Analysis of an improved fractional-order model of boundary formation in the Drosophila large intestine dependent on Delta-Notch pathway

In this paper, an improved fractional-order model of boundary formation in the Drosophila large intestine dependent on Delta-Notch pathway is proposed for the first time. The uniqueness, nonnegativity, and boundedness of solutions are studied. In a two cells model, there are two equilibriums (no-expression of Delta and normal expression of Delta). Local asymptotic stability is proved for both cases. Stability analysis shows that the orders of the fractional-order differential equation model can significantly affect the equilibriums in the two cells model. Numerical simulations are presented to illustrate the conclusions. Next, the sensitivity of model parameters is calculated, and the calculation results show that different parameters have different sensitivities. The most and least sensitive parameters in the two cells model and the 60 cells model are verified by numerical simulations. What is more, we compare the fractional-order model with the integer-order model by simulations, and the results show that the orders can significantly affect the dynamic and the phenotypes.