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Existence of traveling wave solutions for a model of tumor invasion. (English) Zbl 1301.34075

34E15 Singular perturbations for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34E17 Canard solutions to ordinary differential equations
35C07 Traveling wave solutions
35L67 Shocks and singularities for hyperbolic equations
92C17 Cell movement (chemotaxis, etc.)
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[1] E. Barbera, C. Currò, and G. Valenti, A hyperbolic model for the effects of urbanization on air pollution, Appl. Math. Model., 34 (2010), pp. 2192–2202. · Zbl 1193.76027
[2] M. Beck, A. Doelman, and T.J. Kaper, A geometric construction of traveling waves in a bioremediation model, J. Nonlinear Sci., 16 (2006), pp. 329–349. · Zbl 1100.92060
[3] E. Benoit, J.L. Callot, F. Diener, and M. Diener, Chasse au canards, Collect. Math., 31 (1981), pp. 37–119. · Zbl 0529.34046
[4] N.F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, New York, 1986. · Zbl 0602.92001
[5] M. Brøns, M. Krupa, and M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon, Fields Inst. Commun., 49 (2006), pp. 39–63. · Zbl 1228.34063
[6] A. Doelman, P. van Heijster, and T. Kaper, Pulse dynamics in a three-component system: Existence analysis, J. Dynam. Differential Equations, 21 (2009), pp. 73–115. · Zbl 1173.35068
[7] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), pp. 53–98.
[8] R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), pp. 353–369. · JFM 63.1111.04
[9] G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), pp. 347–386. · Zbl 1311.34133
[10] H. Hoshino, Traveling wave analysis for a mathematical model of malignant tumor invasion, Analysis (Munich), 31 (2011), pp. 237–248. · Zbl 1237.34033
[11] C.K.R.T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Lecture Notes in Math. 1609, Springer, Berlin, Heidelberg, 1995, pp. 44–118. · Zbl 0840.58040
[12] D.W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, 3rd ed., Oxford University Press, Oxford, UK, 1999. · Zbl 0955.34001
[13] T.J. Kaper, An introduction to geometric methods and dynamical systems theory for singular perturbation problems, in Analyzing Multiscale Phenomena Using Singular Perturbation Methods, Proc. Sympos. Appl. Math. 56, AMS, Providence, RI, 1999, pp. 85–131.
[14] E.F. Keller and L.A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), pp. 225–234. · Zbl 1170.92307
[15] E.F. Keller and L.A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), pp. 235–248. · Zbl 1170.92308
[16] A. Kolmogoroff, I. Petrovsky, and N. Piscounoff, Étude do l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Moscow Univ. Math. Bull., 1 (1937), pp. 1–25.
[17] M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), pp. 286–314. · Zbl 1002.34046
[18] K.A. Landman, G.J. Pettet, and D.F. Newgreen, Chemotactic cellular migration: Smooth and discontinuous traveling wave solutions, SIAM J. Appl. Math., 63 (2003), pp. 1666–1681. · Zbl 1044.34006
[19] K.A. Landman, M.J. Simpson, and G.J. Pettet, Tactically-driven nonmonotone travelling waves, Phys. D, 237 (2008), pp. 678–691. · Zbl 1145.35403
[20] K.A. Landman, M.J. Simpson, J.L. Slater, and D.F. Newgreen, Diffusive and chemotactic cellular migration: Smooth and discontinuous traveling wave solutions, SIAM J. Appl. Math., 65 (2005), pp. 1420–1442. · Zbl 1077.35117
[21] D.A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math., 34 (1978), pp. 93–103. · Zbl 0373.35036
[22] B.P. Marchant and J. Norbury, Discontinuous travelling wave solutions for certain hyperbolic systems, IMA J. Appl. Math., 67 (2002), pp. 201–224. · Zbl 1076.35071
[23] B.P. Marchant, J. Norbury, and H.M. Byrne, Biphasic behaviour in malignant invasion, Math. Med. Biol., 23 (2006), pp. 173–196. · Zbl 1098.92040
[24] B.P. Marchant, J. Norbury, and A.J. Perumpanani, Traveling shock waves arising in a model of malignant invasion, SIAM J. Appl. Math., 60 (2000), pp. 463–476. · Zbl 0944.34021
[25] B.P. Marchant, J. Norbury, and J.A. Sherratt, Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), pp. 1653–1671. · Zbl 0985.92012
[26] H.P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math., 28 (1975), pp. 323–331. · Zbl 0316.35053
[27] J.D. Meiss, Differential Dynamical Systems, Math. Model. Comput. 14, SIAM, Philadelphia, 2007.
[28] J.D. Murray, Mathematical Biology I: An Introduction, 3rd ed., Springer, New York, 2002. · Zbl 1006.92001
[29] A.J. Perumpanani, J.A. Sherratt, J. Norbury, and H.M. Byrne, A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion, Phys. D, 126 (1999), pp. 145–159. · Zbl 1001.92523
[30] G.J. Pettet, Modelling Wound Healing Angiogenesis and Other Chemotactically Driven Growth Processes, Ph.D. thesis, University of Newcastle, Newcastle, Australia, 1996.
[31] G.J. Pettet, D.L.S McElwain, and J. Norbury, Lotka-Volterra equations with chemotaxis: Walls, barriers and travelling waves, IMA J. Math. Appl. Med., 17 (2000), pp. 395–413. · Zbl 0969.92020
[32] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer-Verlag, New York, 1994. · Zbl 0807.35002
[33] P. Szmolyan and M. Wechselberger, Canards in \(\mathbb{R}^3\), J. Differential Equations, 177 (2001), pp. 419–453. · Zbl 1007.34057
[34] P. Szmolyan and M. Wechselberger, Relaxation oscillations in \(\mathbb{R}^3\), J. Differential Equations, 200 (2004), pp. 69–104. · Zbl 1058.34055
[35] W. van Saarloos, Front propagation into unstable states, Phys. Rep., 386 (2003), pp. 29–222. · Zbl 1042.74029
[36] M. Wechselberger, Singularly Perturbed Folds and Canards in \(\mathbb{R}^3\), Ph.D. thesis, Vienna University of Technology, Vienna, Austria, 1998.
[37] M. Wechselberger, Existence and bifurcation of canards in \(\mathbb{R}^3\) in the case of a folded node, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 101–139. · Zbl 1090.34047
[38] M. Wechselberger, À propos de canards, Trans. Amer. Math. Soc., 304 (2012), pp. 3289–3309. · Zbl 1244.34080
[39] M. Wechselberger and G.J. Pettet, Folds, canards and shocks in advection-reaction-diffusion models, Nonlinearity, 23 (2010), pp. 1949–1969. · Zbl 1213.34073
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