On The Steady State Of Chemotaxis System With Logistic Source Via Bifurcation Theory

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).