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Phase Transition And Hexagonal Patterns In Rich Stimulant Diffusion-Chemotaxis Model

An important component in studying mathematical models in many biochemical systems, such as those found in developmental biology, is phase transition. The purpose of this work is to analyze the phase transition property of a diffusion-chemotaxis model with proliferation source, as a macroscopic model of behavior of mobile species. Along the way, we will discuss that the system exhibits very rich pattern-forming behavior. In particular, a portion of the present work is devoted to the proof of existence of hexagonal patterns as a result of instability of two Fourier modes. It is also shown that they are either saddle points or attracting nodes. Moreover, they belong to an attractor which consists of finite number of steady-state solutions and their connecting heteroclinic orbits. The structure of this attractor will be precisely determined as well.