Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-22T15:59:17.428Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation Approach to Analysis of Travelling Waves in Some Taxis–Cross-Diffusion Models

Published online by Cambridge University Press:  12 June 2013

F. Berezovskaya  and
G. Karev
F. Berezovskaya
Affiliation:
Howard University, Washington, DC 20059, USA
G. Karev*
Affiliation:
National Center for Biotechnology Information, Bethesda, MD 20894, USA
*
⋆ Corresponding author. E-mail: fberezovskaya@howard.edu
Get access

Abstract

An overview of recently obtained authors’ results on traveling wave solutions of some classes of PDEs is presented. The main aim is to describe all possible travelling wave solutions of the equations. The analysis was conducted using the methods of qualitative and bifurcation analysis in order to study the phase-parameter space of the corresponding wave systems of ODEs. In the first part we analyze the wave dynamic modes of populations described by the “growth - taxis - diffusion" polynomial models. It is shown that “suitable" nonlinear taxis can affect the wave front sets and generate non-monotone waves, such as trains and pulses, which represent the exact solutions of the model system. Parametric critical points whose neighborhood displays the full spectrum of possible model wave regimes are identified; the wave mode systematization is given in the form of bifurcation diagrams. In the second part we study a modified version of the FitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, besides giving rise to the typical fast travelling wave solution exhibited in the original “diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where “normal" impulse propagation is possible. In the third part we describe all possible wave solutions for a class of PDEs with cross-diffusion, which fall in a general class of the classical Keller-Segel models describing chemotaxis. Conditions for existence of front-impulse, impulse-front, and front-front traveling wave solutions are formulated. In particular, we show that a non-isolated singular point in the ODE wave system implies existence of free-boundary fronts.

Keywords

traveling wave solutionswave systembifurcation diagramtaxiscross-diffusionKeller-Segel modelFitzHugh-Nagumo equations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 133 - 153
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

N. Tinbergen. Social Behavior in Animals, with Special Reference to Vertebrates. Chapman and Hall, London, 1990.
Patlak, C.S.. Random walk with persistence and external bias. Bull. Math. Biol.. 15 (1953), No.3, 311-338. Google Scholar
Adler, J.. Chemotaxis in bacteria. J. Science 12 (1966), No.153, 708-716. CrossRefGoogle Scholar
A. Okubo. Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin, 1980.
J.D. Murray. Mathematical Biology. Springer, Berlin, 2005.
Keller, E.F., Segel, L.A.J.. Model for chemotaxis. J. Theor. Biol. 30 (1971), 225-234. CrossRefGoogle ScholarPubMed
Kopell, N., Howard, L.N.. Plane wave solutions to reaction-diffusion equations. Studies in Appl. Math. 42, (1973), 291-328. CrossRefGoogle Scholar
A.S. Isaev et al. Dynamics of Forest Insect Populations. Nauka, Novosibirsk, 1984 (in Russian).
Berezovskaya, F.S., Isaev, A.S., Karev, G.P., Khlebopros, R.G.. Role of taxis in forest insect dynamics. Doklady Biological Sci. 365 (1999), 148-151. Google Scholar
Levine, H.A., Sleeman, B.D.. 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
Ivanitsky, G.R., Medvinsky, A.B., Tsyganov, M.A.. From disorder to order as applied to the movement of micro-organisms. Sov. Phys. Usp., 34 (1991), 289-316. CrossRefGoogle Scholar
Othmer, H.G., Stevens, A.. 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
Nagai, T., Ikeda, T.. Travelling waves in a chemotactic model. J. Math. Biol. 30 (1991), No. 2, 169–184. CrossRefGoogle Scholar
Budrene, E.O., Berg, H.C.. Complex patterns formed by motile cells of Escherichia coli. Nature 349 (1991), 630–633. CrossRefGoogle ScholarPubMed
Yu.M. Svirezhev. Nonlinear Waves, Dissipative Structures and Catastrophes in Ecology. Nauka, Moscow, 1987 (in Russian).
A.A. Samarskii, A.P. Mikhailov. Principles of Mathematical Modeling: Ideas, Methods, Examples. Taylor & Francis, London, 2002.
Volpert, V., Petrovskii, S.V.. Reaction–diffusion waves in biology. Physics of life Reviews, 6 (2009), 267-310. CrossRefGoogle Scholar
Fisher, R.A.. The wave of advance of advantageous genes. Ann Eugenics 7 (1937), 353-369. CrossRefGoogle Scholar
Kolmogoroff, A., Petrovsky, I., Piskunoff, N.. 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
Turing, A.M.. The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. London B, 237 (1952), 37-72. CrossRefGoogle Scholar
R. FitzHugh, Mathematical Models of Excitation and Propagation in Nerve, in: Biological Engineering (ed. H. P. Schwan), McGraw-Hill, 1969.
Yu.M. Romanovsky, N.S. Stepanova, D.S. Chernavsky. Mathematical Modeling in Biophysics. Nauka, Moscow, 1975 (in Russian).
Lika, K., Hallam, T.G.. Travelling wave solutions in non-linear reaction-advection equation. J. Math. Biol., 38 (1999), No. 4, 346-358. CrossRefGoogle Scholar
Feltham, D.L., Chaplain, M.A.J.. Travelling waves in a model of species migration. Appl. Math. Lett. 13 (2000), No.7, 67–73. CrossRefGoogle Scholar
Shigesada, N., Kawasaki, K., Teramoto, Ei.. Spatial segregation of interacting species. J. Theor. Biol., 79 (1979), 83-99. CrossRefGoogle ScholarPubMed
Ieda, M., Mimira, M., Ninomia, H.. Diffusion, cross-diffusion and competitive interaction. J. Math. Biol., 53 (2006), 617-641. CrossRefGoogle Scholar
Monk, A., Othmer, H.G.. Cyclic AMP oscillations in suspensions of Dictyostelium discoideum. Phil. Trans. R. Soc. London, 323 (1989), 185-224. CrossRefGoogle ScholarPubMed
Stevens, A.. Trail following and aggregation of myxobacteria. J. Biol. Syst. 3 (1995), No.4, 1059–1068. CrossRefGoogle Scholar
Erban, R., Othmer, H.G.. Taxis equations for amoeboid cells. J. Math. Biol. 54 (2007), No.6, 847–885. CrossRefGoogle Scholar
Kuznetsov, Yu.A., Antonovsky, M.Ya., Biktashev, V.N., Aponina, E.A.. A cross-diffusion model of forest boundary dynamics. J. Math. Biol., 32 (1994), 219-232. CrossRefGoogle Scholar
Sherratt, J.A.. Travelling wave solutions of a mathematical model for tumor capsulation. Siam J. Appl. Math. 60 (1999), No.2, 392–407. CrossRefGoogle Scholar
Tsyganov, M.A., Biktashev, V.N., Brindley, J., Holden, A.V., Ivanitsky, G.R.. Waves in systems with cross-diffusion as a new class of nonlinear waves. Physics-Uspehi, 177 (2007), 3, 275–300. Google Scholar
Berezovskaya, F.S., Karev, G.P.. Travelling waves in cross-diffusion models of the dynamics of populations. Biofizika 45 (2000), No.4, 751–756. Google Scholar
Ni, W.-M.. 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.
D. Henry. Geometric theory of Semilinear Parabolic equations. Springer-Verlag, New York. 1981.
Berezovskaya, F.S., Karev, G.P.. Bifurcations of travelling waves in population models with taxis. Physics-Uspekhi, 42 (1999), No.9, 917-929. CrossRefGoogle Scholar
Berezovskaya, F.S., Novozhilov, A.S., Karev, G.P.. Families of traveling impulses and fronts in some models with cross-diffusion. Nonlinear Anal.: Real World Appl. (2008), 9: 1866–1881 CrossRefGoogle Scholar
Berezovskaya, F., Camacho, E., Wirkus, S., Karev, G.. Traveling wave solutions of FitzHugh model with cross-diffusion. Math. Biol.&Eng., 5 (2008), No.2, 239–260. Google Scholar
Berezovskaya, F.S., Novozhilov, A.S., Karev, G.P.. Traveling fronts, impulses and trains in some taxis models. Neural, Parallel and Scientific Computations 15 (2007), 561-570. Google Scholar
Berezovskaya, F.S., Karev, G.P., Khlebopros, R.G.. 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.
A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier. Qualitative Theory of Second-order Dynamic Systems. Wiley, New-York, 1973.
V.I. Arnold. Geometrical methods in the theory of ODE. Springer-Verlag, 1983.
J. Guckenheimer, P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1983.
Bogdanov, R.I., 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
Dumortier, F., Rossarie, R., Sotomayor, J.. Bifurcations of planar vector fields. Lect. Notes in Mathematics, 1480 (1991), 1-164. CrossRefGoogle Scholar
Turaev, D.. Bifurcations of two-dimensional dynamical systems close to those possessing two separatrix loops. Mathematics survey, 40 (1985), No.6, 203-204. CrossRefGoogle Scholar
Dangelmayr, G., Guckenheimer, J., On a four parameter family of planar vector fields, Arch. Ration. Mech., 97 (1987), 321-352. CrossRefGoogle Scholar
Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, Berlin, 2004.
Khibnik, A., Krauskopf, B., Rousseau, C., Global study of a family of cubic Lienard equations. Nonlinearity, 11 (1998), 1505-1519.
A.D. Bazykin, Yu.A. Kuznetsov, A.I. Khibnik. Portraits of Bifurcations. Znanie, Moscow, 1989 (in Russian).
J.E. Marsden, M. McCracken. The Hopf Bifurcation and Its Applications. Springer-Verlag, New York, 1976.
Maini, P.K., Murray, J.D., Oster, G.F.. A mechanical model for biological pattern formation. A nonlinear bifurcation analysis. Lecture Notes - Mathematics 1151 (1985), Springer-Verlag, Heidelberg, Germany. CrossRefGoogle Scholar
Cartwright, J.H.E., Hermandez-Garcia, E., Piro, O.. Burridge-Knopoff models as elastic excitable media. Phys. Rev. Lett., 79 (1997), No.3, 527-530. CrossRefGoogle Scholar
Ya.B. Zel’dovich, G. Barenblatt, V. Librovich, G. Makhviladze, The Mathematical Theory of Combustion and Explosions. Consultants Bureau, New York, 1985.
FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1 (1961), 445-466. CrossRefGoogle ScholarPubMed
Hodgkin, A.L., Huxley, A.F.. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 (1952), 500-544. CrossRefGoogle Scholar
L. Sherwood. Human Physiology: From Cells to Systems, 4th edition. Brooks and Cole Publishers, 2001.
Volokitin, E.P., Treskov, S.A.. Parameter portrait of FitzHugh system. Mathematical modeling, 6 (1994), No.12, 65-78 (in Russian). Google Scholar
Nagumo, J., Arimoto, S., Yoshisawa, S.. An active pulse transmission line simulating nerve axon. Proc. IRE, 50 (1962), 2061-2070. CrossRefGoogle Scholar
Hastings, S.. On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations. Quart. J. Math. (Oxford) 27 (1976), 123-134. CrossRefGoogle Scholar
Evans, J., Fenichel, N., Feroe, J.. Double impulse solutions in nerve axon equations. SIAM J. Appl. Math., 42 (1982), 219-234. CrossRefGoogle Scholar
Deng, B.. The existence of infinite travelling front and back waves in FitzHugh-Nagumo equation. SIAM J. Math. Anal., 22 (1991), 1631-1650. CrossRefGoogle Scholar
Yu. Kuznetsov, A. Panfilov. Stochastic waves in the FitzHugh-Nagumo system. Preprint of Research Computer Center, Academy of Sci. USSR, 1981 (in Russian).
Sandstede, B.. 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. CrossRefGoogle Scholar
Sanchez-Garduno, F., Maini, P.K.. Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations. J. Math. Biol., 35 (1997), 713-728. Google Scholar
Verzi, D.W., Rheuben, M.B., Baer, S.M.. 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. CrossRefGoogle Scholar
Keller, E.F., Segel, L.A.. Traveling bands of chemotactic bacteria—theoretical analysis. J. Theor. Biol. 30 (1971), No. 2, 235–248. CrossRefGoogle ScholarPubMed
Berezovskaya, F.S., Novozhilov, A.S., Karev, G.P.. Population models with singular equilibrium. Math. Biosci. 208 (2007), 270–299. CrossRefGoogle ScholarPubMed
Gueron, S., Liron, N.. A model of herd grazing as a traveling wave, chemotaxis and stability. J. Math. Biol. 27 (1989), No.5, 595–608.CrossRefGoogle Scholar