Existence Of The Traveling Wave Solutions For A Chemotaxis Equations

In biology,chemotaxis refers to the characteristic movement or orientation of an organism or cell along a chemical concentration gradient either toward or away from the chemical stimulus for finding food or avoidance of noxious, substances to large scale aggregation for survival. Models for chemotaxis have been successfully applied to bacteria, slime molds, skin pigmentation patterns and many other biological systems. The first depicts chemotaxis of the mathematical model is from Keller and Segel in 1970 [1], set up, which we call the classic chemotaxis model. The model was first used to study the mucus of a mold, called Acanthamoeba Polymerization conditions. Experiment in Ader observed: at one end closed and full of nutrients in the test tube; bacteria were injected into the tube. Under the effect of chemokine mechanism, these bacteria groups through continuous movement, and slowly created a wave, and this wave can be stability in the nutrient solution in the mobile. This phenomenon that the disturbance propagate in the form of the limited wave can be described by the traveling wave solution" u(t,x) = u(x-ct)". Therefore, it is important practical significance to study traveling wave solutions of chemotaxis model.A.Boy simplify the classical model through a series of more general assumption ;in general we consider the concentration of bacteria and the impact of the external environment. Here only consider the impact of the concentration of bacteria. Many mathematicians and biologists has already done a lot of discussion about chemotaxis model, such as in [1]-[8],the model has been studied deeply and have been many perfect results.In this paper we will discuss the existence of traveling wave solution of the follow model with f(b) = b^k+b ,D = 0. Where b(x, t) is bacteria concentration. s(x, t) denotes the concentration of chemoatractant and k₀,β₀ are positive constants, Generally, bacteria and nutrient has its own diffusion process. But here we only consider the diffusion process of bacteria and the impact on the nutrition only consider the expendable of bacteria. In this paper we will study the following elliptic-elliptic system which is a kind of simplified Keller-Segel type. We will use some usefull mathmatical methods such as Phase plane, We proved that: (â…°) supposer +α<0The sufficient conditions of the existence of trajectories connecting (S_∞,0), (0,0) is:(â…±) when k + 1 > 0,1 + r≤0,for any B₀ there exist c = (?) such that ODE system has a trajectories connecting(0, B₀)to (S_∞, 0),namely the model has traveling wave solution if r +α= 0.(â…²)(1)k = -1,r +α= 0,c> c^* = (?), then the model has traveling wave solution.(2)If r +α< 0,α< l,r≥-1,k =-1,there exist a unique c =(?)k₀β₀s_∞^r+α such that the model has traveling wave solution.Mansour in [10]study the follow model with only one equation, he get the existence of traveling wave solutions with the constantχ.In this paper,we use phase plane to get above results.