Stability and bifurcation analysis for a fractional-order cancer model with two delays

Considering that healthy tissue cells and tumor cells restrain each other's the average growth rate with a corresponding delay, a fractional-order cancer model with time delays is established. Firstly, by selecting different delays as bifurcation parameters, the delay-independent stability and Hopf bifurcation conditions of such system are derived. Then, some interesting dynamical phenomena of the system under different initial values can be observed, such as a pair of coexisted stable periodic solution, equilibrium point transition, and even chaos. Finally, some numerical simulations are exhibited to substantiate the obtained results. The work reveals that the fractional order and delays play an important role in the stability behavior and bifurcation characteristics of the system, and that a suitable control strategy of the system can be designed by selecting reasonable system parameters (fractional orders and time delays).