Publication: Dynamic transitions and turing patterns of reaction diffusion equations
Abstract
Bu tezde, iki farklı reaksiyon difüzyon denkleminde ortaya çıkan Turing desenlerinin dinamik geçişleri incelenmiştir. Tez kapsamında ele alınan modeller Brusselator modeli ve çapraz difüzyonlu av-avcı modelidir. Ele alınan modellerde zamandan bağımsız bir bariz çözüm bulunmaktadır. Bu bariz çözüm sistemin kontrol parametresinin belli bir kritik eşiğe ulaşmasıyla kararlığını kaybeder. Bu tez kapsamında incelenen durumda, iki öz modun aynı anda kritik hale gelmesiyle ortaya çıkan geçişlere odaklanılır. Göz önünde bulundurulan kararsızlık türü, bariz çözümün difüzyonsuz durumda küçük pertürbasyonlara karşı kararlı olduğu, ancak difüzyonlu durumda küçük uzaysal pertürbasyonlara karşı kararsız hale geldiği Turing kararsızlığıdır. Rulolar ve dikdörtgenler gibi temel desenlerin göreli kararlılıklarını ve altıgen desenlere de yol açan üst üste binmelerini anlamak için iki öz modun aynı anda kararsız hale geldiği durumu incelemek çok önemlidir. Sistemin fiziksel parametreleri, hangi tip geçişlerin ve örüntülerin oluştuğunun belirlenmesinde önemli rol oynamaktadır. Ele alınan problemlerde, geçiş türlerini ve oluşan Turing modellerinin doğasını belirlemek için kullanılan genlik denklemlerini elde etmek için merkez manifold yaklaşımı kullanılmıştır. Elde edilen sonuçlar iki yönlüdür: (1) temel homojen durumlardan modelin dinamik geçişlerinin türlerinin ve yapısının titiz bir karakterizasyonu yapılır ve (2) dinamik geçişler ile desen oluşumları arasındaki ilişki kurulur. Bu çalışmada kullanılan ana araç, yakın zamanda geliştirilen dinamik geçiş teorisidir.
The dynamic transitions of Turing patterns in two distinct reaction diffusion models are studied in this research work. The models are the Brusselator model and the predator-prey model with cross diffusion. In particular, the focus is on the transitions that occurs when two eigenmodes become critical simultaneously as the control parameter of the system attains the critical threshold and the basic steady state loses its stability. The type of instability considered is Turing instability where the homogeneous steady state is stable to small perturbations without diffusion but becomes unstable to small spatial perturbations with diffusion. It is crucial to consider the critical crossing of two eigenmodes in order to describe the relative stabilities of basic patterns (such as rolls and rectangles) and their superposition (leading to hexagonal patterns). The physical parameters of the system play an important role in the determination of the types of transitions and patterns that are formed. In the problems considered, the center manifold approach is used to obtain reduced equations which are used to determine the types of transitions and nature of the Turing patterns formed. The results obtained are in two folds: (1) a rigorous characterization of the types and structure of the dynamic transitions of the model from basic homogeneous states is obtained and (2) the relation between the dynamic transitions and the pattern formations is established. The main framework used in this study is the recently developed dynamic transition theory
The dynamic transitions of Turing patterns in two distinct reaction diffusion models are studied in this research work. The models are the Brusselator model and the predator-prey model with cross diffusion. In particular, the focus is on the transitions that occurs when two eigenmodes become critical simultaneously as the control parameter of the system attains the critical threshold and the basic steady state loses its stability. The type of instability considered is Turing instability where the homogeneous steady state is stable to small perturbations without diffusion but becomes unstable to small spatial perturbations with diffusion. It is crucial to consider the critical crossing of two eigenmodes in order to describe the relative stabilities of basic patterns (such as rolls and rectangles) and their superposition (leading to hexagonal patterns). The physical parameters of the system play an important role in the determination of the types of transitions and patterns that are formed. In the problems considered, the center manifold approach is used to obtain reduced equations which are used to determine the types of transitions and nature of the Turing patterns formed. The results obtained are in two folds: (1) a rigorous characterization of the types and structure of the dynamic transitions of the model from basic homogeneous states is obtained and (2) the relation between the dynamic transitions and the pattern formations is established. The main framework used in this study is the recently developed dynamic transition theory