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Twodimensional KellerSegel model: Optimal critical mass and qualitative properties of the solutions
 J. DIFF. EQNS
, 2006
"... The KellerSegel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative driftdiffusion equation for the cell density coupled to an elliptic equation for the chemoattractant concentration. It is k ..."
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Cited by 128 (15 self)
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The KellerSegel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative driftdiffusion equation for the cell density coupled to an elliptic equation for the chemoattractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blowup occurs. In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blowsup in finite time in the whole euclidean space. Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative. For a solution with subcritical mass, this allows us to give for large times an “intermediate asymptotics ” description of the vanishing. In selfsimilar coordinates, we actually prove a convergence result to a limiting selfsimilar solution which is not a simple reflect of the diffusion.
A user’s guide to PDE models for chemotaxis
 MATHEMATICAL BIOLOGY
, 2009
"... Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying ..."
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Cited by 116 (7 self)
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Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The KellerSegel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225– 234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “autoaggregation”, has led to its prominence as a mechanism for selforganisation of biological systems. This phenomenon has been shown to lead to finitetime blowup under certain formulations of the model, and a large body of work has been devoted to determining when blowup occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.
The diffusion limit of transport equations derived from velocity jump processes
 Siam J. Appl. Math
, 2000
"... 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 c ..."
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Cited by 93 (25 self)
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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 anisotropicdiffusion 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.
Kinetic Models for Chemotaxis and their DriftDiffusion Limits
, 2003
"... Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemoattractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a driftdiusion model is pro ..."
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Cited by 81 (13 self)
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Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemoattractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a driftdiusion model is proven.
Infinite time aggregation for the critical PatlakKellerSegel model in R²
 COMM. PURE APPL. MATH
"... We analyze the twodimensional parabolicelliptic PatlakKellerSegel model in the whole Euclidean space R². Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of ”freeenergy solutions”, namely weak solutions w ..."
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Cited by 80 (13 self)
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We analyze the twodimensional parabolicelliptic PatlakKellerSegel model in the whole Euclidean space R². Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of ”freeenergy solutions”, namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of freeenergy solutions with initial data as before for the critical mass 8pi/χ. Actually, we prove that solutions blowup as a delta dirac at the center of mass when t→ ∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2
Global Solutions of some Chemotaxis and Angiogenesis Systems in high space dimensions
, 2003
"... We consider two simple conservative systems of parabolicelliptic and parabolicdegenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the L d 2 norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global ( ..."
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Cited by 65 (7 self)
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We consider two simple conservative systems of parabolicelliptic and parabolicdegenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the L d 2 norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global (in time) weak solution that stays in all the Lp spaces with max{1; d 2 − 1} ≤ p < ∞. This result is already known for the parabolicelliptic system of chemotaxis, moreover blowup can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolicdegenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equiintegrable in L¹) exist even for large initial data. But breakdown of regularity or propagation of smoothness is an open problem.
The Diffusion Limit of Transport Equations II: Chemotaxis Equations
"... this paper we use the diusionlimit expansion of transport equations developed earlier [23] to study the limiting equation under avariety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modication of the turning rate, the movement speed or t ..."
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Cited by 59 (10 self)
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this paper we use the diusionlimit expansion of transport equations developed earlier [23] to study the limiting equation under avariety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modication of the turning rate, the movement speed or the preferred direction of movement. Depending on the strength of the bias, it leads to anisotropic diusion, to a drift term in the ux or to both, in the parabolic limit. We show that the classical chemotaxis equation  whichwe call the PatlakKellerSegelAlt (PKSA) equation  only arises when the bias is suciently small. Using this general framework, we derive phenomenological models for chemotaxis of agellated bacteria, of slime molds and of myxobacteria. We also show that certain results derived earlier for onedimensional motion can easily be generalized to two or threedimensional motion as well. ## ############# The linear transport equation @ @t p(x; v; t)+v ##p(x; v; t)=#p(x; v; t)+ # # T(v;v # )p(x; v # ;t)dv # ; (1.1) in which p(x; v; t) represents the density of particles at spatial position x # IR # moving with velocity v # V # IR # at time t # 0, arises when the movement of biological organisms is modeled byavelocityjump process [38]. Here the turning rate may be space or velocitydependent, but in other contexts it may also depend on internal variables that evolve in space and time, in which case (1.1) must be generalized. The turning kernel or turn angle distribution T (v; v # ) gives the probabilityofavelocity jump from v # to v if a jump occurs: in general it may also be spacedependent or depend on internal variables. In the present formulation we assume that the `decision' to turn as reected in is not coupled to th...
From individual to collective behavior in bacterial chemotaxis
 SIAM J Appl Math
, 2004
"... Abstract Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individ ..."
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Cited by 46 (7 self)
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Abstract Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time and/or spacedependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time we obtain a single secondorder hyperbolic equation for the populations density, and using a parabolic scaling we obtain the classical chemotaxis equation, wherein the chemotactic sensitivity is now a known function of parameters of the internal dynamics. Numerical simulations show that the solutions of the macroscopic equations agree very well with the results of Monte Carlo simulations of individual movement. 1. Introduction. The