Pattern Formation And Asymptotic Profiles Of Solutions To A Chemotaxis Model With Volume-filling Effect

Chemotaxis usually refers to the process of orientational movement of microorganisms in response to the stimulation of chemicals in the environment.In the 1970 s,E.F.Keller and L.A.Segel established a kind of partial differential equation model to describe the evolution of this process.This equation not only describes chemotactic phenomenon well,but also has complex cross-diffusion structure and rich dynamical properties,so it has become the hot spot of partial differential equations in recent years.In this paper,we consider the patten formation and asymptotic limit of a chemotactic model with volume filling effect and mixed boundary conditions.In this paper,the existence and nonexistence of nontrivial solutions of the model are discussed by using global bifurcation theory in different parameter ranges.Then,based on a series of uniform estimates of solutions,the asymptotic profiles of the solution is further analyzed when different parameters are changed.The results show that the solution of the chemotactic model is a spike at the boundary where nutrient comes when appropriate volume filling effects are considered.Thus,they explain the aggregation phenomenon of bacteria on the boundary of the vessel or on the surface of water droplets appearing in biology.