Modulation equations and parabolic limits of reaction random-walk systems

Math. Models Methods Appl. Sci.

Lutscher

2004

Self Cite

In the Langevin or Ornstein–Uhlenbeck approach to diffusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satisfies a linear partial differential equation of the general form of a transport equation which is hyperbolic with respect to the space variable but parabolic with respect to the velocity variable, the Klein–Kramers or simply Kramers equation. This modeling approach allows for a more detailed description of individual movement and orientation dependent interaction than the frequently used reaction diffusion framework.For the Kramers equation, moments are computed, the infinite system of moment equations is closed at several levels, and telegraph and diffusion equations are derived as approximations. Then nonlinearities are introduced such that the semi-linear reaction Kramers equation describes particles which move and interact on the same time-scale. Also for these nonlinear problems a moment approach is feasible and yields nonlinear damped wave equations as limiting cases.We apply the moment method to the Kramers equation for chemotactic movement and obtain the classical Patlak–Keller–Segel model. We discuss similarities between chemotactic movement of bacteria and gravitational movement of physical particles.

“…with initial conditions Z(0, x) = 0, Z t (0, x) = 0. We now use Theorem 3.2 in Hillen and Müller 32 , which states that…”

Section: Approximation Propertymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Langevin or Kramers Approach to Biological Modeling

Hadeler

Math. Models Methods Appl. Sci.

Lutscher

2004

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“…Nonlinear versions of the one-dimensional telegraph process, which also include birth, death, and interactions of particles, have been studied by Dunbar and Othmer [9], Dunbar [8], Holmes [27], Hadeler [18,19,20], Hillen [23,24,25,26], Müller and Hillen [34], and Schneider and Müller [42]. Asymptotic estimates for classical solutions have been derived and the upper semicontinuity of the attractor has been proven [42].…”

Section: ∂ ∂T P(x V T) + V · ∇P(x V T) = −λP(x V T) + λmentioning

confidence: 99%

The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes

Othmer

2000

SIAM J. Appl. Math.

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323

110

Abstract. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak-Keller-Segel-Alt model for chemotaxis.

Existence of Weak Solutions for a Hyperbolic Model of Chemosensitive Movement

Journal of Mathematical Analysis and Applications

Rohde

Lutscher

2001

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A hyperbolic model for chemotaxis and chemosensitive movement in one space dimension is considered. In contrast to parabolic models for chemotaxis the hyperbolic model allows us to take the dependence of the particle speed on external stimuli explicitly into account. This qualitatively covers recent experiments on chemotaxis in which it has been shown that particles adapt their speed to the surrounding environment. The model presented here consists of two hyperbolic differential equations of first order coupled with an elliptic equation. We assume Ž . that the speed depends on the external stimulus only and not on its gradients . In that case solutions with steep gradients are expected which have the interpretation 1 Supported by the DFG-Project ANUME. 2 Supported by the DFG Project DANSE and by the EU-TMR research network for Ž . Hyperbolic Conservation Laws Project ERBFMRXCT960033 .3 Supported by the DFG-Project DANSE. HILLEN, ROHDE, AND LUTSCHER 174 of moving swarms. A notion of weak solutions for this hyperbolic chemotaxis model is presented and the global existence of weak solutions is shown. The proof relies on the vanishing viscosity method; i.e., we obtain the weak solution as the limit of classical solutions of an associated parabolically regularized problem for vanishing viscosity parameter. Numerical simulations demonstrate phenomena like swarming behaviour and formation of steep gradients. ᮊ