Complex Dynamical Analysis of a Discretized Fractional Order Tumor Dormancy Model with Chaotic Behavior

Abstract

A discrete fractional order model of tumor-immune interaction is proposed to investigate cancer dormancy. The investigation shows that the model undergoes a Neimark-Sacker bifurcation, leading to chaos. The classical predatorprey model is typically used to describe interactions between two biological populations, and it is adapted here to reflect the complex relationship between tumor cells and lymphocytes. Building on earlier continuous models that use fractional calculus to capture memory effects and long-term immune responses, a discrete version of a continuous fractional order model is constructed using piecewise constant arguments, where the continuous time t is divided into equal intervals of length h and replaced by h[t/h]. The function’s value is taken at the start of each interval and held constant until the next one. Then, the stability of the fixed points is analyzed, and the exact conditions under which they exist and remain stable are clearly identified. Using bifurcation theory and center manifold theory, a Neimark-Sacker bifurcation at the positive fixed point is verified under certain parameter conditions. In the feedback control section, control terms are added to reduce tumor growth and stabilize the chaotic behavior of the system. Numerical simulations are then used to show the accuracy of the analytical results. Furthermore, the behavior of the system is investigated as the parameters r, σ , and α vary. At σ ^∗ = 0.432277, the system undergoes a Neimark-Sacker, and the Lyapunov exponent is positive for σ < 0.432277 for certain parameter values, confirming the occurrence of chaos. Finally, the impact of varying the fractional order α on the system’s stability is examined.