Existence Of Traveling Waves Of A Classical Chemotaxis Model

In this article we study the existence of planar traveling wave solutions for a class of Keller-Segel models . The Keller-Segel system is the classical model for Chemotaxis, i.e., the active motion of a population according to a chemical signal. A reactive control of the chemical allows the population to feedback into the taxis mechanism. Such a phenomenon is commonly faced in microbiology. The system for a population density u and the chemical v readsHere d denotes the diffusion coefficient, possibly density dependent, and H the sensitivity function. The feedback reaction is described by g ( u ,v ). As the first equation is of convergence form, the total mass of u is conserved meaning that the population is in a non-proliferating stage.Shortly after the proposition of the model Keller and Segel searched for traveling wave solutions of the system allowing to describe the experimental observed traveling bands of certain bacteria . For a simplified model, in particular, neglecting diffusion of the chemical, i.e.,μ= 0, they could show the existence of traveling wave solutions for the sensitivity H ( v ) = log( v). Subsequently, Rosen generalized the feedback g and Keller, Odell the sensitivity to H = ? v?p. But still the class of feedback reactions was very restricted and no diffusion of the chemical could be included by their methods. Much later, Nagai and Ikeda could consider a diffusing chemical together with a simplified feedback g = ? u.Recently, Horstmann and Stevens proposed a constructive approach to relate several sensitivity and feedback functions, which do allow for traveling wave solutions.Schwetlick and Hartmut proved a singular sensitivity function is necessary for the existence of bounded traveling waves and gave conditions for the existence and nonexistence of traveling waves with g =γv ?Γuαvβ.Moreover they introduced growth and annihilation effects in the equation of the cell density u , traveling waves are possible non-singular, physically reasonable sensitivity functions, e.g., H ( v )= v or H ( v ) =βv+ v. Then Li Yong discussed the existence of traveling wave solutions with ( )g u ,v = ?κ0vαu whenμ= 0 orμ≠0.In this paper we study the existence of traveling wave solutions of the Keller-Segel model, a general model of chemotaxis,where the species do not reproduce. In the case of logarithmic sensitivity and a diffusing chemical,we show that various functions modeling the reactive feedback on the chemo-attractant do allow for traveling waves . We can get pulse solutions in the density of the bacteria. Meanwhile we can also get the existence of traveling wave solutions on some conditions when neglecting the diffusion of the chemical. Comparing with each other, we find that the probability of the existence of traveling wave solutions becomes smaller with the same parameters when neglecting the diffusion of the chemical.