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 A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shengmao Fu ◽  
Ji Liu
Keyword(s):  
Boundary Value Problem ◽  
Source Term ◽  
Nonlinear Evolution ◽  
Initial Perturbation ◽  
Neumann Boundary ◽  
Chemical Signal ◽  
Linear Dynamics ◽  
Neumann Boundary Value Problem ◽  
Time Period ◽  
Signal Substance

This paper deals with a Neumann boundary value problem for a Keller-Segel model with a cubic source term in ad-dimensional box(d=1,2,3), which describes the movement of cells in response to the presence of a chemical signal substance. 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 order ln(1/δ). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model.

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

Patterns in a generalized volume-filling chemotaxis model with cell proliferation

2016 ◽  
Vol 15 (01) ◽  
pp. 83-106 ◽  
Author(s):  
Manjun Ma ◽  
Zhian Wang
Keyword(s):  
Neumann Boundary ◽  
Index Theory ◽  
Normal Vector ◽  
Interesting Phenomenon ◽  
Chemotaxis Model ◽  
Volume Filling ◽  
Outward Unit ◽  
Theoretical Results ◽  
Outward Unit Normal Vector ◽  
Iteration Technique

In this paper, we consider the following system [Formula: see text] which corresponds to the stationary system of a generalized volume-filling chemotaxis model with logistic source in a bounded domain in [Formula: see text] with zero Neumann boundary conditions. Here the parameters [Formula: see text] are positive and [Formula: see text], and [Formula: see text] denotes the outward unit normal vector of [Formula: see text]. With the priori positive lower- and upper-bound solutions derived by the Moser iteration technique and maximum principle, we apply the degree index theory in an annulus to show that if the chemotactic coefficient [Formula: see text] is suitably large, the system with [Formula: see text] admits pattern solutions under certain conditions. Numerical simulations of the pattern formation are shown to illustrate the theoretical results and predict the interesting phenomenon for further studies.

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

scholarly journals Nonlinear Instability for a Leslie-Gower Predator-Prey Model with Cross Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Lina Zhang ◽  
Shengmao Fu
Keyword(s):  
Finite Number ◽  
Nonlinear Evolution ◽  
Early Stage ◽  
Nonlinear Instability ◽  
Spatial And Temporal Patterns ◽  
Linear Dynamics ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model

A rigorous mathematical characterization for early-stage spatial and temporal patterns formation in a Leslie-Gower predator-prey model with cross diffusion is investigated. Given any general perturbation near an unstable constant equilibrium, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of the fastest growing modes.

scholarly journals POSITIVE SOLUTIONS OF A SECOND-ORDER NEUMANN BOUNDARY VALUE PROBLEM WITH A PARAMETER

2012 ◽  
Vol 86 (2) ◽  
pp. 244-253 ◽  
Author(s):  
YANG-WEN ZHANG ◽  
HONG-XU LI
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Existence And Uniqueness Theorem ◽  
Neumann Boundary Value Problem ◽  
Image Position ◽  
Continuous Dependence Of Solutions

AbstractIn this paper, we consider the Neumann boundary value problem with a parameter λ∈(0,∞): By using fixed point theorems in a cone, we obtain some existence, multiplicity and nonexistence results for positive solutions in terms of different values of λ. We also prove an existence and uniqueness theorem and show the continuous dependence of solutions on the parameter λ.

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.

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