Bifurcation analysis in a discrete time square root response function of predator-prey system with fractional order

Stability and bifurcation analysis for a two spices discrete fractional order system by introducing square root response function of the form x ( j + 1 ) = x ( j ) + h α Γ ( 1 + α ) [ μ x ( j ) − μ x 2 ( j ) − σ x ( j ) y ( j ) ] y ( j + 1 ) = y ( j ) + h α Γ ( 1 + α ) [ η x ( j ) y ( j ) − β y ( j ) ] is examined in the closed first quadrant ℝ + 2 . The model parameters h , α , β , μ , σ , η are biologically feasible positive real numbers. Piecewise constant arguments method is applied to obtain the discrete fractional order system, particularly to study the rich dynamical behavior of the proposed system. Because the system has square root response function, the variation matrix of trivial and axial equilibrium states are in-determinant. In order to study the stability of the trivial and semi trivial equilibrium states, the change of variables x ( j ) = X 2 ( j ) and y ( j ) = Y 2 ( j ) is considered. Moreover, we determine the criteria for stability of the interior equilibrium state using jury conditions. The numerical performance is shown for various parameter values and the time series and phase line diagrams are obtained. Bifurcation theory is applied to check whether the system undergoes periodic doubling bifurcations at its semi trivial and interior equilibrium states. Also we explore the periodic halving bifurcation which occurs for the fractional order as a bifurcation parameter. Numerical examples are presented to show the validity and feasibility of the obtained stability criterion in both species, including periodic doubling, periodic - 2, 4, 8, 16, periodic windows and Non periodic orbit (i.e., chaos).