Stationary Solutions of a Volume-Filling Chemotaxis Model with Logistic Growth and Their Stability

2012 ◽  
Vol 72 (3) ◽  
pp. 740-766 ◽  
Author(s):  
Manjun Ma ◽  
Chunhua Ou ◽  
Zhi-An Wang
Keyword(s):  
Stationary Solutions ◽  
Logistic Growth ◽  
Chemotaxis Model ◽  
Volume Filling

Global Bifurcation of Stationary Solutions for a Volume-Filling Chemotaxis Model with Logistic Growth

2020 ◽  
Vol 30 (13) ◽  
pp. 2050182
Author(s):  
Yaying Dong ◽  
Shanbing Li
Keyword(s):  
Bifurcation Theory ◽  
Global Bifurcation ◽  
Neumann Boundary ◽  
Stationary Solutions ◽  
Logistic Growth ◽  
Fredholm Operators ◽  
Chemotaxis Model ◽  
Volume Filling ◽  
Constant Solution ◽  
Global Bifurcation Theory

In this paper, we show how the global bifurcation theory for nonlinear Fredholm operators (Theorem 4.3 of [Shi & Wang, 2009]) and for compact operators (Theorem 1.3 of [Rabinowitz, 1971]) can be used in the study of the nonconstant stationary solutions for a volume-filling chemotaxis model with logistic growth under Neumann boundary conditions. Our results show that infinitely many local branches of nonconstant solutions bifurcate from the positive constant solution [Formula: see text] at [Formula: see text]. Moreover, for each [Formula: see text], we prove that each [Formula: see text] can be extended into a global curve, and the projection of the bifurcation curve [Formula: see text] onto the [Formula: see text]-axis contains [Formula: see text].

scholarly journals Pattern formation for a volume-filling chemotaxis model with logistic growth

2017 ◽  
Vol 448 (2) ◽  
pp. 885-907 ◽  
Author(s):  
Yazhou Han ◽  
Zhongfang Li ◽  
Jicheng Tao ◽  
Manjun Ma
Keyword(s):  
Pattern Formation ◽  
Logistic Growth ◽  
Chemotaxis Model ◽  
Volume Filling

scholarly journals Nonlinear Instability for a Volume-Filling Chemotaxis Model with Logistic Growth

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Haiyan Gao ◽  
Shengmao Fu
Keyword(s):  
Nonlinear Evolution ◽  
Initial Perturbation ◽  
Neumann Boundary ◽  
Logistic Growth ◽  
Nonlinear Instability ◽  
Chemotaxis Model ◽  
Linear Dynamics ◽  
Volume Filling ◽  
Neumann Boundary Value Problem ◽  
Time Period

This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in ad-dimensional boxTd=(0,π)d  (d=1,2,3). It is proved that given any general perturbation of magnitudeδ, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the orderln⁡(1/δ). Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.

scholarly journals Wavefront invasion for a volume-filling chemotaxis model with logistic growth

2016 ◽  
Vol 71 (2) ◽  
pp. 471-478 ◽  
Author(s):  
Yazhou Han ◽  
Zhongfang Li ◽  
Shutao Zhang ◽  
Manjun Ma
Keyword(s):  
Logistic Growth ◽  
Chemotaxis Model ◽  
Volume Filling

Long-time behaviour of solutions to a chemotaxis model with volume-filling effect

2006 ◽  
Vol 136 (2) ◽  
pp. 431-444 ◽  
Author(s):  
Dariusz Wrzosek
Keyword(s):  
Stationary Solutions ◽  
Upper Bounds ◽  
Typical Case ◽  
Time Behaviour ◽  
Quasilinear Parabolic System ◽  
Lower And Upper Bounds ◽  
Chemotaxis Model ◽  
Volume Filling ◽  
Long Time Behaviour ◽  
Long Time

The Lyapunov functional is constructed for a quasilinear parabolic system which models chemotaxis and takes into account a volume-filling effect. For some typical case it is proved that the ω-limit set of any trajectory consists of regular stationary solutions. Some lower and upper bounds on the stationary solutions are found. For a given range of parameters there are stationary solutions which are inhomogeneous in space.

scholarly journals Recent results for the logarithmic Keller-Segel-Fisher/KPP System

2019 ◽  
Vol 38 (7) ◽  
pp. 37-48
Author(s):  
Yanni Zeng ◽  
Kun Zhao
Keyword(s):  
Ground State ◽  
Decay Rates ◽  
Optimal Time ◽  
Logistic Growth ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Chemotaxis Model ◽  
Time Decay Rates ◽  
Long Time ◽  
Logarithmic Sensitivity

We consider a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth. It is a 2 by 2 system describing the interaction of cells and a chemical signal. We study Cauchy problem with finite initial data, i.e., without the commonly used smallness assumption on  initial perturbations around a constant ground state. We survey a sequence of recent results by the authors on  the existence of global-in-time solution,  long-time behavior, vanishing coefficient limit and optimal time decay rates of the solution.

A 1-d quasilinear nonuniform parabolic chemotaxis model with volume-filling effect

2013 ◽  
Vol 83 (1-2) ◽  
pp. 101-125 ◽  
Author(s):  
Yanyan Zhang ◽  
Songmu Zheng
Keyword(s):  
Chemotaxis Model ◽  
Volume Filling

scholarly journals Stationary solutions for some shadow system of the Keller-Segel model with logistic growth

2015 ◽  
Vol 8 (5) ◽  
pp. 1023-1034 ◽  
Author(s):  
Tohru Tsujikawa ◽  
◽  
Kousuke Kuto ◽  
Yasuhito Miyamoto ◽  
Hirofumi Izuhara ◽  
...  
Keyword(s):  
Stationary Solutions ◽  
Logistic Growth ◽  
Shadow System

scholarly journals Traveling waves in a chemotaxis model with logistic growth

2019 ◽  
Vol 24 (12) ◽  
pp. 6465-6480
Author(s):  
Tong Li ◽  
◽  
Jeungeun Park
Keyword(s):  
Traveling Waves ◽  
Logistic Growth ◽  
Chemotaxis Model
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