Dynamics in a Fractional Order Predator-Prey Model Involving Michaelis-Menten type Functional Response and both Unequal Delays

The interrelationship between predator populations and prey populations is a central problem in biology and mathematics. Setting up appropriate predator-prey models to portray the development law of predator populations and prey populations has aroused widespread interest in many scholars. In this work, we propose a new fractional order predator-prey system involving Michaelis-Menten-type functional response and both unequal delays. Utilizing the contraction mapping theorem, we prove the existence and uniqueness of the solution to the considered fractional order predator-prey system. By virtue of some mathematical analysis techniques, nonnegativeness of the solution to the involved fractional order predator-prey system is analyzed. By constructing a suitable function, the boundedness of the solution to the considered fractional order predator-prey system is explored. Making use of Laplace transform, we derive the characteristic equation of the involved fractional order predator-prey system, then by means of the stability principle and the bifurcation theory of fractional order dynamical system, a series of novel delay-independent stability criteria and bifurcation conditions ensuring the stability of the equilibrium point and the creation of Hopf bifurcation of the considered fractional order predator-prey system, are built. The global stability of the involved fractional order predator-prey system is analyzed in detail. The role of time delay in controlling the stability and the creation of Hopf bifurcation is revealed. To check the legitimacy of the derived key results, software simulation results are effectively presented. The obtained results in this work are completely novel and play a significant role in maintaining ecological balance.