Traveling waves in a chemotactic model

Nagai, Toshitaka; Ikeda, Tsutomu

doi:10.1007/bf00160334

Cited by 91 publications

(98 citation statements)

References 11 publications

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“…Thus for δ > 1 and 0 β k( √ δ −1) 2 /c 2 the origin is a node for system (21) which implies that in the initial system (20) there exist a family of bounded orbits which tent to (0, 0) for y → ∞ or y → −∞. This family corresponds to a family of traveling wave solution of the system (1) where u-profile is an impulse and v-profile is a free-boundary front.…”

Section: Impulse-impulse Solutionsmentioning

confidence: 96%

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Families of traveling impulses and fronts in some models with cross-diffusion

Berezovskaya¹,

Novozhilov²,

Karev³

2008

Nonlinear Analysis: Real World Applications

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An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller-Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of the initial PDE. Special attention is paid to families of traveling wave solutions. Conditions for existence of front-impulse, impulse-front, and front-front travelling wave solutions are formulated. In particular, the simplest mathematical model is presented that has an impulse-impulse solution; we also show that a non-isolated singular point in the ordinary differential equation (ODE) wave system implies existence of free-boundary fronts. The results can be used for construction and analysis of different mathematical models describing systems with chemotaxis.

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Section: Impulse-impulse Solutionsmentioning

confidence: 96%

“…8b it can be seen that bacteria (u(x, t)) seek an optimal environment: the bacteria avoid low concentrations and move preferentially toward higher concentrations of some critical substrate (v(x, t)). The stability of the traveling solutions found was studied analytically in [19,21].…”

Section: The Phase Plane Analysis Of the Keller-segel Modelmentioning

confidence: 99%

Families of traveling impulses and fronts in some models with cross-diffusion

Berezovskaya¹,

Novozhilov²,

Karev³

2008

Nonlinear Analysis: Real World Applications

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“…Note that such kind of problems were studied to model the movement of travelling bands of Escherichia coli [12][13][14], amoeba clustering [28,29], insect invasion in a forest [33,41], species migration [24], tumor encapsulation and tissue invasion [31], for a survey see, e.g. [32] and references therein.…”

Section: Families Of Travelling Wave Solutionsmentioning

confidence: 99%

“…Self-similar solutions, in particular those of "travelling wave" type, are of particular interest. Such solutions correspond to spatially heterogeneous distributions of various types that propagate with a definite velocity (see, for instance, [5][6][7][13][14][15]). After the fundamental works of Fisher [18] , Kolmogorov, Petrovskii, and Piskunov [19] and Turing [20] the "growth-diffusion" equations have become major tools in various mathematical problems of biology and biophysics [4,5,10,11,[21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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

Berezovskaya

Karev

2013

Math. Model. Nat. Phenom.

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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 "growthtaxis -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.

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“…the overview in [37] and references cited therein) and also their stability has been investigated ( [19], [27]). In spite of the rich literature concerned with travelling wave solutions (for such solutions to related systems see also [24], [25], [20], or [11], [26]), little is known about existence of solutions for more general initial data (see below).…”

Section: Introductionmentioning

confidence: 99%