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Kinetic models for chemotaxis: Hydrodynamic limits and the back-of-the-wave problem
- Journal of Mathematical Biology
, 2005
"... Abstract We study kinetic models for amoebal chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. After having chosen an appropriate scaling of time and space, we carry out a formal hyperbolic limit, constru ..."
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Cited by 35 (3 self)
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Abstract We study kinetic models for amoebal chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. After having chosen an appropriate scaling of time and space, we carry out a formal hyperbolic limit, constructing a diffusion term as a higher order correction. The resulting macroscopic equations are coupled with a simple model for production and propagation of the chemoattractant. To test our model, we perform numerical experiments. 1 Introduction Chemotaxis can be defined as the biased migration of cells in the direction of a chemical gradient. Many examples for chemotaxis are found in nature: it plays an important role in embryonal morphogenesis, the immune response of the body, wound healing and metastasis. Unicellular organisms such as bacteria or amoeba use chemotaxis to avoid harmful substances or to form cell aggregates. There has been a large number of attempts to describe chemotaxis mathematically. The earliest model for chemotaxis has been developed by Patlak [16] and Keller and Segel [11]. There, the evolution of the cell density ae(t; x) at position x 2 IR
Spatial dynamics of the diffusive logistic equation with a sedentary compartment,
- Canadian Appl. Math. Quart,
, 2002
"... ABSTRACT. We study an extension of the diffusive logistic equation or Fisher's equation for a situation where one part of the population is sedentary and reproducing, and the other part migrating and subject to mortality. We show that this system is essentially equivalent to a semi-linear wave ..."
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Cited by 11 (4 self)
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ABSTRACT. We study an extension of the diffusive logistic equation or Fisher's equation for a situation where one part of the population is sedentary and reproducing, and the other part migrating and subject to mortality. We show that this system is essentially equivalent to a semi-linear wave equation with viscous damping. With respect to persistence in bounded domains with absorbing boundary conditions and with respect to the rate of spread of a locally introduced population, there are two distinct scenarios, depending on the choice of parameters. In the first scenario the population can survive in sufficiently large domains and the linearization at the leading edge of the front yields a unique candidate for the spread rate. In the second scenario the population can survive in arbitrarily small domains and there are two possible candidates for the spread rate. Analysis shows it is the larger candidate which gives the correct spread rate. The phenomenon of spread is also investigated using travelling wave theory. Here the minimal speed of possible travelling front solutions equals the previously calculated spread rate. The results are explained in biological terms.
The Langevin or Kramers approach . . .
, 2004
"... 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 respec ..."
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Cited by 1 (0 self)
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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 non-linear problems a moment approach is feasible and yields non-linear 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 pyhsical particles.
Coupled Dynamics and Quiescent Phases
- In: Math Everywhere- Deterministic and Stochastic Modelling in Biomedicine, Economy and Industry. Proc. Conf. Milano 2005
, 2006
"... Summary. We analyze diffusively coupled dynamical systems, which are con-structed from two dynamical systems in continuous time by switching between the two dynamics. If one of the vector fields is zero we call it a quiescent phase. We present a detailed analysis of coupled systems and of systems wi ..."
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Summary. We analyze diffusively coupled dynamical systems, which are con-structed from two dynamical systems in continuous time by switching between the two dynamics. If one of the vector fields is zero we call it a quiescent phase. We present a detailed analysis of coupled systems and of systems with quiescent phase and we prove results on scaling limits, singular perturbations, attractors, gradient fields, stability of stationary points and amplitudes of periodic orbits. In particular we show that introducing a quiescent phase is always stabilizing. 1
Received (Day Month Year)
, 2004
"... Communicated by (xxxxxxxxxx) In the Langevin or Ornstein-Uhlenbeck approach to diusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satis es a linear partial dierential equation of the general form of a transport equation which is ..."
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Communicated by (xxxxxxxxxx) In the Langevin or Ornstein-Uhlenbeck approach to diusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satis es a linear partial dierential 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 diusion framework. For the Kramers equation, moments are computed, the innite system of moment equations is closed at several levels, and telegraph and diusion 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 non-linear problems a moment approach is feasible and yields non-linear damped wave equations as limiting cases. We apply the moment method to the Kramers equation for chemotactic move-ment and obtain the classical Patlak-Keller-Segel model. We discuss similarities between chemotactic movement of bacteria and gravitational movement of pyhsical particles.
On the L²-moment closure of transport equations: the Cattaneo approximation
- DISCRETE CONTIN DYN. SYST. SER. B
, 2004
"... We consider the moment-closure approach to transport equations which arise in Mathematical Biology. We show that the negative L²-norm is an entropy in the sense of thermodynamics, and it satisfies an H-theorem. With an L²-norm minimization procedure we formally close the moment hierarchy for the f ..."
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We consider the moment-closure approach to transport equations which arise in Mathematical Biology. We show that the negative L²-norm is an entropy in the sense of thermodynamics, and it satisfies an H-theorem. With an L²-norm minimization procedure we formally close the moment hierarchy for the first two moments. The closure leads to semilinear Cattaneo systems, which are closely related to damped wave equations. In the linear case we derive estimates for the accuracy of this moment approximation. The method is used to study reaction-transport models and transport models for chemosensitive movement. With this method also order one perturbations of the turning kernel can be treated- in extension of an earlier theory on the parabolic limit of transport equations (Hillen and Othmer 2000). Moreover, this closure procedure allows us to derive appropriate boundary conditions for the Cattaneo approximation. Finally, we illustrate that the Cattaneo system is the gradient flow of a weighted Dirichlet integral and we show simulations. The moment closure for higher order moments and for general transport models will be studied in a second paper. Introduction. Flagellated
hadeler-hillen-lutscher˙19-04-04 Mathematical Models and Methods in Applied Sciences c ○ World Scientific Publishing Company The Langevin or Kramers Approach to Biological Modeling
, 2004
"... 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 ..."
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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 non-linear problems a moment approach is feasible and yields non-linear 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
UNIQUENESS OF TRAVELLING FRONTS FOR BISTABLE NONLINEAR TRANSPORT EQUATIONS
, 2005
"... We consider a nonlinear transport equation as a hyperbolic generalisation of the well-known reaction diffusion equation. The model is based on earlier work of K.P. Hadeler attempting to include run-and-tumble motion into the mathematical description. Previously, we proved the existence of strictly ..."
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We consider a nonlinear transport equation as a hyperbolic generalisation of the well-known reaction diffusion equation. The model is based on earlier work of K.P. Hadeler attempting to include run-and-tumble motion into the mathematical description. Previously, we proved the existence of strictly monotone travelling fronts for the three main types of the nonlinearity: the positive source term, the combustion law, and the bistable case. Here we revisit the latter case, showing that bistable fronts are unique up to translation.
Additional services for European Journal of Applied Mathematics:
, 2012
"... Anisotropic diffusion in oriented environments can lead to singularity formation ..."
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Anisotropic diffusion in oriented environments can lead to singularity formation
