Abstract. A (Patlak-) Keller-Segel model in two space dimensions with an additional crossdiffusion term in the equation for the chemical signal is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical substance. This allows one to prove, for arbitrarily small cross diffusion, the global existence of ...

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 Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399-415, 1970;...

In this paper, we give a new construction of $u_0\in B^\sigma_{p,\infty}$ such that the corresponding solution to hyperbolic Keller-Segel model starting from $u_0$ is discontinuous at $t = 0$ in metric $B^\sigma_{p,\infty}(\R^d)$ with $d\geq1$ and $1\leq p\leq\infty$, which implies ill-posedness for equation $B^\sigma_{p,\infty}$. Our result generalizes recent work \cite{Zhang01} (J. Differ. Eq...

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as ...

We show that the critical mass Mc = 8π of bacterial populations in two dimensions in the chemotactic problem is the counterpart of the critical temperature Tc = GMm/4kB of self-gravitating Brownian particles in two-dimensional gravity. We obtain these critical values by using the Virial theorem or by considering stationary solutions of the Keller-Segel model and Smoluchowski-Poisson system. We ...

We derive a continuous limit of a two-dimensional stochastic cellular Potts model (CPM) describing cells moving in a medium and reacting to each other through direct contact, cell-cell adhesion, and long-range chemotaxis. All coefficients of the general macroscopic model in the form of a Fokker-Planck equation describing evolution of the cell probability density function are derived from parame...

A theory, which should have widespread application, is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio Rayleigh-B~nard boxes. For case of presentation the theory is developed using model equations. Wavenumber selection, the shape of patterns, stability and the time dependence resulting from long wavelength instabilities are disc...

In this work, we investigate the dynamics of a non-local model describing spontaneous cell polarisation. It consists in a drift-diffusion equation set in the half-space, with the coupling involving the trace value on the boundary. We characterize the following behaviours in the one-dimensional case: solutions are global if the mass is below the critical mass and they blow-up in finite time abov...

A pathway-based mean-field theory (PBMFT) that incorporated the most recent quantitatively measured signaling pathway was recently proposed for the E. coli chemotaxis in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101]. In this paper, we formally derive a new kinetic system of PBMFT under the assumption that the methylation level is locally concentrated, whose turning o...