title: Stability and bifurcation analysis of models for zebrafish somitogenesis.
abstract: Notch-Delta signaling is indispensable for somitogenesis, which controls the vertebrate segmentation during embryonic development. Several theoretical models have been proposed to explain this interesting process. In zebrafish somitogenesis, genes her1, her7, delta, and their proteins plays the important roles. However, an auto-repression model with time delay involved only by her1/her7 is able to explain zebrafish somitogenesis. This paper will systematically study the dynamics of this model. Specifically we investigate its stability, bifurcation (oscillation), and stability of oscillation. First, the conditions for both stability and bifurcation are presented based on the linearized model. Then three indices for bifurcation of this nonlinear model are derived by using linear functional operator analysis. Finally, the numerical simulations are carried out to illustrate the theoretical results developed in this study.
authors.affiliation: School of Information, Beijing Wuzi University, Beijing, China. bmitian@163.com
authors.forename: Li-Ping
authors.initials: LP
authors.lastname: Tian
authors.forename: Fang-Xiang
authors.initials: FX
authors.lastname: Wu
mesh_terms: D000818:Animals; D019521:Body Patterning / Q000235:genetics; D002453:Cell Cycle; D047108:Embryonic Development / Q000235:genetics; D005260:Female; D018507:Gene Expression Regulation, Developmental; D008958:Models, Molecular; D017711:Nonlinear Dynamics; D013997:Time Factors; D015027:Zebrafish / Q000196:embryology*