Aggregation, blowup, and collapse: the ABC's of taxis in reinforced random walks

A Stevens, HG Othmer - SIAM Journal on Applied Mathematics, 1997 - SIAM
In many biological systems, movement of an organism occurs in response to a diffusible or
otherwise transported signal, and in its simplest form this can be modeled by diffusion …

Aggregating pattern dynamics in a chemotaxis model including growth

M Mimura, T Tsujikawa - Physica A: Statistical Mechanics and its …, 1996 - Elsevier
A population model including diffusion, chemotaxis and growth is studied. Assuming that the
diffusion rate and the chemotactic rate are both very small compared with the growth rate …

Non-local concepts and models in biology

CT Lee, MF Hoopes, J Diehl, W Gilliland… - Journal of theoretical …, 2001 - Elsevier
In this paper, we consider local and non-local spatially explicit mathematical models for
biological phenomena. We show that, when rate differences between fast and slow local …

Cellular pattern formation during Dictyostelium aggregation

T Höfer, JA Sherratt, PK Maini - Physica D: Nonlinear Phenomena, 1995 - Elsevier
The development of multicellularity in the life cycle of Dictyostelium discoideum provides a
paradigm model system for biological pattern formation. Previously, mathematical models …

Singular limit of a reaction-diffusion equation with a spatially inhomogeneous reaction term

KI Nakamura, H Matano, D Hilhorst… - Journal of statistical …, 1999 - Springer
We study reaction-diffusion equations with a spatially inhomogeneous reaction term. If the
coefficient of these reaction term is much larger than the diffusion coefficient, a sharp …

Pattern formation (I): the Keller–Segel model

Y Guo, HJ Hwang - Journal of Differential Equations, 2010 - Elsevier
We investigate nonlinear dynamics near an unstable constant equilibrium in the classical
Keller–Segel model. Given any general perturbation of magnitude δ, we prove that its …

Singular limit of a chemotaxis-growth model

A Bonami, D Hilhorst, E Logak, M Mimura - 2001 - projecteuclid.org
We consider a reaction-diffusion-advection system which is a model for chemotaxis with
growth. An appropriate singular limit of this system yields a free-boundary problem where …

[PDF][PDF] Singular limit analysis of planar equilibrium solutions to a chemotaxis model equation with growth

T Tsujikawa - Methods and Applications of Analysis, 1996 - intlpress.com
We consider the stability of planar localized equilibrium solutions of a chemotaxis model
equation with growth term, which describes aggregation of biological individuals. Assuming …

Asymptotic properties of reaction-diffusion systems modeling chemotaxis

MA Herrero - Applied and Industrial Mathematics, Venice—2, 1998 …, 2000 - Springer
We discuss in the sequel on the aggregation properties of some systems of nonlinear
parabolic equations which have been extensively used as models for chemotaxis. In …

Chemotaxis-growth under the influence of lateral inhibition in a three-component reaction–diffusion system

S Kawaguchi - Nonlinearity, 2011 - iopscience.iop.org
In this study, we consider the effects of chemotaxis and lateral inhibition on an activator in a
three-component reaction–diffusion system. Simulation results show that spot, planar and …