Free Access
Issue |
Math. Model. Nat. Phenom.
Volume 8, Number 3, 2013
Front Propagation
|
|
---|---|---|
Page(s) | 133 - 153 | |
DOI | https://doi.org/10.1051/mmnp/20138309 | |
Published online | 12 June 2013 |
- N. Tinbergen. Social Behavior in Animals, with Special Reference to Vertebrates. Chapman and Hall, London, 1990. [Google Scholar]
- C.S. Patlak. Random walk with persistence and external bias. Bull. Math. Biol.. 15 (1953), No.3, 311-338. [Google Scholar]
- J. Adler. Chemotaxis in bacteria. J. Science 12 (1966), No.153, 708-716. [Google Scholar]
- A. Okubo. Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin, 1980. [Google Scholar]
- J.D. Murray. Mathematical Biology. Springer, Berlin, 2005. [Google Scholar]
- E.F. Keller, L.A.J. Segel. Model for chemotaxis. J. Theor. Biol. 30 (1971), 225-234. [CrossRef] [PubMed] [Google Scholar]
- N. Kopell, L.N. Howard. Plane wave solutions to reaction-diffusion equations. Studies in Appl. Math. 42, (1973), 291-328. [Google Scholar]
- A.S. Isaev et al. Dynamics of Forest Insect Populations. Nauka, Novosibirsk, 1984 (in Russian). [Google Scholar]
- F.S. Berezovskaya, A.S. Isaev, G.P. Karev, R.G. Khlebopros. Role of taxis in forest insect dynamics. Doklady Biological Sci. 365 (1999), 148-151. [Google Scholar]
- H.A. Levine, B.D. Sleeman. A system of reaction diffusion equations arising in the theory of reinforced random walks, Siam J. Appl. Math. 57 (1997), No.3, 683–730. [Google Scholar]
- G.R. Ivanitsky, A.B. Medvinsky, M.A. Tsyganov. From disorder to order as applied to the movement of micro-organisms. Sov. Phys. Usp., 34 (1991), 289-316. [CrossRef] [Google Scholar]
- H.G. Othmer, A. Stevens. Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. Siam J. Appl. Math. 57 (1997), No.4, 1044–1081. [Google Scholar]
- T. Nagai, T. Ikeda. Travelling waves in a chemotactic model. J. Math. Biol. 30 (1991), No. 2, 169–184. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- E.O. Budrene, H.C. Berg. Complex patterns formed by motile cells of Escherichia coli. Nature 349 (1991), 630–633. [CrossRef] [PubMed] [Google Scholar]
- Yu.M. Svirezhev. Nonlinear Waves, Dissipative Structures and Catastrophes in Ecology. Nauka, Moscow, 1987 (in Russian). [Google Scholar]
- A.A. Samarskii, A.P. Mikhailov. Principles of Mathematical Modeling: Ideas, Methods, Examples. Taylor & Francis, London, 2002. [Google Scholar]
- V. Volpert, S.V. Petrovskii. Reaction–diffusion waves in biology. Physics of life Reviews, 6 (2009), 267-310. [CrossRef] [PubMed] [Google Scholar]
- R.A. Fisher. The wave of advance of advantageous genes. Ann Eugenics 7 (1937), 353-369. [Google Scholar]
- A. Kolmogoroff, I. Petrovsky, N. Piskunoff. Etude de lequation de la diffusion avec croissamce de la quantite de matiere et son application a un problem biologique. Moscow. Univ. Bull. Math., 1 (1937), 1-25. [Google Scholar]
- A.M. Turing. The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. London B, 237 (1952), 37-72. [CrossRef] [Google Scholar]
- R. FitzHugh, Mathematical Models of Excitation and Propagation in Nerve, in: Biological Engineering (ed. H. P. Schwan), McGraw-Hill, 1969. [Google Scholar]
- Yu.M. Romanovsky, N.S. Stepanova, D.S. Chernavsky. Mathematical Modeling in Biophysics. Nauka, Moscow, 1975 (in Russian). [Google Scholar]
- K. Lika, T.G. Hallam. Travelling wave solutions in non-linear reaction-advection equation. J. Math. Biol., 38 (1999), No. 4, 346-358. [Google Scholar]
- D.L. Feltham, M.A.J. Chaplain. Travelling waves in a model of species migration. Appl. Math. Lett. 13 (2000), No.7, 67–73. [Google Scholar]
- N. Shigesada, K. Kawasaki, Ei. Teramoto. Spatial segregation of interacting species. J. Theor. Biol., 79 (1979), 83-99. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- M. Ieda, M. Mimira, H. Ninomia. Diffusion, cross-diffusion and competitive interaction. J. Math. Biol., 53 (2006), 617-641. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- A. Monk, H.G. Othmer. Cyclic AMP oscillations in suspensions of Dictyostelium discoideum. Phil. Trans. R. Soc. London, 323 (1989), 185-224. [Google Scholar]
- A. Stevens. Trail following and aggregation of myxobacteria. J. Biol. Syst. 3 (1995), No.4, 1059–1068. [Google Scholar]
- R. Erban, H.G. Othmer. Taxis equations for amoeboid cells. J. Math. Biol. 54 (2007), No.6, 847–885. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- Yu.A. Kuznetsov, M.Ya. Antonovsky, V.N. Biktashev, E.A. Aponina. A cross-diffusion model of forest boundary dynamics. J. Math. Biol., 32 (1994), 219-232. [Google Scholar]
- J.A. Sherratt. Travelling wave solutions of a mathematical model for tumor capsulation. Siam J. Appl. Math. 60 (1999), No.2, 392–407. [Google Scholar]
- M.A. Tsyganov, V.N. Biktashev, J. Brindley, A.V. Holden, G.R. Ivanitsky. Waves in systems with cross-diffusion as a new class of nonlinear waves. Physics-Uspehi, 177 (2007), 3, 275–300. [Google Scholar]
- F.S. Berezovskaya, G.P. Karev. Travelling waves in cross-diffusion models of the dynamics of populations. Biofizika 45 (2000), No.4, 751–756. [PubMed] [Google Scholar]
- W.-M. Ni. Diffusion, cross-diffusion and their spike-layer steady states. Not. Am. Math. Soc. 45 (1998), No.1, 9–18. [Google Scholar]
- A.I. Volpert, V.A. Volpert, V.A. Volpert. Travelling Wave Solutions of Parabolic Systems. AMS, Providence, RI, 1994. [Google Scholar]
- D. Henry. Geometric theory of Semilinear Parabolic equations. Springer-Verlag, New York. 1981. [Google Scholar]
- F.S. Berezovskaya, G.P. Karev. Bifurcations of travelling waves in population models with taxis. Physics-Uspekhi, 42 (1999), No.9, 917-929. [CrossRef] [Google Scholar]
- F.S. Berezovskaya, A.S. Novozhilov, G.P. Karev. Families of traveling impulses and fronts in some models with cross-diffusion. Nonlinear Anal.: Real World Appl. (2008), 9: 1866–1881 [CrossRef] [MathSciNet] [Google Scholar]
- F. Berezovskaya, E. Camacho, S. Wirkus, G. Karev. Traveling wave solutions of FitzHugh model with cross-diffusion. Math. Biol.&Eng., 5 (2008), No.2, 239–260. [Google Scholar]
- F.S. Berezovskaya, A.S. Novozhilov, G.P. Karev. Traveling fronts, impulses and trains in some taxis models. Neural, Parallel and Scientific Computations 15 (2007), 561-570. [MathSciNet] [Google Scholar]
- F.S. Berezovskaya, G.P. Karev, R.G. Khlebopros. The models of insects-phytophagan populations with taxis: travelling waves and stability. Problems of Ecological Monitoring and modeling of ecosystems, XVII (2000), 17-33 (in Russian). [Google Scholar]
- A.D. Bazykin. Non-linear dynamics of interacting populations. World Scientific, Singapore, 1999. [Google Scholar]
- A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier. Qualitative Theory of Second-order Dynamic Systems. Wiley, New-York, 1973. [Google Scholar]
- V.I. Arnold. Geometrical methods in the theory of ODE. Springer-Verlag, 1983. [Google Scholar]
- J. Guckenheimer, P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1983. [Google Scholar]
- R.I. Bogdanov, Versal deformation of a singular point on the plane in the case of zero eigenvalues. Selecta Math. Soviet. 1 (1976), No.4, 373-388. [Google Scholar]
- F. Dumortier, R. Rossarie, J. Sotomayor. Bifurcations of planar vector fields. Lect. Notes in Mathematics, 1480 (1991), 1-164. [CrossRef] [Google Scholar]
- D. Turaev. Bifurcations of two-dimensional dynamical systems close to those possessing two separatrix loops. Mathematics survey, 40 (1985), No.6, 203-204. [Google Scholar]
- G. Dangelmayr, J. Guckenheimer, On a four parameter family of planar vector fields, Arch. Ration. Mech., 97 (1987), 321-352. [CrossRef] [Google Scholar]
- Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, Berlin, 2004. [Google Scholar]
- A. Khibnik, B. Krauskopf, C. Rousseau, Global study of a family of cubic Lienard equations. Nonlinearity, 11 (1998), 1505-1519. [Google Scholar]
- A.D. Bazykin, Yu.A. Kuznetsov, A.I. Khibnik. Portraits of Bifurcations. Znanie, Moscow, 1989 (in Russian). [Google Scholar]
- J.E. Marsden, M. McCracken. The Hopf Bifurcation and Its Applications. Springer-Verlag, New York, 1976. [Google Scholar]
- P.K. Maini, J.D. Murray, G.F. Oster. A mechanical model for biological pattern formation. A nonlinear bifurcation analysis. Lecture Notes - Mathematics 1151 (1985), Springer-Verlag, Heidelberg, Germany. [Google Scholar]
- J.H.E. Cartwright, E. Hermandez-Garcia, O. Piro. Burridge-Knopoff models as elastic excitable media. Phys. Rev. Lett., 79 (1997), No.3, 527-530. [Google Scholar]
- Ya.B. Zel’dovich, G. Barenblatt, V. Librovich, G. Makhviladze, The Mathematical Theory of Combustion and Explosions. Consultants Bureau, New York, 1985. [Google Scholar]
- R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1 (1961), 445-466. [Google Scholar]
- A.L. Hodgkin, A.F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (1952), 500-544. [CrossRef] [PubMed] [Google Scholar]
- L. Sherwood. Human Physiology: From Cells to Systems, 4th edition. Brooks and Cole Publishers, 2001. [Google Scholar]
- E.P. Volokitin, S.A. Treskov. Parameter portrait of FitzHugh system. Mathematical modeling, 6 (1994), No.12, 65-78 (in Russian). [Google Scholar]
- J. Nagumo, S. Arimoto, S. Yoshisawa. An active pulse transmission line simulating nerve axon. Proc. IRE, 50 (1962), 2061-2070. [Google Scholar]
- S. Hastings. On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations. Quart. J. Math. (Oxford) 27 (1976), 123-134. [CrossRef] [Google Scholar]
- J. Evans, N. Fenichel, J. Feroe. Double impulse solutions in nerve axon equations. SIAM J. Appl. Math., 42 (1982), 219-234. [Google Scholar]
- B. Deng. The existence of infinite travelling front and back waves in FitzHugh-Nagumo equation. SIAM J. Math. Anal., 22 (1991), 1631-1650. [CrossRef] [MathSciNet] [Google Scholar]
- Yu. Kuznetsov, A. Panfilov. Stochastic waves in the FitzHugh-Nagumo system. Preprint of Research Computer Center, Academy of Sci. USSR, 1981 (in Russian). [Google Scholar]
- B. Sandstede. Stability of N-fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equations. SIAM J. Math. Anal., 29 (1998), 183-207. [CrossRef] [MathSciNet] [Google Scholar]
- F. Sanchez-Garduno, P.K. Maini. Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations. J. Math. Biol., 35 (1997), 713-728. [Google Scholar]
- D.W. Verzi, M.B. Rheuben, S.M. Baer. Impact of time-dependent changes in spine density and spine change on the input-output properties of a dendric branch: a computational study. J. Neurophysiol. 93 (2005), 2073-2089. [CrossRef] [PubMed] [Google Scholar]
- E.F. Keller, L.A. Segel. Traveling bands of chemotactic bacteria—theoretical analysis. J. Theor. Biol. 30 (1971), No. 2, 235–248. [CrossRef] [PubMed] [Google Scholar]
- F.S. Berezovskaya, A.S. Novozhilov, G.P. Karev. Population models with singular equilibrium. Math. Biosci. 208 (2007), 270–299. [Google Scholar]
- S. Gueron, N. Liron. A model of herd grazing as a traveling wave, chemotaxis and stability. J. Math. Biol. 27 (1989), No.5, 595–608. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.