# A Simple Predator-Prey Population Model with Rich Dynamics
## 1 Introduction
Two types of harvest strategies are: with a nonzero constant and a linear harvesting rate. The choice of idealized harvest strategies will contribute to a qualitative understanding of the properties of different harvesting strategies. [PERSON] and [PERSON] [16] considered the dynamical properties of the ratio-dependent predator-prey model with constant prey harvesting. There existed numerous kinds of bifurcations, such as the saddle-node bifurcation, the subcritical and supercritical Hopf bifurcations. There also existed a limit cycle, a homoclinic or heteroclinic orbit satisfying different parameter values. In Ref. [17], the ratio-dependent predator-prey model with constant predator harvesting was focused on. [PERSON] _et al._[26] also discussed two predator-prey models with linear or nonzero constant predator harvesting. The above two types of harvesting rates seemingly have their own advantages as well as disadvantages in fitting the harvest in the real world. When the density of the predator or prey is rather low, the nonzero constant harvesting rate is not as reasonable as of the proportional type [13, 22, 26]; while if the predator or prey is abundant, linear harvesting rate is less possible than the constant harvesting rate [16]. In Ref. [22], to compare the stability properties of the system with two harvest strategies, they applied linear or constant harvesting rate, respectively. In that comparative study, the authors demonstrated that switching from linear to constant harvesting rate may turn a stable stationary state to a periodic or chaotic oscillatory mode from a mathematical perspective. However, when deciding the constant level of harvesting, the instability of the constant harvest strategy calls for great care. In Ref. [29], [PERSON] _et al._ introduced a more realistic smooth harvesting function in which the fishing effort is limited upwards because the constant harvest cannot be achieved for small populations of the fish. The adjustment of a harvesting function can prevent extinction and increase the stability to some extent. Moreover, the dramatic increase of the predator or prey challenges the normal ecological balance and capacity of harvest. Thus, it is interesting to construct a new kind of harvesting rate that combines the advantages from both linear and constant harvesting rates. Motivated by these ideas, in this paper, we consider a predator-prey model with a novel harvesting rate. Our ideas to develop the harvesting rate are derived from the capacity of treatments of diseases that had been well studied in the dynamical epidemic models [30; 31; 32]. Using model Equation (1) as our baseline model, we assume that harvesting takes place, but only the predator population is under harvesting and introduce harvesting function \(H(P)\) of the predator to prey-predator model Equation (1) for discussing its dynamical features. The interactive dynamics are governed by the following system  
\[ \begin{array}{ll}\frac{dN}{dt}=&rN\left(1-\frac{N}{K}\right)-aNP\\ \frac{dP}{dt}=&caNP-dP-H(P)\end{array}  \tag{3}\]  
Following the methods in [14; 15; 16; 17; 30], we investigate the existence and stability of multiple equilibria, bifurcations, and limit cycles, and study the effects of switched harvest on the dynamics of the predator-prey model. This paper is organized as follow. Sections 2 and 3.1 represent the boundedness of model Equation (3) and existence of multiple equilibria. In Section 3.2, we study the stability of equilibria, bifurcations, and the existence and stability of a limit cycle. In Section 4, we give numerical simulations to verify our results. Brief discussions are presented finally in Section 5.