  This paper is focused on local and global stability of a fractional-order
predator-prey model with habitat complexity constructed in the Caputo sense and
corresponding discrete fractional-order system. Mathematical results like
positivity and boundedness of the solutions in fractional-order model is
presented. Conditions for local and global stability of different equilibrium
points are proved. It is shown that there may exist fractional-order-dependent
instability through Hopf bifurcation for both fractional-order and
corresponding discrete systems. Dynamics of the discrete fractional-order model
is more complex and depends on both step length and fractional-order. It shows
Hopf bifurcation, flip bifurcation and more complex dynamics with respect to
the step size. Several examples are presented to substantiate the analytical
results.
