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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 the corresponding discrete fractional-order system. Mathematical results like positivity and boundedness of the solutions of fractional-order predator–prey 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. We have determined an extra stability region in the lower range of habitat complexity where all populations coexist in stable state for some fractional-order values but unstable for integer-order value. Dynamics of the discrete fractional-order model is shown to be more complex and depends on both the step-size and fractional-order. It shows Hopf bifurcation, flip bifurcation and chaos with respect to the step-size. Several examples are presented to substantiate the analytical results.

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system