Mathematical models of restoration and control of a single species with Allee effect

According to the initial density of a single species with Allee effect and corresponding management strategy, three kinds of mathematical models are presented to describe the evolutionary process of the species by impulsive differential equations. When the initial density of the species is larger than economic injury level (EIL) (or economical threshold, ET), impulsive harvest control is considered in a finite time to decrease the population of the species. The feasibility of the impulsive harvest control in a finite time is given by the existence of solution of the model with initial and boundary value problem. When the initial density of the species is less than EIL (or ET), the model with state feedback control is formulated according to the state of the species. The existence and stability of periodic solution of the model with state feedback control are discussed. When the initial density of the species is less than the Allee threshold and the species tends to extinction, the model with impulsive release at fixed moments is presented to study the restoration of the species. The conditions for the feasibility of periodic restoration of the species are given. Finally, some discussions are given.