The Stability Of Constant Steady States Of Some Chemotaxis System

A characteristic feature of living systems is that they sense the environment in which they reside and respond to it.The response frequently involves movement toward or away from an external stimulus,and such a response is called a taxis.The purposes of taxis range from movement toward food and avoidance of noxious substances to large-scale aggregation for the purpose of survival.Many different types of taxis are known,including aerotaxis,chemotaxis,geotaxis,haptotaxis,and others. And chemotaxis being more noticed in biological and mathematical since it is crucial in biological processes where are numerous examples.Chemotaxis is the phenomenon whereby living organisms respond to chemical substance by motion and rearrangement(taxis).They may move toward the higher concentration of the chemical substance(positive taxis),or away from it(negative taxis),they may aggregate,or they may disperse.A model that leads to aggregation of certain types of bacteria has been set up by Keller and Segel[1,2],called keller-Segel model.K-S model opened the windows of the chemotaxis systems research.From then on,lots of mathematicians and biologists devoted themselves to this intersting field. There are however many instances in which the organism modifies its environment in a strictly local manner,and there is little or no transport of the modifying substance. Examples include the myxobacteria,which produce slime over which their cohorts can easily move.A novel approach to modeling the behaviour of myxobacteria is due to Othmer and Stevens[2].We call it O-S model.We will consider the O-S model.Their qualitative properties have been studied by Yang Yin,Chen Hua,Liu Wei an,Levine and Sleeman.Rascle studies a special case of the O-S systems and appears to be the first to consider local existence and uniqueness of solution.And Yang Yin,Chen Hua,Liu Wei An consider a development of these indeas to address the fundamental problems of existence and uniqueness of solutions for O-S models.Also they have shown us that the solutions of the problem are extremely sensetive of their initial data.On the stability Analytically,Nagal consider a special case of O-S models,and give a stability criterion.And Avner,Friedman,J.Ignacio proved that the stationary solutions of a kind of O-S models are asymtotically stable. In this paper,we will disscussed the stability of the positive constant steady states to two kind of O-S models,we will give two stability criterious and the results are illustrate in application to a number of partial differential equation models arising in biology.