. A New
Approach to Problems of Self-Organization and Turbulence, Kluwer Academic Publishers (1994)
3949:
3811:
3148:
995:{\displaystyle \partial _{t}{\tilde {u}}=D\partial _{x}^{2}{\tilde {u}}-U(x){\tilde {u}},\qquad U(x)=-R^{\prime }(u){\Big |}_{u=u_{0}(x)}.}
4110:
Sherratt, J. A.; Nowak, M. A. (June 22, 1992). "Oncogenes, anti-oncogenes and the immune response to cancer : a mathematical model".
2619:
2587:
3584:
Vanag, Vladimir K.; Epstein, Irving R. (March 24, 2004). "Stationary and
Oscillatory Localized Patterns, and Subcritical Bifurcations".
3138:
3231:
2953:
region where the pattern coexists with the ground state. Other frequently encountered structures comprise pulse trains (also known as
3007:
2603:
384:
4556:
Bode, M.; Purwins, H.-G. (1995). "Pattern formation in reaction-diffusion systems - dissipative solitons in physical systems".
2114:
257:
4800:
3750:
H.-G. Purwins et al. in: Dissipative
Solitons, Lectures Notes in Physics, Ed. N. Akhmediev and A. Ankiewicz, Springer (2005)
3635:
Lobanova, E. S.; Ataullakhanov, F. I. (August 26, 2004). "Running Pulses of Complex Shape in a Reaction-Diffusion Model".
3093:
2321:
4810:
3213:
3872:
Duran-Nebreda, Salva; Pla, Jordi; Vidiella, Blai; Piñero, Jordi; Conde-Pueyo, Nuria; Solé, Ricard (January 15, 2021).
333:. If the reaction term vanishes, then the equation represents a pure diffusion process. The corresponding equation is
3055:
have been investigated. Third, the propagation of running nerve pulses is modelled using reaction–diffusion systems.
4283:
P. Grindrod, Patterns and Waves: The Theory and Applications of Reaction-Diffusion Equations, Clarendon Press (1991)
4270:
2966:
4016:
Murray, James D.; Stanley, E. A.; Brown, D. L. (November 22, 1986). "On the spatial spread of rabies among foxes".
2978:
3170:
3006:
For a variety of systems, reaction–diffusion equations with more than two components have been proposed, e.g. the
3123:
2954:
3317:
Segel, Lee A. (August 14, 1969). "Distant side-walls cause slow amplitude modulation of cellular convection".
1337:{\displaystyle D\partial _{\xi }^{2}{\hat {u}}(\xi )+c\partial _{\xi }{\hat {u}}(\xi )+R({\hat {u}}(\xi ))=0.}
4411:
Rotermund, H. H.; Jakubith, S.; von Oertzen, A.; Ertl, G. (June 10, 1991). "Solitons in a surface reaction".
4598:
E. Schöll, Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors, Cambridge University Press (2001)
3223:
1130:
4657:
4499:"A quantitative description of membrane current and its application to conduction and excitation in nerve"
61:. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in
4462:
Graham, Michael D.; Lane, Samuel L.; Luss, Dan (1993). "Temperature pulse dynamics on a catalytic ring".
4075:
Chaplain, M. A. J. (1995). "Reaction–diffusion prepatterning and its potential role in tumour invasion".
3266:
Newell, Alan C.; Whitehead, J. A. (September 3, 1969). "Finite bandwidth, finite amplitude convection".
3068:
1195:
should have at least one zero, and for a non-monotonic stationary solution the corresponding eigenvalue
718:
209:
1133:
where negative eigenvalues result in the instability of the solution. Due to translational invariance
4649:
4565:
4420:
4375:
4320:
4183:
4025:
3769:
3699:
3644:
3593:
3486:
3434:
3326:
3275:
3160:
2946:
519:
357:
4662:
3128:
3052:
239:". Each function, for which a reaction diffusion differential equation holds, represents in fact a
232:
3030:
In recent times, reaction–diffusion systems have attracted much interest as a prototype model for
2961:
Other patterns found in the above two-component reaction–diffusion system of Fitzhugh–Nagumo type.
4344:
4252:
4201:
4143:
4057:
3998:
3990:
3350:
3299:
3118:
3103:
2560:
1400:
A linear stability analysis however shows that when linearizing the general two-component system
37:
4745:
4727:
4705:
4581:
4538:
4520:
4479:
4444:
4436:
4393:
4336:
4244:
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4135:
4127:
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3982:
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3939:
3911:
3893:
3845:
3807:
3785:
3733:
3725:
3668:
3660:
3617:
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3566:
3522:
3504:
3452:
3342:
3291:
3227:
3113:
3098:
3031:
2110:
1378:
1209:
of a moving front, one may go to a moving coordinate system and look at stationary solutions:
228:
225:
47:
4735:
4717:
4667:
4573:
4528:
4510:
4471:
4428:
4383:
4328:
4228:
4191:
4119:
4084:
4033:
3974:
3901:
3885:
3837:
3777:
3715:
3707:
3652:
3601:
3558:
3512:
3494:
3442:
3334:
3283:
3217:
2564:
2522:
422:
3076:
3071:. There are existing several numerical treatments in research literature. Also for complex
3549:
Kopell, N.; Howard, L. N. (1973). "Plane Wave Solutions to Reaction-Diffusion Equations".
3408:
A. S. Mikhailov, Foundations of Synergetics I. Distributed Active Systems, Springer (1990)
1393:
is that a state that is stable in the local system can become unstable in the presence of
221:
205:
1119:{\displaystyle {\hat {H}}\psi =\lambda \psi ,\qquad {\hat {H}}=-D\partial _{x}^{2}+U(x),}
4653:
4569:
4424:
4379:
4324:
4187:
4029:
3773:
3703:
3648:
3597:
3490:
3438:
3427:
Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences
3330:
3279:
820:
is an infinitesimally perturbed solution, linear stability analysis yields the equation
251:
The simplest reaction–diffusion equation is in one spatial dimension in plane geometry,
4740:
4533:
4498:
3906:
3873:
3517:
3474:
3209:
3143:
3011:
236:
4364:"Excitation waves in reaction-diffusion media with non-monotonic dispersion relations"
3499:
3079:
or ReaDDy which employ for example reversible interacting-particle reaction dynamics.
4794:
4706:"Simulation tools for particle-based reaction-diffusion dynamics in continuous space"
4687:"Numerical methods for solving the reactive diffusion equation in complex geometries"
4577:
4219:
Sherratt, J. A.; Murray, J. D. (July 23, 1990). "Models of epidermal wound healing".
4205:
4061:
3804:
Dissipative Solitons in Reaction Diffusion Systems. Mechanism, Dynamics, Interaction.
3354:
3205:
3035:
3015:
1158:
334:
4388:
4363:
4256:
4147:
4002:
3303:
4515:
4348:
3196:
388:
3862:
L.G. Harrison, Kinetic Theory of Living Pattern, Cambridge University Press (1993)
3656:
3605:
36:
A simulation of two virtual chemicals reacting and diffusing on a Torus using the
32:
17:
3393:
3377:
4780:
4432:
3781:
3418:
3048:
1390:
4196:
4171:
3806:
Volume 70 of Springer Series in Synergetics, Springer, Berlin Heidelberg 2013,
4722:
4088:
3889:
3841:
3338:
3287:
3192:
2950:
1634:
1162:
426:
361:
4731:
4585:
4524:
4483:
4440:
4397:
4340:
4240:
4131:
4096:
4045:
3986:
3897:
3874:"Synthetic Lateral Inhibition in Periodic Pattern Forming Microbial Colonies"
3849:
3789:
3729:
3664:
3613:
3570:
3508:
3456:
3346:
3295:
1394:
1374:
657:{\displaystyle {\mathfrak {L}}=\int _{-\infty }^{\infty }\left\,{\text{d}}x}
58:
51:
4749:
4542:
4448:
4232:
4123:
4037:
3928:
H. Meinhardt, Models of Biological Pattern Formation, Academic Press (1982)
3915:
3737:
3672:
3621:
3562:
3526:
3475:"Impulses and Physiological States in Theoretical Models of Nerve Membrane"
3447:
3422:
3367:
Y. B. Zeldovich and D. A. Frank-Kamenetsky, Acta Physicochim. 9 (1938): 341
4248:
4139:
4053:
2133:
is supposed to be the most unstable one, the Jacobian must have the signs
506:{\displaystyle \partial _{t}u=-{\frac {\delta {\mathfrak {L}}}{\delta u}}}
77:. Mathematically, reaction–diffusion systems take the form of semi-linear
4786:
RD Tool: an interactive web application for reaction-diffusion simulation
4704:
Schöneberg, Johannes; Ullrich, Alexander; Noé, Frank (October 24, 2014).
4292:
J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer (1994)
4018:
Proceedings of the Royal Society of London. Series B. Biological Sciences
3088:
3072:
4475:
2567:
to a globally oscillating homogeneous state with a dominant wave number
1381:). This phenomenon leads to the so-called curvature-driven instability.
3994:
3687:
74:
70:
66:
62:
4785:
4775:
Reaction–Diffusion by the Gray–Scott Model: Pearson's parameterization
4671:
3720:
3711:
4781:
A thesis on reaction–diffusion patterns with an overview of the field
4777:
a visual map of the parameter space of Gray–Scott reaction diffusion.
4774:
4617:
4332:
3133:
1347:
This equation has a nice mechanical analogue as the motion of a mass
3978:
54:
which causes the substances to spread out over a surface in space.
3539:
J. Nagumo et al., Proc. Inst. Radio Engin. Electr. 50 (1962): 2061
717:
516:
and therefore describes a permanent decrease of the "free energy"
31:
2125:
of the reaction function. In particular, if a finite wave vector
447:
that is sometimes referred to as the Zeldovich equation as well.
360:
that was originally used to describe the spreading of biological
3686:
Osborne, A. G.; Recktenwald, G. D.; Deinert, M. R. (June 2012).
2945:
If the bifurcation is subcritical, often localized structures (
3067:
A reaction–diffusion system can be solved by using methods of
4172:"Stability instability and Hopf bifurcation in fission waves"
50:
in which the substances are transformed into each other, and
4160:
R.A. Gatenby and E.T. Gawlinski, Cancer Res. 56 (1996): 5745
2904:
2766:
2012:
939:
3944:. Springer Science & Business Media. pp. 436–450.
3378:
Travelling Waves in Nonlinear Diffusion Convection Reaction
4607:
S.Tang et al., J.Austral.Math.Soc. Ser.B 35(1993): 223–243
3692:
Chaos: An Interdisciplinary Journal of Nonlinear Science
1202:
cannot be the lowest one, thereby implying instability.
319:{\displaystyle \partial _{t}u=D\partial _{x}^{2}u+R(u),}
231:
like stripes, hexagons or more intricate structure like
4221:
Proceedings of the Royal Society B: Biological Sciences
4112:
Proceedings of the Royal Society B: Biological Sciences
722:
A travelling wave front solution for Fisher's equation.
3394:
Mathematical Aspects of Reacting and Diffusing Systems
3002:
Three- and more-component reaction–diffusion equations
2268:
2228:
2188:
2148:
2022:
1911:
1799:
1691:
1569:
1514:
1464:
1415:
584:
4170:
Osborne, Andrew G.; Deinert, Mark R. (October 2021).
3022:
the properties of the underlying system is given in.
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2332:
2142:
1793:
1646:
1409:
1218:
1033:
829:
549:
522:
459:
260:
90:
1784:
of the stationary homogeneous solution will satisfy
57:
Reaction–diffusion systems are naturally applied in
4470:(29). American Chemical Society (ACS): 7564–7571.
4419:(23). American Physical Society (APS): 3083–3086.
3768:(19). American Physical Society (APS): 3781–3784.
2934:
2487:
2304:
2098:
1773:
1622:
1336:
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994:
656:
532:
505:
318:
164:
4497:Hodgkin, A. L.; Huxley, A. F. (August 28, 1952).
4362:Hamik, Chad T; Steinbock, Oliver (June 6, 2003).
2996:Stationary localized pulse (dissipative soliton).
956:
776:, there is a simple proof for this statement: if
3325:(1). Cambridge University Press (CUP): 203–224.
3274:(2). Cambridge University Press (CUP): 279–303.
429:theory, and its particular degenerate case with
3592:(12). American Physical Society (APS): 128301.
4083:(4). World Scientific Pub Co Pte Lt: 929–936.
3643:(9). American Physical Society (APS): 098303.
81:. They can be represented in the general form
2113:of systems characterized by the signs of the
8:
3468:
3466:
3404:
3402:
2109:Turing's idea can only be realized in four
364:, the Newell–Whitehead-Segel equation with
4271:"A Math Theory for Why People Hallucinate"
3109:Autocatalytic reactions and order creation
1385:Two-component reaction–diffusion equations
247:One-component reaction–diffusion equations
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224:and wave-like phenomena as well as other
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135:
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109:
101:
95:
89:
4806:Parabolic partial differential equations
3688:"Propagation of a solitary fission wave"
194:represents the unknown vector function,
79:parabolic partial differential equations
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2959:
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2071:
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2007:
1980:
1938:
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3386:
331:Kolmogorov–Petrovsky–Piskunov equation
3823:
3821:
3819:
3423:"The chemical basis of morphogenesis"
3256:R. A. Fisher, Ann. Eug. 7 (1937): 355
7:
4118:(1323). The Royal Society: 261–271.
4024:(1255). The Royal Society: 111–150.
3166:Zeldovich–Frank-Kamenetskii equation
3149:Multi-state modeling of biomolecules
1024:we arrive at the eigenvalue problem
389:Zeldovich–Frank-Kamenetskii equation
3139:The Chemical Basis of Morphogenesis
552:
525:
487:
4319:(6477). Springer Nature: 215–218.
4227:(1300). The Royal Society: 29–36.
3938:Murray, James D. (March 9, 2013).
2457:
2421:
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1845:
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281:
262:
235:. Such patterns have been dubbed "
126:
92:
25:
4464:The Journal of Physical Chemistry
3433:(641). The Royal Society: 37–72.
3201:Turing's theory of morphogenesis
2989:
2977:
2965:
2618:
2602:
2586:
2825:
2679:
2525:travels through a nerve. Here,
2315:This class of systems is named
2262:
2222:
2182:
1061:
915:
533:{\displaystyle {\mathfrak {L}}}
73:(neutron diffusion theory) and
4558:Physica D: Nonlinear Phenomena
4516:10.1113/jphysiol.1952.sp004764
3551:Studies in Applied Mathematics
3376:B. H. Gilding and R. Kersner,
2922:
2909:
2815:
2809:
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2771:
2669:
2663:
2398:
2392:
2082:
2076:
2064:
2050:
2044:
2032:
1991:
1985:
1973:
1949:
1943:
1931:
1879:
1873:
1861:
1837:
1831:
1819:
1737:
1731:
1725:
1712:
1706:
1700:
1680:
1666:
1654:
1609:
1597:
1587:
1575:
1377:(this can be seen by going to
1325:
1322:
1316:
1310:
1301:
1292:
1286:
1280:
1255:
1249:
1243:
1110:
1104:
1068:
1040:
984:
978:
950:
944:
925:
919:
906:
897:
891:
879:
846:
637:
631:
310:
304:
156:
148:
1:
4176:Cell Reports Physical Science
4077:Journal of Biological Systems
3657:10.1103/physrevlett.93.098303
3606:10.1103/physrevlett.92.128301
3500:10.1016/s0006-3495(61)86902-6
3094:Diffusion-controlled reaction
3026:Applications and universality
3008:Belousov–Zhabotinsky reaction
790:is a stationary solution and
4622:, GollyGang, August 20, 2024
4578:10.1016/0167-2789(95)00087-k
3830:Chemical Engineering Journal
2593:Noisy initial conditions at
1357:in the course of the "time"
4564:(1–2). Elsevier BV: 53–63.
4433:10.1103/physrevlett.66.3083
3782:10.1103/physrevlett.78.3781
3485:(6). Elsevier BV: 445–466.
329:is also referred to as the
4827:
4197:10.1016/j.xcrp.2021.100588
3473:FitzHugh, Richard (1961).
3319:Journal of Fluid Mechanics
3268:Journal of Fluid Mechanics
3014:, fission waves or planar
2625:Almost converged state at
2317:activator-inhibitor system
1205:To determine the velocity
385:Rayleigh–Bénard convection
44:Reaction–diffusion systems
27:Type of mathematical model
4723:10.1186/s13628-014-0011-5
4503:The Journal of Physiology
4389:10.1088/1367-2630/5/1/358
4374:(1). IOP Publishing: 58.
4089:10.1142/s0218339095000824
3890:10.1021/acssynbio.0c00318
3842:10.1016/j.cej.2008.08.025
3339:10.1017/s0022112069000127
3288:10.1017/s0022112069000176
2955:periodic travelling waves
2949:) can be observed in the
4685:Linker, Patrick (2016).
3216:; Sprevak, Mark (2017).
3124:Periodic travelling wave
2555:are positive constants.
2322:FitzHugh–Nagumo equation
540:given by the functional
4413:Physical Review Letters
3762:Physical Review Letters
3637:Physical Review Letters
3586:Physical Review Letters
3224:Oxford University Press
2609:State of the system at
2521:which describes how an
4368:New Journal of Physics
4233:10.1098/rspb.1990.0061
4124:10.1098/rspb.1992.0071
4038:10.1098/rspb.1986.0078
3563:10.1002/sapm1973524291
3448:10.1098/rstb.1952.0012
2936:
2489:
2306:
2100:
1775:
1624:
1338:
1120:
996:
723:
658:
534:
507:
320:
241:concentration variable
210:diffusion coefficients
166:
40:
4801:Mathematical modeling
4509:(4). Wiley: 500–544.
3878:ACS Synthetic Biology
3557:(4). Wiley: 291–328.
3171:FitzHugh–Nagumo model
3069:numerical mathematics
2937:
2490:
2307:
2101:
1776:
1625:
1339:
1121:
997:
721:
659:
535:
508:
321:
167:
35:
3941:Mathematical Biology
3063:Numerical treatments
2947:dissipative solitons
2641:
2330:
2140:
1791:
1644:
1407:
1216:
1031:
827:
547:
520:
457:
258:
233:dissipative solitons
88:
4811:Reaction mechanisms
4654:2006SJSC...28...47I
4642:SIAM J. Sci. Comput
4570:1995PhyD...86...53B
4476:10.1021/j100131a028
4425:1991PhRvL..66.3083R
4380:2003NJPh....5...58H
4325:1994Natur.369..215L
4188:2021CRPS....200588O
4030:1986RSPSB.229..111M
3973:(1). Wiley: 17–29.
3774:1997PhRvL..78.3781S
3704:2012Chaos..22b3148O
3649:2004PhRvL..93i8303L
3598:2004PhRvL..92l8301V
3491:1961BpJ.....1..445F
3479:Biophysical Journal
3439:1952RSPTB.237...37T
3421:(August 14, 1952).
3380:, Birkhäuser (2004)
3331:1969JFM....38..203S
3280:1969JFM....38..279N
3129:Stochastic geometry
2883:
2852:
2808:
2740:
2712:
2662:
2454:
2371:
2111:equivalence classes
1375:radius of curvature
1236:
1097:
872:
577:
387:, the more general
294:
3210:Bowen, Jonathan P.
3119:Patterns in nature
3104:Phase space method
3053:catalytic surfaces
2932:
2930:
2869:
2838:
2794:
2726:
2698:
2648:
2576:Turing bifurcation
2485:
2483:
2440:
2357:
2302:
2293:
2253:
2213:
2173:
2096:
2087:
1996:
1884:
1771:
1742:
1620:
1614:
1555:
1503:
1450:
1334:
1222:
1116:
1083:
992:
858:
724:
654:
593:
560:
530:
503:
316:
280:
162:
122:
118:
48:chemical reactions
41:
18:Reaction diffusion
4672:10.1137/040605060
3951:978-3-662-08539-4
3812:978-3-642-31250-2
3712:10.1063/1.4729927
3396:, Springer (1979)
3206:Copeland, B. Jack
3161:Fisher's equation
3114:Pattern formation
3099:Chemical kinetics
3032:pattern formation
2972:Rotating spiral.
2864:
2801:
2724:
2688:
2655:
2629: = 100.
2613: = 10.
2067:
2035:
1976:
1934:
1864:
1822:
1728:
1703:
1657:
1379:polar coordinates
1313:
1283:
1246:
1071:
1043:
909:
882:
849:
667:with a potential
649:
592:
501:
425:) that arises in
358:Fisher's equation
335:Fick's second law
111:
110:
16:(Redirected from
4818:
4763:
4760:
4754:
4753:
4743:
4725:
4701:
4695:
4694:
4682:
4676:
4675:
4665:
4637:
4631:
4630:
4629:
4627:
4614:
4608:
4605:
4599:
4596:
4590:
4589:
4553:
4547:
4546:
4536:
4518:
4494:
4488:
4487:
4459:
4453:
4452:
4408:
4402:
4401:
4391:
4359:
4353:
4352:
4333:10.1038/369215a0
4308:
4302:
4299:
4293:
4290:
4284:
4281:
4275:
4274:
4273:. July 30, 2018.
4267:
4261:
4260:
4216:
4210:
4209:
4199:
4167:
4161:
4158:
4152:
4151:
4107:
4101:
4100:
4072:
4066:
4065:
4013:
4007:
4006:
3962:
3956:
3955:
3935:
3929:
3926:
3920:
3919:
3909:
3869:
3863:
3860:
3854:
3853:
3825:
3814:
3800:
3794:
3793:
3757:
3751:
3748:
3742:
3741:
3723:
3683:
3677:
3676:
3632:
3626:
3625:
3581:
3575:
3574:
3546:
3540:
3537:
3531:
3530:
3520:
3502:
3470:
3461:
3460:
3450:
3415:
3409:
3406:
3397:
3390:
3381:
3374:
3368:
3365:
3359:
3358:
3314:
3308:
3307:
3263:
3257:
3254:
3248:
3244:
3238:
3237:
3219:The Turing Guide
3189:
2993:
2984:Target pattern.
2981:
2969:
2941:
2939:
2938:
2933:
2931:
2921:
2920:
2908:
2907:
2893:
2892:
2882:
2877:
2865:
2863:
2862:
2861:
2851:
2846:
2827:
2824:
2807:
2802:
2799:
2783:
2782:
2770:
2769:
2755:
2754:
2745:
2741:
2739:
2734:
2725:
2717:
2711:
2706:
2689:
2681:
2678:
2661:
2656:
2653:
2622:
2606:
2597: = 0.
2590:
2573:
2565:Hopf bifurcation
2563:may be either a
2554:
2548:
2523:action potential
2520:
2494:
2492:
2491:
2486:
2484:
2465:
2464:
2453:
2448:
2429:
2428:
2382:
2381:
2370:
2365:
2346:
2345:
2311:
2309:
2308:
2303:
2298:
2297:
2258:
2257:
2218:
2217:
2178:
2177:
2132:
2124:
2105:
2103:
2102:
2097:
2092:
2091:
2075:
2074:
2069:
2068:
2060:
2043:
2042:
2037:
2036:
2028:
2016:
2015:
2010:
2001:
2000:
1984:
1983:
1978:
1977:
1969:
1965:
1964:
1942:
1941:
1936:
1935:
1927:
1923:
1922:
1905:
1904:
1889:
1888:
1872:
1871:
1866:
1865:
1857:
1853:
1852:
1830:
1829:
1824:
1823:
1815:
1811:
1810:
1780:
1778:
1777:
1772:
1770:
1769:
1768:
1760:
1747:
1746:
1730:
1729:
1721:
1705:
1704:
1696:
1673:
1665:
1664:
1659:
1658:
1650:
1629:
1627:
1626:
1621:
1619:
1618:
1560:
1559:
1549:
1548:
1529:
1528:
1508:
1507:
1500:
1499:
1476:
1475:
1455:
1454:
1444:
1443:
1427:
1426:
1368:
1364:
1361:under the force
1360:
1356:
1350:
1343:
1341:
1340:
1335:
1315:
1314:
1306:
1285:
1284:
1276:
1273:
1272:
1248:
1247:
1239:
1235:
1230:
1208:
1201:
1194:
1170:
1156:
1131:Schrödinger type
1125:
1123:
1122:
1117:
1096:
1091:
1073:
1072:
1064:
1045:
1044:
1036:
1023:
1005:With the ansatz
1001:
999:
998:
993:
988:
987:
977:
976:
960:
959:
943:
942:
911:
910:
902:
884:
883:
875:
871:
866:
851:
850:
842:
839:
838:
819:
789:
775:
767:
763:
749:
715:
713:
711:
710:
704:
701:
677:
663:
661:
660:
655:
650:
647:
644:
640:
624:
623:
618:
614:
610:
609:
594:
585:
576:
571:
556:
555:
539:
537:
536:
531:
529:
528:
512:
510:
509:
504:
502:
500:
492:
491:
490:
480:
469:
468:
446:
423:Zeldovich number
420:
409:
382:
355:
325:
323:
322:
317:
293:
288:
270:
269:
222:travelling waves
219:
203:
202:
193:
171:
169:
168:
163:
155:
147:
139:
134:
133:
123:
105:
100:
99:
38:Gray–Scott model
21:
4826:
4825:
4821:
4820:
4819:
4817:
4816:
4815:
4791:
4790:
4771:
4766:
4761:
4757:
4703:
4702:
4698:
4684:
4683:
4679:
4663:10.1.1.105.2369
4639:
4638:
4634:
4625:
4623:
4619:GollyGang/ready
4616:
4615:
4611:
4606:
4602:
4597:
4593:
4555:
4554:
4550:
4496:
4495:
4491:
4461:
4460:
4456:
4410:
4409:
4405:
4361:
4360:
4356:
4310:
4309:
4305:
4300:
4296:
4291:
4287:
4282:
4278:
4269:
4268:
4264:
4218:
4217:
4213:
4169:
4168:
4164:
4159:
4155:
4109:
4108:
4104:
4074:
4073:
4069:
4015:
4014:
4010:
3979:10.2307/1939378
3964:
3963:
3959:
3952:
3937:
3936:
3932:
3927:
3923:
3871:
3870:
3866:
3861:
3857:
3827:
3826:
3817:
3801:
3797:
3759:
3758:
3754:
3749:
3745:
3685:
3684:
3680:
3634:
3633:
3629:
3583:
3582:
3578:
3548:
3547:
3543:
3538:
3534:
3472:
3471:
3464:
3417:
3416:
3412:
3407:
3400:
3391:
3384:
3375:
3371:
3366:
3362:
3316:
3315:
3311:
3265:
3264:
3260:
3255:
3251:
3245:
3241:
3234:
3204:
3191:Wooley, T. E.,
3190:
3186:
3182:
3157:
3085:
3065:
3045:
3028:
3004:
2997:
2994:
2985:
2982:
2973:
2970:
2929:
2928:
2912:
2899:
2894:
2884:
2853:
2831:
2821:
2791:
2790:
2774:
2761:
2756:
2746:
2697:
2693:
2675:
2639:
2638:
2630:
2623:
2614:
2607:
2598:
2591:
2568:
2550:
2538:
2531:
2526:
2499:
2482:
2481:
2456:
2433:
2420:
2414:
2413:
2373:
2350:
2337:
2328:
2327:
2292:
2291:
2286:
2280:
2279:
2274:
2264:
2252:
2251:
2246:
2240:
2239:
2234:
2224:
2212:
2211:
2206:
2200:
2199:
2194:
2184:
2172:
2171:
2166:
2160:
2159:
2154:
2144:
2138:
2137:
2126:
2117:
2086:
2085:
2057:
2054:
2053:
2025:
2018:
2005:
1995:
1994:
1966:
1956:
1953:
1952:
1924:
1914:
1907:
1896:
1883:
1882:
1854:
1844:
1841:
1840:
1812:
1802:
1795:
1789:
1788:
1748:
1741:
1740:
1716:
1715:
1687:
1647:
1642:
1641:
1613:
1612:
1591:
1590:
1565:
1554:
1553:
1537:
1534:
1533:
1517:
1510:
1502:
1501:
1491:
1489:
1483:
1482:
1477:
1467:
1460:
1449:
1448:
1435:
1432:
1431:
1418:
1411:
1405:
1404:
1387:
1366:
1362:
1358:
1352:
1348:
1264:
1214:
1213:
1206:
1196:
1188:
1181:
1172:
1165:
1150:
1143:
1134:
1029:
1028:
1006:
968:
953:
934:
830:
825:
824:
801:
791:
783:
777:
770:
765:
751:
728:
705:
702:
691:
690:
688:
679:
668:
601:
600:
596:
595:
582:
578:
545:
544:
518:
517:
493:
481:
460:
455:
454:
430:
411:
392:
365:
338:
261:
256:
255:
249:
237:Turing patterns
213:
206:diagonal matrix
196:
195:
176:
125:
91:
86:
85:
28:
23:
22:
15:
12:
11:
5:
4824:
4822:
4814:
4813:
4808:
4803:
4793:
4792:
4789:
4788:
4783:
4778:
4770:
4769:External links
4767:
4765:
4764:
4755:
4710:BMC Biophysics
4696:
4677:
4632:
4609:
4600:
4591:
4548:
4489:
4454:
4403:
4354:
4303:
4294:
4285:
4276:
4262:
4211:
4182:(10): 100588.
4162:
4153:
4102:
4067:
4008:
3957:
3950:
3930:
3921:
3884:(2): 277–285.
3864:
3855:
3836:(3): 399–411.
3815:
3795:
3752:
3743:
3678:
3627:
3576:
3541:
3532:
3462:
3410:
3398:
3382:
3369:
3360:
3309:
3258:
3249:
3239:
3233:978-0198747826
3232:
3199:, Chapter 34,
3183:
3181:
3178:
3177:
3176:
3173:
3168:
3163:
3156:
3153:
3152:
3151:
3146:
3144:Turing pattern
3141:
3136:
3131:
3126:
3121:
3116:
3111:
3106:
3101:
3096:
3091:
3084:
3081:
3064:
3061:
3044:
3041:
3027:
3024:
3012:blood clotting
3003:
3000:
2999:
2998:
2995:
2988:
2986:
2983:
2976:
2974:
2971:
2964:
2962:
2943:
2942:
2927:
2924:
2919:
2915:
2911:
2906:
2902:
2898:
2895:
2891:
2887:
2881:
2876:
2872:
2868:
2860:
2856:
2850:
2845:
2841:
2837:
2834:
2830:
2822:
2820:
2817:
2814:
2811:
2806:
2797:
2793:
2792:
2789:
2786:
2781:
2777:
2773:
2768:
2764:
2760:
2757:
2753:
2749:
2744:
2738:
2733:
2729:
2723:
2720:
2715:
2710:
2705:
2701:
2696:
2692:
2687:
2684:
2676:
2674:
2671:
2668:
2665:
2660:
2651:
2647:
2646:
2632:
2631:
2624:
2617:
2615:
2608:
2601:
2599:
2592:
2585:
2583:
2536:
2529:
2496:
2495:
2480:
2477:
2474:
2471:
2468:
2463:
2459:
2452:
2447:
2443:
2439:
2436:
2434:
2432:
2427:
2423:
2419:
2416:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2380:
2376:
2369:
2364:
2360:
2356:
2353:
2351:
2349:
2344:
2340:
2336:
2335:
2313:
2312:
2301:
2296:
2290:
2287:
2285:
2282:
2281:
2278:
2275:
2273:
2270:
2269:
2267:
2261:
2256:
2250:
2247:
2245:
2242:
2241:
2238:
2235:
2233:
2230:
2229:
2227:
2221:
2216:
2210:
2207:
2205:
2202:
2201:
2198:
2195:
2193:
2190:
2189:
2187:
2181:
2176:
2170:
2167:
2165:
2162:
2161:
2158:
2155:
2153:
2150:
2149:
2147:
2107:
2106:
2095:
2090:
2084:
2081:
2078:
2073:
2066:
2063:
2056:
2055:
2052:
2049:
2046:
2041:
2034:
2031:
2024:
2023:
2021:
2014:
2009:
2004:
1999:
1993:
1990:
1987:
1982:
1975:
1972:
1963:
1959:
1955:
1954:
1951:
1948:
1945:
1940:
1933:
1930:
1921:
1917:
1913:
1912:
1910:
1903:
1899:
1895:
1892:
1887:
1881:
1878:
1875:
1870:
1863:
1860:
1851:
1847:
1843:
1842:
1839:
1836:
1833:
1828:
1821:
1818:
1809:
1805:
1801:
1800:
1798:
1782:
1781:
1767:
1763:
1759:
1755:
1751:
1745:
1739:
1736:
1733:
1727:
1724:
1718:
1717:
1714:
1711:
1708:
1702:
1699:
1693:
1692:
1690:
1685:
1682:
1679:
1676:
1672:
1668:
1663:
1656:
1653:
1631:
1630:
1617:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1570:
1568:
1563:
1558:
1552:
1547:
1544:
1540:
1536:
1535:
1532:
1527:
1524:
1520:
1516:
1515:
1513:
1506:
1498:
1494:
1490:
1488:
1485:
1484:
1481:
1478:
1474:
1470:
1466:
1465:
1463:
1458:
1453:
1447:
1442:
1438:
1434:
1433:
1430:
1425:
1421:
1417:
1416:
1414:
1386:
1383:
1351:with position
1345:
1344:
1333:
1330:
1327:
1324:
1321:
1318:
1312:
1309:
1303:
1300:
1297:
1294:
1291:
1288:
1282:
1279:
1271:
1267:
1263:
1260:
1257:
1254:
1251:
1245:
1242:
1234:
1229:
1225:
1221:
1186:
1177:
1148:
1139:
1127:
1126:
1115:
1112:
1109:
1106:
1103:
1100:
1095:
1090:
1086:
1082:
1079:
1076:
1070:
1067:
1060:
1057:
1054:
1051:
1048:
1042:
1039:
1003:
1002:
991:
986:
983:
980:
975:
971:
967:
964:
958:
952:
949:
946:
941:
937:
933:
930:
927:
924:
921:
918:
914:
908:
905:
899:
896:
893:
890:
887:
881:
878:
870:
865:
861:
857:
854:
848:
845:
837:
833:
799:
781:
665:
664:
653:
643:
639:
636:
633:
630:
627:
622:
617:
613:
608:
604:
599:
591:
588:
581:
575:
570:
567:
563:
559:
554:
527:
514:
513:
499:
496:
489:
484:
478:
475:
472:
467:
463:
327:
326:
315:
312:
309:
306:
303:
300:
297:
292:
287:
283:
279:
276:
273:
268:
264:
248:
245:
226:self-organized
173:
172:
161:
158:
154:
150:
146:
142:
138:
132:
128:
121:
117:
114:
108:
104:
98:
94:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4823:
4812:
4809:
4807:
4804:
4802:
4799:
4798:
4796:
4787:
4784:
4782:
4779:
4776:
4773:
4772:
4768:
4759:
4756:
4751:
4747:
4742:
4737:
4733:
4729:
4724:
4719:
4715:
4711:
4707:
4700:
4697:
4692:
4688:
4681:
4678:
4673:
4669:
4664:
4659:
4655:
4651:
4647:
4643:
4636:
4633:
4621:
4620:
4613:
4610:
4604:
4601:
4595:
4592:
4587:
4583:
4579:
4575:
4571:
4567:
4563:
4559:
4552:
4549:
4544:
4540:
4535:
4530:
4526:
4522:
4517:
4512:
4508:
4504:
4500:
4493:
4490:
4485:
4481:
4477:
4473:
4469:
4465:
4458:
4455:
4450:
4446:
4442:
4438:
4434:
4430:
4426:
4422:
4418:
4414:
4407:
4404:
4399:
4395:
4390:
4385:
4381:
4377:
4373:
4369:
4365:
4358:
4355:
4350:
4346:
4342:
4338:
4334:
4330:
4326:
4322:
4318:
4314:
4307:
4304:
4298:
4295:
4289:
4286:
4280:
4277:
4272:
4266:
4263:
4258:
4254:
4250:
4246:
4242:
4238:
4234:
4230:
4226:
4222:
4215:
4212:
4207:
4203:
4198:
4193:
4189:
4185:
4181:
4177:
4173:
4166:
4163:
4157:
4154:
4149:
4145:
4141:
4137:
4133:
4129:
4125:
4121:
4117:
4113:
4106:
4103:
4098:
4094:
4090:
4086:
4082:
4078:
4071:
4068:
4063:
4059:
4055:
4051:
4047:
4043:
4039:
4035:
4031:
4027:
4023:
4019:
4012:
4009:
4004:
4000:
3996:
3992:
3988:
3984:
3980:
3976:
3972:
3968:
3961:
3958:
3953:
3947:
3943:
3942:
3934:
3931:
3925:
3922:
3917:
3913:
3908:
3903:
3899:
3895:
3891:
3887:
3883:
3879:
3875:
3868:
3865:
3859:
3856:
3851:
3847:
3843:
3839:
3835:
3831:
3824:
3822:
3820:
3816:
3813:
3809:
3805:
3802:A. W. Liehr:
3799:
3796:
3791:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3756:
3753:
3747:
3744:
3739:
3735:
3731:
3727:
3722:
3717:
3713:
3709:
3705:
3701:
3698:(2): 023148.
3697:
3693:
3689:
3682:
3679:
3674:
3670:
3666:
3662:
3658:
3654:
3650:
3646:
3642:
3638:
3631:
3628:
3623:
3619:
3615:
3611:
3607:
3603:
3599:
3595:
3591:
3587:
3580:
3577:
3572:
3568:
3564:
3560:
3556:
3552:
3545:
3542:
3536:
3533:
3528:
3524:
3519:
3514:
3510:
3506:
3501:
3496:
3492:
3488:
3484:
3480:
3476:
3469:
3467:
3463:
3458:
3454:
3449:
3444:
3440:
3436:
3432:
3428:
3424:
3420:
3419:Turing, A. M.
3414:
3411:
3405:
3403:
3399:
3395:
3389:
3387:
3383:
3379:
3373:
3370:
3364:
3361:
3356:
3352:
3348:
3344:
3340:
3336:
3332:
3328:
3324:
3320:
3313:
3310:
3305:
3301:
3297:
3293:
3289:
3285:
3281:
3277:
3273:
3269:
3262:
3259:
3253:
3250:
3243:
3240:
3235:
3229:
3225:
3221:
3220:
3215:
3214:Wilson, Robin
3211:
3207:
3202:
3198:
3194:
3188:
3185:
3179:
3175:Wrinkle paint
3174:
3172:
3169:
3167:
3164:
3162:
3159:
3158:
3154:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3086:
3082:
3080:
3078:
3074:
3070:
3062:
3060:
3056:
3054:
3050:
3042:
3040:
3037:
3036:morphogenesis
3033:
3025:
3023:
3019:
3017:
3016:gas discharge
3013:
3009:
3001:
2992:
2987:
2980:
2975:
2968:
2963:
2960:
2958:
2956:
2952:
2948:
2925:
2917:
2913:
2900:
2896:
2889:
2885:
2879:
2874:
2870:
2866:
2858:
2854:
2848:
2843:
2839:
2835:
2832:
2828:
2818:
2812:
2804:
2795:
2787:
2779:
2775:
2762:
2758:
2751:
2747:
2742:
2736:
2731:
2727:
2721:
2718:
2713:
2708:
2703:
2699:
2694:
2690:
2685:
2682:
2672:
2666:
2658:
2649:
2637:
2636:
2635:
2628:
2621:
2616:
2612:
2605:
2600:
2596:
2589:
2584:
2581:
2579:
2577:
2571:
2566:
2562:
2556:
2553:
2547:
2543:
2539:
2532:
2524:
2519:
2515:
2511:
2507:
2503:
2478:
2475:
2472:
2469:
2466:
2461:
2450:
2445:
2441:
2437:
2435:
2430:
2425:
2417:
2410:
2407:
2404:
2401:
2395:
2389:
2386:
2383:
2378:
2367:
2362:
2358:
2354:
2352:
2347:
2342:
2326:
2325:
2324:
2323:
2318:
2299:
2294:
2288:
2283:
2276:
2271:
2265:
2259:
2254:
2248:
2243:
2236:
2231:
2225:
2219:
2214:
2208:
2203:
2196:
2191:
2185:
2179:
2174:
2168:
2163:
2156:
2151:
2145:
2136:
2135:
2134:
2131:
2130:
2122:
2121:
2116:
2112:
2093:
2088:
2079:
2061:
2047:
2029:
2019:
2002:
1997:
1988:
1970:
1961:
1957:
1946:
1928:
1919:
1915:
1908:
1901:
1897:
1893:
1890:
1885:
1876:
1858:
1849:
1834:
1816:
1807:
1796:
1787:
1786:
1785:
1761:
1753:
1749:
1743:
1734:
1722:
1709:
1697:
1688:
1683:
1677:
1674:
1640:
1639:
1638:
1637:perturbation
1636:
1615:
1606:
1603:
1600:
1594:
1584:
1581:
1578:
1572:
1566:
1561:
1556:
1550:
1545:
1542:
1530:
1525:
1522:
1511:
1504:
1496:
1492:
1486:
1479:
1472:
1468:
1461:
1456:
1451:
1445:
1440:
1428:
1423:
1412:
1403:
1402:
1401:
1398:
1396:
1392:
1384:
1382:
1380:
1376:
1370:
1355:
1331:
1328:
1319:
1307:
1298:
1295:
1289:
1277:
1269:
1261:
1258:
1252:
1240:
1232:
1227:
1219:
1212:
1211:
1210:
1203:
1199:
1192:
1185:
1180:
1175:
1168:
1164:
1160:
1159:eigenfunction
1157:is a neutral
1154:
1147:
1142:
1137:
1132:
1113:
1107:
1101:
1098:
1093:
1088:
1080:
1077:
1074:
1065:
1058:
1055:
1052:
1049:
1046:
1037:
1027:
1026:
1025:
1021:
1017:
1013:
1009:
989:
981:
973:
969:
965:
962:
947:
935:
931:
928:
922:
916:
912:
903:
894:
888:
885:
876:
868:
863:
855:
852:
843:
835:
823:
822:
821:
817:
813:
809:
805:
798:
794:
787:
780:
773:
762:
758:
754:
747:
743:
739:
735:
731:
720:
716:
709:
699:
695:
686:
682:
675:
671:
651:
641:
634:
628:
625:
620:
615:
611:
606:
597:
589:
586:
579:
565:
561:
557:
543:
542:
541:
497:
494:
482:
476:
473:
470:
465:
453:
452:
451:
448:
445:
441:
437:
433:
428:
424:
419:
415:
407:
403:
399:
395:
390:
386:
380:
376:
372:
368:
363:
359:
353:
349:
345:
341:
337:. The choice
336:
332:
313:
307:
301:
298:
295:
290:
285:
277:
274:
271:
266:
254:
253:
252:
246:
244:
242:
238:
234:
230:
227:
223:
218:
217:
211:
207:
201:
200:
191:
187:
186:
181:
180:
159:
140:
130:
119:
115:
106:
96:
84:
83:
82:
80:
76:
72:
68:
64:
60:
55:
53:
49:
45:
39:
34:
30:
19:
4758:
4713:
4709:
4699:
4691:The Winnower
4690:
4680:
4648:(1): 47–74.
4645:
4641:
4635:
4626:September 4,
4624:, retrieved
4618:
4612:
4603:
4594:
4561:
4557:
4551:
4506:
4502:
4492:
4467:
4463:
4457:
4416:
4412:
4406:
4371:
4367:
4357:
4316:
4312:
4306:
4297:
4288:
4279:
4265:
4224:
4220:
4214:
4179:
4175:
4165:
4156:
4115:
4111:
4105:
4080:
4076:
4070:
4021:
4017:
4011:
3970:
3966:
3960:
3940:
3933:
3924:
3881:
3877:
3867:
3858:
3833:
3829:
3803:
3798:
3765:
3761:
3755:
3746:
3695:
3691:
3681:
3640:
3636:
3630:
3589:
3585:
3579:
3554:
3550:
3544:
3535:
3482:
3478:
3430:
3426:
3413:
3392:P. C. Fife,
3372:
3363:
3322:
3318:
3312:
3271:
3267:
3261:
3252:
3242:
3218:
3200:
3197:Maini, P. K.
3193:Baker, R. E.
3187:
3066:
3057:
3046:
3029:
3020:
3005:
2944:
2633:
2626:
2610:
2594:
2575:
2569:
2557:
2551:
2545:
2541:
2534:
2527:
2517:
2513:
2509:
2505:
2501:
2497:
2316:
2314:
2128:
2127:
2119:
2118:
2108:
1783:
1632:
1399:
1388:
1371:
1353:
1346:
1204:
1197:
1190:
1183:
1178:
1173:
1166:
1152:
1145:
1140:
1135:
1128:
1019:
1015:
1011:
1007:
1004:
815:
811:
807:
803:
796:
792:
785:
778:
771:
760:
756:
752:
745:
741:
737:
733:
729:
725:
707:
697:
693:
684:
680:
673:
669:
666:
515:
449:
443:
439:
435:
431:
417:
413:
405:
401:
397:
393:
383:to describe
378:
374:
370:
366:
351:
347:
343:
339:
328:
250:
240:
215:
214:
198:
197:
189:
184:
183:
178:
177:
174:
56:
43:
42:
29:
3049:temperature
3043:Experiments
2561:bifurcation
1391:Alan Turing
362:populations
4795:Categories
3721:2152/43281
3180:References
3073:geometries
3051:pulses on
2951:hysteretic
1635:plane wave
1163:eigenvalue
678:such that
427:combustion
4732:2046-1682
4716:(1): 11.
4658:CiteSeerX
4586:0167-2789
4525:0022-3751
4484:0022-3654
4441:0031-9007
4398:1367-2630
4341:0028-0836
4241:0962-8452
4206:240589650
4132:0962-8452
4097:0218-3390
4062:129301761
4046:2053-9193
3987:0012-9658
3898:2161-5063
3850:1385-8947
3790:0031-9007
3730:1054-1500
3665:0031-9007
3614:0031-9007
3571:0022-2526
3509:0006-3495
3457:2054-0280
3355:122764449
3347:0022-1120
3296:0022-1120
3018:systems.
2905:′
2829:κ
2767:′
2722:τ
2686:τ
2476:−
2458:∇
2422:∂
2418:τ
2405:σ
2402:−
2375:∇
2339:∂
2277:−
2272:−
2244:−
2232:−
2209:−
2204:−
2169:−
2157:−
2065:~
2033:~
2013:′
1974:~
1932:~
1894:−
1862:~
1846:∂
1820:~
1804:∂
1762:⋅
1726:~
1701:~
1655:~
1539:∂
1519:∂
1437:∂
1420:∂
1395:diffusion
1320:ξ
1311:^
1290:ξ
1281:^
1270:ξ
1266:∂
1253:ξ
1244:^
1228:ξ
1224:∂
1161:with the
1085:∂
1078:−
1069:^
1056:ψ
1053:λ
1047:ψ
1041:^
940:′
932:−
907:~
886:−
880:~
860:∂
847:~
832:∂
626:−
603:∂
574:∞
569:∞
566:−
562:∫
495:δ
483:δ
477:−
462:∂
282:∂
263:∂
127:∇
120:_
116:_
93:∂
59:chemistry
52:diffusion
4750:25737778
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