Bifurcation Analysis About a Mathematical Model of Somitogenesis Based on the Runge–Kutta Method
- 57 Downloads
In this paper, we investigated the 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 cannot. Here, we mainly studied the non-diffusing case. We have used the Hopf bifurcation theorem, the Center manifold theorem and Runge–Kutta method in our investigation. First, we investigate its dynamical behaviors and put forward a sufficient condition for the oscillation of the small network. Then, we give the mathematical simulation based on the Runge–Kutta method. In the process of solving ordinary differential equations, the four order Runge–Kutta method has the advantages of high accuracy, convergence and stability (under certain conditions), which can change the step size and do not need to calculate higher order derivatives. Therefore, it has become the most commonly used numerical solution. At the same time, we get the sufficient condition in which the bistable state of the system exists and give the numerical simulation. Because somitogenesis occupies an important position in the process of biological development, 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. In addition, it is beneficial to study the clone animal variation problem of spinal bone number and is of great help to the treatment and prevention of defects of human spine disease.
KeywordsHopf bifurcation Bistable Somitogenesis Runge–Kutta method
This work is supported by National Natural Science Foundation of China (No. 11772291, 11572278), Innovation Scientists and Technicians Troop Construction Projects of Henan Province (No. 2017JR0013).
Compliance with ethical standards
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
- 6.Meinhardt, H. (1996). Model of biological pattern formation: Common mechanism in plant and animal development. International Journal of Developmental Biology, 40, 123–134.Google Scholar
- 16.Polezhayev, A. A. (1995). Mathematical model of segmentation in somitogenesis in vertebrates. Biophysics, 3(40), 583–589.Google Scholar
- 18.Munoz Alicea, R. (2011). Introduction to bifurcations and the Hopf bifurcation theorem for planar systems. Opreation for Mathematics, 640, 11.Google Scholar
- 22.Sarmah, H. K., Baishya, T. K., & Das, M. C. (2014). Hopf-bifurcation in a two dimensional nonlinear differential equation. International Journal of Modern Engineering and Research Technology, 4, 2249–6645.Google Scholar
- 23.Das D., Banerjee, D. & Bhattacharjee, J. K. (2013). Super-critical and Sub-Critical Hopf bifurcations in two and three dimensions. arXiv:1309.5470v1.
- 25.Shimono, Y., Yamaguchi, I., Ishimatsu, K., Akao, A., Ogawa, Y., Jimbo, Y., & Kotani, K. (2018) Evaluation of heuristic reductions of a model for the segmentation clock in zebrafish. In IEEJ transactions on electrical and electronic engineering.Google Scholar