Mohamed Al E, Elshahed Mahmoud Moustafa
(2018)
Dynamical Analysis Of
Fractional-Order
Rosenzweig-Macarthur Models.
Masters thesis, Universiti Sains Malaysia.
Abstract
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.
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