Transition of patterns in the cell-chemotaxis system with proliferation source

Phase transition analysis is a fundamental component in understanding pattern-forming behavior of many biochemical systems, such as those found in developmental biology. The purpose of this work is to analyze the phase transition property of a diffusion–chemotaxis model with proliferation source, a macroscopic model of mobile species aggregation. Along the way, we will show that the system exhibits a very rich pattern-forming behavior, resulting in a competition between hexagonal, roll, and rectangular patterns. In particular, we show existence of regular hexagonal patterns in a confined spatial geometry where only two modes become unstable. It is also shown that they are either saddle points or attracting nodes. Moreover, we show that the competition happens inside a local attractor which consists of a finite number of stationary solutions and their connecting heteroclinic orbits. The structure of this attractor will be precisely determined as well.