Researching On The Dynamic Behaviours During Somitogenesis In Embryos

Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental pattern-ing processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. In this paper, we investigate a modified two dimensional model which can explain somite patterning in embryos. It is suitable for exploring a design space of somitogenesis and can explain aspects of somitogenesis that previous models can-not. First, we studied the dynamical behavior of the non-diffusing case by using the Hopf bifurcation theorem and the Center manifold theorem. Then, through the linear stability analysis of local equilibrium we obtain the condition how the Hopf bifurcation, Turing bifurcation and the oscillatory instability boundaries arise with the effect of diffusion. By using the weak nonlinear multiple scales analysis and Taylor series expansion, we derive the equations for the amplitude of the stationary patterns. The analysis of amplitude equations shows the occurrence of different complex phenomena, including Spot pattern,mixture of spot-stripe patterns?labyrinthine and so on. Finally, numerical simulation are given to verify the correctness of our theoretical results. Somitogenesis in the process of biological development occupies an important position, and as a pattern process can be used to study pattern formation and many aspects of embryogenesis. So our study have a great help for Embryonic development, gene expression, cell differentiation. What's more,it is helpful to study bone associated congenital diseases. So it is of great significance for the diagnosis and treatment of human diseases to study the related knowledge of model biology.