Pattern Formation Of Chemotaxis Models With Nonlinear Diffusion In Mathematical Biology

The biological chemotaxis model describes the phenomenon of aggregation of organisms(such as cells or bacteria)which is common in ecology.This phenomenon corresponds to the nonconstant steady state solution of the model,I.E.Model formation.In this paper,we consider the existence and morphology of nonconstant steady state solutions for a class of biological chemotactic models with nonlinear diffusion.First,we obtain the existence of nontrivial solutions by using the local bifurcation and global bifurcation theories;Then,the monotonicity of the steady-state solution is obtained by using the continuity method;Finally,based on Sturm-Liouville Oscillation Theorem and Helly compactness theorem,the asymptotic behavior of the steady state solution is obtained when the chemotaxis coefficient tends to infinity.The results show that the steady-state solution is a peak or a transition layer(corresponding to different sensitivity functions),so it describes the aggregation of organisms.