Dynamical Analysis Of Fractional-Order Rosenzweig-Macarthur Models - Mohamed Al E, Elshahed Mahmoud Moustafa (2018)
In this thesis, three extended fractional order Rosenzweig-MacArthur (R-M) models are considered: i) a two-species R-M model incorporating a prey refuge; ii) a three species R-M model with a prey refuge; iii) a three-species R-M model with stage structure and a prey refuge. The models are constructed and analyzed in detail. The existence, uniqueness, non-negativity and boundedness of the solutions as well as the local and global asymptotic stability of the equilibrium points are studied. Sufficient conditions for the stability and the occurrence of Hopf bifurcation for these fractional order R-M models are demonstrated. The impacts of fractional order and prey refuge on the stability of these systems are also studied both theoretically and by using numerical simulations. The results indicate that the outcomes of R-M fractional order model are more stable than its integer counterpart model because the domain of stability in the fractional order model is larger than the domain for the corresponding integer order model. Rosenzweig in a paper published in 1971 highlighted that increasing the carrying capacity of the prey (i.e. enriching the systems) may lead to destroy the steady state. This is known as the paradox of enrichment. In this study, it was found that the introduction of fractional order to the R-M models can lead to stabilization of the species ecosystems and thus resolve the paradox of enrichment.