Dynamical analysis of a class of Caputo hetero-order fractional differential systems with double time delays

This paper innovatively constructs a class of Caputo hetero-order fractional predator-prey systems incorporating cannibalism, fear effect, and double time delays. By differentially setting the fractional orders of prey and predators, the heterogeneous memory effects exhibited by both species during their evolutionary processes are characterized. The model simultaneously integrates the cannibalistic behavior of prey and the fear effect induced by predators, and introduces double time delays (fear effect delay and gestation delay), breaking through the limitation of traditional same-order fractional models in describing the memory differences among species. The research adopts a progressive analysis approach: First, for the non-time-delay system, the uniqueness and boundedness of the system's solutions are proved, the existence conditions of the positive equilibrium point are given, and the local stability criterion is established based on the characteristic equation. Furthermore, with the cannibalism rate and fear parameter as bifurcation parameters, the Hopf bifurcation mechanism is analyzed. Second, for the time-delay system, the focus is on analyzing various combinations of time delays; with time delay as the bifurcation parameter, the stability of the equilibrium point and the conditions for Hopf bifurcation are derived. Finally, the correctness of the theoretical results is verified through multiple sets of numerical simulations.