【作者】 李琳；

【导师】 周风；

【作者基本信息】 华东师范大学 ， 应用数学， 2014， 硕士

【摘要】 许多生物体都具有一种特性,它们能够通过探测、整合和处理其生存环境中的各种内部和外部的信号来决定自身的移动.这种移动是生物体的一种本能反应,它使得生物体在生存上具有竞争优势.在这一类特性中,我们将生物体根据其生存环境中的化学信号主动进行趋向运动的现象称为趋化性(Chemotaxis).近几十年来,趋化性因其存在广泛,变化多样而在生物学领域和数学领域内成为了一个热门的研究问题.本文主要研究定义在一维区域(0,l)上的带生物体Logistic增长项的趋化性系统：其中其中u,u分别表示在区域[0,l]内生物体的密度及化学信号的浓度,系数D,χ,m,α,β均为大于零的常数.我们将重要的趋化系数χ作为分支参数：利用分支理论方法去研究趋化性系统(0.1)正的非常值稳态解的存在性问题：1.我们证明了：如果m,D满足q2π2/l2<m/D<(q+1)2π2/l2,q=0,1,2,,趋化性系统(0.1)存在正的非常值稳态解2.更进一步,利用Shi J-P和Wang X-F给出的单边分支理论,在特定假设下,我们还证明了：对于任意的χ>(Dπ2+ml2)(π2+al2)/βmπ2l2,趋化性系统(0.1)都存在正的非常值稳态解.更多还原

【Abstract】 Many organisms have an ability to navigate within a complex environment through the detection, integration and processing of a variety of internal and ex-ternal signals. This ability is an instinctive reaction, it makes the organism has an advantage in the competitive with others. In this kind of ability, The directed movement of organisms in response to chemical gradients, called chemotaxis, has attracted significant interest from biologist and mathematician due to its critical role in a wide range of biological phenomena.In this paper, we study the chemotaxis system with Logistic source in a one-dimensional domain: where u, v denotes the cell (or organism) density and the concentration of the chemical signal respectively, the coefficients D, χ, in, α, β are positive constants.We set the chemotactic coefficient χ as the bifurcation parameter, apply the bifurcation theory directly to the chemotaxis system to study the existence of nonconstant steady states:1. We proved if m,D satisfy<q2π2/l2<m/D<(q+1)2π2/l2,q=0,,2,…, then there exist positive nonconstant steady states of the chemotaxis system(0.2);2. Moreover, using the "unilateral" global bifurcation theorem which is proved by Shi J-P and Wang X-F, under some special situation, we proved that if χ>(Dπ2+ml2)(π2+al)/βmπ2l2, then there exist positive nonconstant steady states of the chemotaxis system(0.2).更多还原