Classical solutions and pattern formation for a volume filling chemotaxis model

2007 ◽  
Vol 17 (3) ◽  
pp. 037108 ◽  
Author(s):  
Zhian Wang ◽  
Thomas Hillen
Keyword(s):  
Pattern Formation ◽  
Classical Solutions ◽  
Chemotaxis Model ◽  
Volume Filling

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

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].

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 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.

Traveling wavefronts for a reaction-diffusion-chemotaxis model with volume-filling effect

2017 ◽  
Vol 32 (1) ◽  
pp. 108-116
Author(s):  
Man-jun Ma ◽  
Hui Li ◽  
Mei-yan Gao ◽  
Ji-cheng Tao ◽  
Ya-zhou Han
Keyword(s):  
Reaction Diffusion ◽  
Chemotaxis Model ◽  
Volume Filling

A dual-gradient chemotaxis system modeling the spontaneous aggregation of microglia in Alzheimer’s disease

2018 ◽  
Vol 16 (03) ◽  
pp. 307-338
Author(s):  
Hai-Yang Jin ◽  
Zhi-An Wang
Keyword(s):  
Alzheimer’S Disease ◽  
Alzheimer's Disease ◽  
Pattern Formation ◽  
System Modeling ◽  
Neumann Boundary ◽  
Classical Solutions ◽  
Open Problems ◽  
Global Classical Solutions ◽  
Spontaneous Aggregation ◽  
Dual Gradient

In this paper, we consider the following dual-gradient chemotaxis model [Formula: see text] with [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text] and [Formula: see text]. The model was proposed to interpret the spontaneous aggregation of microglia in Alzheimer’s disease due to the interaction of attractive and repulsive chemicals released by the microglia. It has been shown in the literature that, when [Formula: see text], the solution of the model with homogeneous Neumann boundary conditions either blows up or asymptotically decays to a constant in multi-dimensions depending on the sign of [Formula: see text], which means there is no pattern formation. In this paper, we shall show as [Formula: see text], the uniformly-in-time bounded global classical solutions exist in multi-dimensions and hence pattern formation can develop. This is significantly different from the results for the case [Formula: see text]. We perform the numerical simulations to illustrate the various patterns generated by the model, verify our analytical results and predict some unsolved questions. Biological applications of our results are discussed and open problems are presented.

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