This note studies the Monge–Ampère Keller–Segel equation in a periodic domain Td(d ≥ 2), a fully nonlinear modification of the Keller–Segel equation where the Monge–Ampère equation det(I + ∇2v) = u + 1 substitutes for the usual Poisson equation ∆v = u. The existence of global weak solutions is obtained for this modified equation. Moreover, we prove the regularity in L∞  0, T ;L∞ ∩W 1,1+γ(Td) ...

The cellular Potts model (CPM) has been used for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. We derive a continuous limit of a discrete one-dimensional CPM with the chemotactic interactions between cells in t...

Chemotaxis plays a crucial role in a variety of processes in biology and ecology. In many instances, processes involving chemical attraction take place in fluids. One of the most studied PDE models of chemotaxis is given by Keller-Segel equation, which describes a population density of bacteria or mold which attract chemically to substance they secrete. Solution of Keller-Segel equation can exh...

We present PDE (partial differential equation) model hierarchies for the chemotactically driven motion of biological cells. Starting from stochastic differential models we derive a kinetic formulation of cell motion coupled to diffusion equations for the chemoattractants. Also we derive a fluid dynamic (macroscopic) Keller-Segel type chemotaxis model by scaling limit procedures. We review rigor...

In this paper, we introduce a random particle blob method for the Keller-Segel equation (with dimension d ≥ 2) and establish a rigorous convergence analysis.

A population-level model of bacterial chemotaxis is derived from a simple bacteriallevel model of behavior. This model, to be contrasted with the Keller-Segel equations, exhibits behavior we refer to as the “volcano effect”: steady-state bacterial aggregation forming a ring of higher density some distance away from an optimal environment. The model is derived, as in Erban and Othmer [7], from a...

We will show how the critical mass classical Keller-Segel system and the critical displacement convex fast-diffusion equation in two dimensions are related. On one hand, the critical fast diffusion entropy functional helps to show global existence around equilibrium states of the critical mass Keller-Segel system. On the other hand, the critical fast diffusion flow allows to show functional ine...

The dynamics of an interface connecting a stationary stripe pattern with a homogeneous state is studied. The conventional approach which describes this interface, Newell–Whitehead–Segel amplitude equation, does not account for the rich dynamics exhibited by these interfaces. By amending this amplitude equation with a nonresonate term, we can describe this interface and its dynamics in a unified...

We introduce three new examples of kinetic models for chemotaxis, where a kinetic equation for the phase-space density is coupled to a parabolic or elliptic equation for the chemo-attractant, in two or three dimensions. We prove that these models have global-in-time existence and rigorously converge, in the drift-diffusion limit to the Keller–Segel model. Furthermore, the cell density is unifor...