Global dynamics of a nonlinear state-dependent feedback control ecological model with a multiple-hump discrete map

An ecological model with nonlinear state-dependent feedback control is proposed and studied. The Poincaré map defined in the phase set integrates the properties of continuous, discrete and impulsive dynamical systems. The map has complex behavior including single hump and multiple-hump functions and multiple discontinuous points. The properties of various cases of the Poincaré map are systematically studied including monotonicity, continuity and the existence of a fixed point and a transcritical bifurcation, which correspond to the existence and global stability of the order-1 periodic solutions or limit cycles of the proposed model. Moreover, the conditions under which the discontinuity occurs and the number of discontinuous points are discussed, which can result in the coexistence of multiple order-1 periodic solutions and complex dynamics. Furthermore, some sufficient conditions for the global stability of the fixed point (order-1 limit cycle) have been provided even when there are multiple peaks and valleys or multiple discontinuous points for the Poincaré map.