Presentation Information
[POS-67]Turing instability and dynamic bifurcation in the Gray-Scott model
*Taeyoung Ha1, Youngho Woo1, Yuncherl Choi2, Jongmin Han3, Sewoong Kim4, Doo Seok Lee5 (1. National Institute for Mathematical Sciences (Korea), 2. Kwangwoon University (Korea), 3. Kyung Hee University (Korea), 4. Samsung Fire \& Marine Insurance (Korea), 5. Daegu Gyeongbuk Institute of Science and Technology (Korea))
Keywords:
bifurcation,turing instability,Gray-scott equation
Pattern formation arises from various reaction–diffusion systems for physical, chemical, and biological phenomena. Such phenomena happen when a system undergoes phase transitions, the change of phase from one state to another. Turing instability plays a crucial role in such a phase transition dynamics. Ahomogeneous steady-state solution is destabilized due to the diffusion effect and the system leads to a bifurcation that accompanies pattern formation. There have been lots of studies about the conditions for the onset of instabilities and the ensuing pattern formations.
In this paper, e study the dynamic bifurcation in the Gray-Scott model by taking the diffusion coefficient $\lambda$ of the reactor as a bifurcation parameter. We define a parameter space $\Sigma$ of $(k,F)$ for which the Turing instability may happen. Then, we show that it really occurs below the critical number $\lambda_0$ and obtain rigorous formula for the bifurcated stable patterns. And, we find the conditions to happen supercritical bifurcation and subcritical bifurcation. We also provide numerical results that illustrate the main Theorems.
In this paper, e study the dynamic bifurcation in the Gray-Scott model by taking the diffusion coefficient $\lambda$ of the reactor as a bifurcation parameter. We define a parameter space $\Sigma$ of $(k,F)$ for which the Turing instability may happen. Then, we show that it really occurs below the critical number $\lambda_0$ and obtain rigorous formula for the bifurcated stable patterns. And, we find the conditions to happen supercritical bifurcation and subcritical bifurcation. We also provide numerical results that illustrate the main Theorems.