March  2013, 8(1): 79-114. doi: 10.3934/nhm.2013.8.79

Traveling fronts guided by the environment for reaction-diffusion equations

1. 

CAMS, UMR 8557, EHESS, 190-198 avenue de France, 75244 Paris Cedex 13, France

2. 

LATP, UMR 7353, Aix-Marseille Université, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France

Received  May 2012 Revised  March 2013 Published  April 2013

This paper deals with the existence of traveling fronts for the reaction-diffusion equation: $$ \frac{\partial u}{\partial t} - \Delta u =h(u,y) \qquad t\in \mathbb{R}, \; x=(x_1,y)\in \mathbb{R}^N. $$ We first consider the case $h(u,y)=f(u)-\alpha g(y)u$ where $f$ is of KPP or bistable type and $\lim_{|y|\rightarrow +\infty}g(y)=+\infty$. This equation comes from a model in population dynamics in which there is spatial spreading as well as phenotypic mutation of a quantitative phenotypic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of threshold value $\alpha_0$ and of a nonzero asymptotic profile (a stationary limiting solution) $V(y)$ if and only if $\alpha<\alpha_0$. When this condition is met, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case.
    We also study here the case where $h(y,u)=f(u)$ for $|y|\leq L_1$ and $h(y,u) \approx - \alpha u$ for $|y|>L_2\geq L_1$. This equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if $L_1$ is large enough and the non-existence if $L_2$ is too small.
Citation: Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations. Networks & Heterogeneous Media, 2013, 8 (1) : 79-114. doi: 10.3934/nhm.2013.8.79
References:
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Matthieu Alfaro, Jérôme Coville and Gaël Raoul, Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait,, , (2013).   Google Scholar

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Tobias Back, Jochen G. Hirsch, Kristina Szabo and Achim Gass, Failure to demonstrate peri-infarct depolarizations by repetitive mr diffusion imaging in acute human stroke,, Stroke, 31 (2000), 2901.  doi: 10.1161/01.STR.31.12.2901.  Google Scholar

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H. Berestycki and P.-L. Lions, Une méthode locale pour l'existence de solutions positives de problèmes semi-linéaires elliptiques dans $R^N$,, J. Analyse Math., 38 (1980), 144.   Google Scholar

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H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method,, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1.  doi: 10.1007/BF01244896.  Google Scholar

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Henri Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques,, J. Funct. Anal., 40 (1981), 1.  doi: 10.1016/0022-1236(81)90069-0.  Google Scholar

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Henri Berestycki and Guillemette Chapuisat, A numerical study of a non local reaction-diffusion equation in population dynamics,, In preparation., (2013).   Google Scholar

[7]

Henri Berestycki and Guillemette Chapuisat, Propagation in a non homogeneous kpp equation arising in cancer modeling,, In preparation, (2013).   Google Scholar

[8]

Henri Berestycki, Françcois Hamel and Hiroshi Matano, Bistable traveling waves passing an obstacle,, Comm. Pures and Appl. Math., 62 (2009), 729.  doi: 10.1002/cpa.20275.  Google Scholar

[9]

Henri Berestycki and François Hamel, Fronts and invasions in general domains,, C. R. Math. Acad. Sci. Paris, 343 (2006), 711.  doi: 10.1016/j.crma.2006.09.036.  Google Scholar

[10]

Henri Berestycki and François Hamel, Generalized travelling waves for reaction-diffusion equations,, in, 446 (2007), 101.  doi: 10.1090/conm/446.  Google Scholar

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Henri Berestycki and François Hamel, Generalized transition waves and their properties,, Comm. Pure Appl. Math., (2012).  doi: 10.1002/cpa.21389.  Google Scholar

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Henri Berestycki and Pierre-Louis Lions, Some applications of the method of super and subsolutions,, in, 782 (1980), 16.  doi: 10.1007/BFb0090426.  Google Scholar

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Henri Berestycki, Grégoire Nadin, Benoit Perthame and Lenya Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states,, Nonlinearity, 22 (2009), 2813.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[14]

Henri Berestycki and Louis Nirenberg, Travelling fronts in cylinders,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497.   Google Scholar

[15]

Henri Berestycki and Luca Rossi, On the principal eigenvalue of elliptic operators in $\mathbbR^N$ and applications,, J. Eur. Math. Soc. (JEMS), 8 (2006), 195.  doi: 10.4171/JEMS/47.  Google Scholar

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Henri Berestycki and Luca Rossi, Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space,, Discrete Contin. Dyn. Syst., 21 (2008), 41.  doi: 10.3934/dcds.2008.21.41.  Google Scholar

[17]

Guillemette Chapuisat, Existence and nonexistence of curved front solution of a biological equation,, J. Differential Equations, 236 (2007), 237.  doi: 10.1016/j.jde.2007.01.021.  Google Scholar

[18]

Guillemette Chapuisat and Emmanuel Grenier, Existence and nonexistence of traveling wave solutions for a bistable reaction-diffusion equation in an infinite cylinder whose diameter is suddenly increased,, Comm. Partial Differential Equations, 30 (2005), 1805.  doi: 10.1080/03605300500300006.  Google Scholar

[19]

Guillemette Chapuisat, Emmanuel Grenier, Marie-Aim\'ee Dronne, Marc Hommel and Jean-Pierre Boissel, A global model of ischemic stroke with stress on spreading depression,, Progress in Biophysics and Molecular Biology, 97 (2008), 4.  doi: 10.1016/j.pbiomolbio.2007.10.004.  Google Scholar

[20]

Guillemette Chapuisat and Romain Joly, Asymptotic profiles for a traveling front solution of a biological equation,, Math. Models Methods Appl. Sci., 21 (2011), 2155.  doi: 10.1142/S0218202511005696.  Google Scholar

[21]

Jacques De Keyser, Geert Sulter and Paul G. Luiten, Clinical trials with neuroprotective drugs in acute ischaemic stroke: are we doing the right thing?,, Trends Neurosci., 22 (1999), 535.   Google Scholar

[22]

Laurent Desvillettes, Régis Ferrières and Céline Prevost, Infinite dimensional reaction-diffusion for population dynamics,, Prépublication du CMLA No. 2003-04, (2003), 2003.   Google Scholar

[23]

Lawrence C. Evans, "Partial Differential Equations,", 19 of Graduate Studies in Mathematics, 19 (1998).   Google Scholar

[24]

Laurent Excoffier, Matthieu Foll and Rémy J Petit, Genetic consequences of range expansions,, Annual Review of Ecology Evolution and Systematics, 40 (2009), 481.  doi: 10.1146/annurev.ecolsys.39.110707.173414.  Google Scholar

[25]

Ido Filin, Robert D. Holt and Michael Barfield, The relation of density regulation to habitat specialization, evolution of a species' range, and the dynamics of biological invasions,, Am. Nat., 172 (2008), 233.  doi: 10.1086/589459.  Google Scholar

[26]

Ali Gorji, Dieter Scheller, Heidrun Straub, Frank Tegtmeier, Rüdiger Köhling, Jörg Michael Höhling, Ingrid Tuxhorn, Alois Ebner, Peter Wolf, Hans Werner Panneck, Falk Oppel and Erwin Josef Speckmann, Spreading depression in human neocortical slices,, Brain Res., 906 (2001), 74.  doi: 10.1016/S0006-8993(01)02557-4.  Google Scholar

[27]

Oskar Hallatschek, Pascal Hersen, Sharad Ramanathan and David R. Nelson, Genetic drift at expanding frontiers promotes gene segregation,, Proceedings of the National Academy of Sciences, 104 (2007), 19926.  doi: 10.1073/pnas.0710150104.  Google Scholar

[28]

Oskar Hallatschek and David R. Nelson, Gene surfing in expanding populations,, Theoretical Population Biology, 73 (2008), 158.  doi: 10.1016/j.tpb.2007.08.008.  Google Scholar

[29]

Oskar Hallatschek and David R. Nelson, Life at the front of an expanding population,, Evolution, 64 (2010), 193.  doi: 10.1111/j.1558-5646.2009.00809.x.  Google Scholar

[30]

Robert D. Holt, Michael Barfield, Ido Filin and Samantha Forde, Predation and the evolutionary dynamics of species ranges,, Am. Nat., 178 (2011), 488.  doi: 10.1086/661909.  Google Scholar

[31]

Mark Kirkpatrick and Nicholas H. Barton, Evolution of a species' range,, Am. Nat., 150 (1997), 1.  doi: 10.1086/286054.  Google Scholar

[32]

Jean-François Mallordy and Jean-Michel Roquejoffre, A parabolic equation of the KPP type in higher dimensions,, SIAM J. Math. Anal., 26 (1995), 1.  doi: 10.1137/S0036141093246105.  Google Scholar

[33]

Hiroshi Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401.  doi: 10.2977/prims/1195188180.  Google Scholar

[34]

Hiroshi Matano, Traveling waves in spatially inhomogeneous diffusive media - the non-periodic case,, Preprint, (2012).   Google Scholar

[35]

Avraham Mayevsky, Avi Doron, Tamar Manor, Sigal Meilin, Nili Zarchin and George E. Ouaknine, Cortical spreading depression recorded from the human brain using a multiparmetric monitoring system,, Brain Res., 740 (1996), 268.  doi: 10.1016/S0006-8993(96)00874-8.  Google Scholar

[36]

Maiken Nedergaard, Arthur J. Cooper and Steven A. Goldman, Gap junctions are required for the propagation of spreading depression,, J. Neurobiol., 28 (1995), 433.  doi: 10.1002/neu.480280404.  Google Scholar

[37]

James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik and Andrej Zlatoš, Existence and non-existence of fisher-kpp transition fronts,, Arch. Ration. Mech. Anal., (2012).  doi: 10.1007/s00205-011-0449-4.  Google Scholar

[38]

L. A. Peletier and James Serrin, Uniqueness of nonnegative solutions of semilinear equations in $R^n$,, J. Differential Equations, 61 (1986), 380.  doi: 10.1016/0022-0396(86)90112-9.  Google Scholar

[39]

Cristina Pocci, Ayman Moussa, Florence Hubert and Guillemette Chapuisat., Numerical study of the stopping of aura during migraine,, in, 30 (2009), 44.  doi: 10.1051/proc/2010005.  Google Scholar

[40]

Paul H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations,, Indiana Univ. Math. J., 23 (): 173.  doi: 10.1512/iumj.1973.23.23014.  Google Scholar

[41]

Lionel Roques, Jimmy Garnier, François Hamel and Etienne K. Klein, Allee effect promotes diversity in traveling waves of colonization,, Proceedings of the National Academy of Sciences, 109 (2012), 8828.  doi: 10.1073/pnas.1201695109.  Google Scholar

[42]

Laurent Schwartz, "Analyse Hilbertienne,", Collection Méthodes. Hermann, (1979).   Google Scholar

[43]

Bruno Shapiro, Osmotic forces and gap junctions in spreading depression: A computational model,, J. Comput. Neurosci., 10 (2001), 99.   Google Scholar

[44]

Wenxian Shen, Dynamical systems and traveling waves in almost periodic structures,, J. Differential Equations, 169 (2001), 493.  doi: 10.1006/jdeq.2000.3906.  Google Scholar

[45]

George G. Somjen, "Ions in the Brain: Normal Function, Seizures, and Stroke,", Oxford University Press, (2004).   Google Scholar

[46]

Anthony J. Strong, Martin Fabricius, Martyn G. Boutelle, Stuart J. Hibbins, Sarah E. Hopwood, Robina Jones, Mark C. Parkin and Martin Lauritzen, Spreading and synchronous depressions of cortical activity in acutely injured human brain,, Stroke, 33 (2002), 2738.  doi: 10.1161/01.STR.0000043073.69602.09.  Google Scholar

[47]

Henry C. Tuckwell, Predictions and properties of a model of potassium and calcium ion movements during spreading cortical depression,, Int. J. Neurosci., 10 (1980), 145.  doi: 10.3109/00207458009160493.  Google Scholar

[48]

José M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains,, Comm. Partial Differential Equations, 18 (1993), 505.  doi: 10.1080/03605309308820939.  Google Scholar

[49]

Marcel O. Vlad, L. Luca Cavalli-Sforza and John Ross, Enhanced (hydrodynamic) transport induced by population growth in reactiondiffusion systems with application to population genetics,, Proceedings of the National Academy of Sciences, 101 (2004), 10249.   Google Scholar

[50]

Andrej Zlatoš, Generalized traveling waves in disordered media: Existence, uniqueness, and stability,, Preprint, (2012).   Google Scholar

show all references

References:
[1]

Matthieu Alfaro, Jérôme Coville and Gaël Raoul, Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait,, , (2013).   Google Scholar

[2]

Tobias Back, Jochen G. Hirsch, Kristina Szabo and Achim Gass, Failure to demonstrate peri-infarct depolarizations by repetitive mr diffusion imaging in acute human stroke,, Stroke, 31 (2000), 2901.  doi: 10.1161/01.STR.31.12.2901.  Google Scholar

[3]

H. Berestycki and P.-L. Lions, Une méthode locale pour l'existence de solutions positives de problèmes semi-linéaires elliptiques dans $R^N$,, J. Analyse Math., 38 (1980), 144.   Google Scholar

[4]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method,, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1.  doi: 10.1007/BF01244896.  Google Scholar

[5]

Henri Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques,, J. Funct. Anal., 40 (1981), 1.  doi: 10.1016/0022-1236(81)90069-0.  Google Scholar

[6]

Henri Berestycki and Guillemette Chapuisat, A numerical study of a non local reaction-diffusion equation in population dynamics,, In preparation., (2013).   Google Scholar

[7]

Henri Berestycki and Guillemette Chapuisat, Propagation in a non homogeneous kpp equation arising in cancer modeling,, In preparation, (2013).   Google Scholar

[8]

Henri Berestycki, Françcois Hamel and Hiroshi Matano, Bistable traveling waves passing an obstacle,, Comm. Pures and Appl. Math., 62 (2009), 729.  doi: 10.1002/cpa.20275.  Google Scholar

[9]

Henri Berestycki and François Hamel, Fronts and invasions in general domains,, C. R. Math. Acad. Sci. Paris, 343 (2006), 711.  doi: 10.1016/j.crma.2006.09.036.  Google Scholar

[10]

Henri Berestycki and François Hamel, Generalized travelling waves for reaction-diffusion equations,, in, 446 (2007), 101.  doi: 10.1090/conm/446.  Google Scholar

[11]

Henri Berestycki and François Hamel, Generalized transition waves and their properties,, Comm. Pure Appl. Math., (2012).  doi: 10.1002/cpa.21389.  Google Scholar

[12]

Henri Berestycki and Pierre-Louis Lions, Some applications of the method of super and subsolutions,, in, 782 (1980), 16.  doi: 10.1007/BFb0090426.  Google Scholar

[13]

Henri Berestycki, Grégoire Nadin, Benoit Perthame and Lenya Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states,, Nonlinearity, 22 (2009), 2813.  doi: 10.1088/0951-7715/22/12/002.  Google Scholar

[14]

Henri Berestycki and Louis Nirenberg, Travelling fronts in cylinders,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 497.   Google Scholar

[15]

Henri Berestycki and Luca Rossi, On the principal eigenvalue of elliptic operators in $\mathbbR^N$ and applications,, J. Eur. Math. Soc. (JEMS), 8 (2006), 195.  doi: 10.4171/JEMS/47.  Google Scholar

[16]

Henri Berestycki and Luca Rossi, Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space,, Discrete Contin. Dyn. Syst., 21 (2008), 41.  doi: 10.3934/dcds.2008.21.41.  Google Scholar

[17]

Guillemette Chapuisat, Existence and nonexistence of curved front solution of a biological equation,, J. Differential Equations, 236 (2007), 237.  doi: 10.1016/j.jde.2007.01.021.  Google Scholar

[18]

Guillemette Chapuisat and Emmanuel Grenier, Existence and nonexistence of traveling wave solutions for a bistable reaction-diffusion equation in an infinite cylinder whose diameter is suddenly increased,, Comm. Partial Differential Equations, 30 (2005), 1805.  doi: 10.1080/03605300500300006.  Google Scholar

[19]

Guillemette Chapuisat, Emmanuel Grenier, Marie-Aim\'ee Dronne, Marc Hommel and Jean-Pierre Boissel, A global model of ischemic stroke with stress on spreading depression,, Progress in Biophysics and Molecular Biology, 97 (2008), 4.  doi: 10.1016/j.pbiomolbio.2007.10.004.  Google Scholar

[20]

Guillemette Chapuisat and Romain Joly, Asymptotic profiles for a traveling front solution of a biological equation,, Math. Models Methods Appl. Sci., 21 (2011), 2155.  doi: 10.1142/S0218202511005696.  Google Scholar

[21]

Jacques De Keyser, Geert Sulter and Paul G. Luiten, Clinical trials with neuroprotective drugs in acute ischaemic stroke: are we doing the right thing?,, Trends Neurosci., 22 (1999), 535.   Google Scholar

[22]

Laurent Desvillettes, Régis Ferrières and Céline Prevost, Infinite dimensional reaction-diffusion for population dynamics,, Prépublication du CMLA No. 2003-04, (2003), 2003.   Google Scholar

[23]

Lawrence C. Evans, "Partial Differential Equations,", 19 of Graduate Studies in Mathematics, 19 (1998).   Google Scholar

[24]

Laurent Excoffier, Matthieu Foll and Rémy J Petit, Genetic consequences of range expansions,, Annual Review of Ecology Evolution and Systematics, 40 (2009), 481.  doi: 10.1146/annurev.ecolsys.39.110707.173414.  Google Scholar

[25]

Ido Filin, Robert D. Holt and Michael Barfield, The relation of density regulation to habitat specialization, evolution of a species' range, and the dynamics of biological invasions,, Am. Nat., 172 (2008), 233.  doi: 10.1086/589459.  Google Scholar

[26]

Ali Gorji, Dieter Scheller, Heidrun Straub, Frank Tegtmeier, Rüdiger Köhling, Jörg Michael Höhling, Ingrid Tuxhorn, Alois Ebner, Peter Wolf, Hans Werner Panneck, Falk Oppel and Erwin Josef Speckmann, Spreading depression in human neocortical slices,, Brain Res., 906 (2001), 74.  doi: 10.1016/S0006-8993(01)02557-4.  Google Scholar

[27]

Oskar Hallatschek, Pascal Hersen, Sharad Ramanathan and David R. Nelson, Genetic drift at expanding frontiers promotes gene segregation,, Proceedings of the National Academy of Sciences, 104 (2007), 19926.  doi: 10.1073/pnas.0710150104.  Google Scholar

[28]

Oskar Hallatschek and David R. Nelson, Gene surfing in expanding populations,, Theoretical Population Biology, 73 (2008), 158.  doi: 10.1016/j.tpb.2007.08.008.  Google Scholar

[29]

Oskar Hallatschek and David R. Nelson, Life at the front of an expanding population,, Evolution, 64 (2010), 193.  doi: 10.1111/j.1558-5646.2009.00809.x.  Google Scholar

[30]

Robert D. Holt, Michael Barfield, Ido Filin and Samantha Forde, Predation and the evolutionary dynamics of species ranges,, Am. Nat., 178 (2011), 488.  doi: 10.1086/661909.  Google Scholar

[31]

Mark Kirkpatrick and Nicholas H. Barton, Evolution of a species' range,, Am. Nat., 150 (1997), 1.  doi: 10.1086/286054.  Google Scholar

[32]

Jean-François Mallordy and Jean-Michel Roquejoffre, A parabolic equation of the KPP type in higher dimensions,, SIAM J. Math. Anal., 26 (1995), 1.  doi: 10.1137/S0036141093246105.  Google Scholar

[33]

Hiroshi Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations,, Publ. Res. Inst. Math. Sci., 15 (1979), 401.  doi: 10.2977/prims/1195188180.  Google Scholar

[34]

Hiroshi Matano, Traveling waves in spatially inhomogeneous diffusive media - the non-periodic case,, Preprint, (2012).   Google Scholar

[35]

Avraham Mayevsky, Avi Doron, Tamar Manor, Sigal Meilin, Nili Zarchin and George E. Ouaknine, Cortical spreading depression recorded from the human brain using a multiparmetric monitoring system,, Brain Res., 740 (1996), 268.  doi: 10.1016/S0006-8993(96)00874-8.  Google Scholar

[36]

Maiken Nedergaard, Arthur J. Cooper and Steven A. Goldman, Gap junctions are required for the propagation of spreading depression,, J. Neurobiol., 28 (1995), 433.  doi: 10.1002/neu.480280404.  Google Scholar

[37]

James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik and Andrej Zlatoš, Existence and non-existence of fisher-kpp transition fronts,, Arch. Ration. Mech. Anal., (2012).  doi: 10.1007/s00205-011-0449-4.  Google Scholar

[38]

L. A. Peletier and James Serrin, Uniqueness of nonnegative solutions of semilinear equations in $R^n$,, J. Differential Equations, 61 (1986), 380.  doi: 10.1016/0022-0396(86)90112-9.  Google Scholar

[39]

Cristina Pocci, Ayman Moussa, Florence Hubert and Guillemette Chapuisat., Numerical study of the stopping of aura during migraine,, in, 30 (2009), 44.  doi: 10.1051/proc/2010005.  Google Scholar

[40]

Paul H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations,, Indiana Univ. Math. J., 23 (): 173.  doi: 10.1512/iumj.1973.23.23014.  Google Scholar

[41]

Lionel Roques, Jimmy Garnier, François Hamel and Etienne K. Klein, Allee effect promotes diversity in traveling waves of colonization,, Proceedings of the National Academy of Sciences, 109 (2012), 8828.  doi: 10.1073/pnas.1201695109.  Google Scholar

[42]

Laurent Schwartz, "Analyse Hilbertienne,", Collection Méthodes. Hermann, (1979).   Google Scholar

[43]

Bruno Shapiro, Osmotic forces and gap junctions in spreading depression: A computational model,, J. Comput. Neurosci., 10 (2001), 99.   Google Scholar

[44]

Wenxian Shen, Dynamical systems and traveling waves in almost periodic structures,, J. Differential Equations, 169 (2001), 493.  doi: 10.1006/jdeq.2000.3906.  Google Scholar

[45]

George G. Somjen, "Ions in the Brain: Normal Function, Seizures, and Stroke,", Oxford University Press, (2004).   Google Scholar

[46]

Anthony J. Strong, Martin Fabricius, Martyn G. Boutelle, Stuart J. Hibbins, Sarah E. Hopwood, Robina Jones, Mark C. Parkin and Martin Lauritzen, Spreading and synchronous depressions of cortical activity in acutely injured human brain,, Stroke, 33 (2002), 2738.  doi: 10.1161/01.STR.0000043073.69602.09.  Google Scholar

[47]

Henry C. Tuckwell, Predictions and properties of a model of potassium and calcium ion movements during spreading cortical depression,, Int. J. Neurosci., 10 (1980), 145.  doi: 10.3109/00207458009160493.  Google Scholar

[48]

José M. Vega, Travelling wavefronts of reaction-diffusion equations in cylindrical domains,, Comm. Partial Differential Equations, 18 (1993), 505.  doi: 10.1080/03605309308820939.  Google Scholar

[49]

Marcel O. Vlad, L. Luca Cavalli-Sforza and John Ross, Enhanced (hydrodynamic) transport induced by population growth in reactiondiffusion systems with application to population genetics,, Proceedings of the National Academy of Sciences, 101 (2004), 10249.   Google Scholar

[50]

Andrej Zlatoš, Generalized traveling waves in disordered media: Existence, uniqueness, and stability,, Preprint, (2012).   Google Scholar

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