Comparison of dynamical behavior between fractional order delayed and discrete conformable fractional order tumor-immune system

Abstract

In this paper, we analyze the dynamical behavior of the delayed fractional-order tumor model with Caputo sense and discretized conformable fractional-order tumor model. The model is constituted with the group of nonlinear differential equations having effector and tumor cells. First of all, stability and bifurcation analysis of the delayed fractional-order tumor model in the sense of Caputo fractional derivative is studied, and the existence of Hopf bifurcation depending on the time delay parameter is proved by using center manifold and bifurcation theory. Applying the discretization process based on using the piecewise constant arguments to the conformable version of the model gives a two-dimensional discrete system. Stability and Neimark–Sacker bifurcation analysis of the discrete system are demonstrated using the Schur-Cohn criterion and projection method. This study reveals that the delay parameter τ in the model with Caputo fractional derivative and the discretization parameter h in the discrete-time conformable fractional-order model have similar effects on the dynamical behavior of corresponding systems. Moreover, the effect of the order of fractional derivative on the dynamical behavior of the systems is discussed. Finally, all results obtained are interpreted biologically, and numerical simulations are presented to illustrate and support theoretical results.