PDE Models for Chemotactic Movements: Parabolic, Hyperbolic and Kinetic

Abstract

Modeling the movement of cells (bacteria, amoeba) is a long standing subject and Partial Differential equations have been used several times. The most classical and successful system was proposed by Patlak [55] and Keller & Segel [39] and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on bacteria like Escherichia Coli or amoeba like Dictyostelium Discoïdeum exhibiting pointwise concentrations.For human endothelial cells, several experiments show the formation of networks that can be interpreted as the initiation of angiogenesis. To recover such patterns a hydrodynamical model seems better adapted.The two systems can be unified by a kinetic approach that was proposed for Escherichia Coli, based on more precise experiments showing a movement by 'jump and tumble'. This nonlinear kinetic model is interesting by itself and the existence theory is not complete. It is also interesting from a scaling point of view; in a diffusion limit one recovers the Keller-Segel model and in a hydrodynamical limit one recovers the model proposed for human endothelial cells.We also mention the mathematical interest of a analyzing another degenerate parabolic system (exhibiting different properties) proposed to describe the angiogenesis phenomena i.e. the formation of capillary blood vessels. 1

Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ2

“…The literature on these models is huge and it is out of the scope of this paper to give exhaustive references. For further bibliography on the Patlak-Keller-Segel system and related models we refer the interested reader to the surveys [28,29,48].…”

Kinetic and hydrodynamic models of chemotactic aggregation

“…The second term is a stochastic term where R α (t) is a white noise satisfying R α (t) = 0 and R i,α (t)R j,β (t ′ ) = δ ij δ α,β δ(t−t ′ ) (where α = 1, ..., N refer to the particles and i = 1, ..., d to the space coordinates) and D * is a diffusion coefficient. The diffusion, that is observed for several biological organisms, can have different origins depending on the system under consideration [27]. In the case of small organisms moving in a fluid (matrigel), it can be due to the repeated impact of the molecules of the fluid on the particles like in ordinary Brownian motion for colloidal suspensions.…”

“…It should be stressed that the damped Euler equations (27)- (29) remain heuristic because their derivation is based on the Local Thermodynamic Equilibrium (L.T.E.) condition (24) which is not rigorously justified.…”

Numerical Simulations of Kinetic Models for Chemotaxis

“…Several models, depending on the level of description, have been developed mathematically for the collective motion of cells. We refer to [5,24,25] for a review on parabolic, hyperbolic and kinetic models and to [3,27,28] for traveling waves drivn by growth and chemotaxis. Among them the kinetic model introduced by Othmer, Dunbar and Alt [2,22], describes a population of bacteria in motion (for instance the E. Coli) in interactions with a chemoattractant concentration [10].…”