# A Simple Predator-Prey Population Model with Rich Dynamics
## 1 Introduction  
Mathematical modeling of predator-prey interactions have attracted wide attention since the original work by [PERSON] and [PERSON] in 1920s, and there have been extensively studied for their rich dynamics [1; 2; 3]. Since the rich and complex dynamics for interactive species are common in the real world, many researchers have investigated the processes that affect the dynamics of prey-predator models and wanted to know what models can best represent species interactions. As a simplest form, the interaction between a predator and prey may be modeled by a pair of differential equations [1; 3; 4; 5],  
\[ \begin{split}\frac{dN}{dt}&=rN(1-\frac{N}{K})-aNP\\ \frac{dP}{dt}&=caNP-dP\end{split}  \tag{1}\]where \(N\) and \(P\) represent the prey and predator species, respectively; \(r,K,a,c\), and \(d\) are positive constants. In the absence of the predation, the prey grows logistically with intrinsic growth rate \(r\) and carrying capacity \(K\). In the presence of the predator, the prey species decreases at a rate proportional to the functional response \(aN\), where \(a\) presents the rate of predation. The factor \(c\) denotes the efficiency of predation which divides a maximum per capita birth rate of the predators into a maximum per capita consumption rate. Without the prey, no predation occurs and the predator species decreases exponentially with mortality rate \(d\). To consider the dynamics of model Equation (1), it is shown that the origin \((0,0)\) is a saddle point. Define the net reproductive number of the predator population \(n_{0}\), _i.e._, the expected number of a predator individual producing as the predator population is introduced into a stable prey population [6, 7], as  
\[n_{0}=\frac{caK}{d} \tag{2}\]  
Then if \(n_{0}<1\), the boundary equilibrium \((K,0)\) with the predator going extinct is globally asymptotically stable and there exists no positive equilibrium for the prey-predator interaction. If \(n_{0}>1\), the boundary equilibrium \((K,0)\) is a saddle point and there exists a positive coexistence equilibrium \((N_{*},P_{*})=\left(\frac{K}{n_{0}},\frac{r}{a}\left(1-\frac{1}{n_{0}}\right)\right)\), which is globally asymptotically stable. The model dynamics are relatively simple. Using \(n_{0}\) as bifurcation parameter, we have a transcritical bifurcation at \(n_{0}=1\) as \((N_{*},P_{*})\) is bifurcated. Notice that model Equation (1) is a special case of the prey-predator model in [8]. Apparently, model Equation (1) fails to show the complicated dynamics of the predator-prey interactions in the real world. Later, many researchers improve and enrich model Equation (1) by incorporating some other elements, for example, stage-structure [9, 10, 11, 12, 13], nonlinear functional response function [14, 15, 16, 17, 18], dispersal among patchers [19], delays [9, 10, 11, 20], or impulsive effects [18, 21]. In the real world, from the point of view of predators' needs, the exploitation of biological resources and harvest are commonly practiced in fishery, forestry, and wildlife management. There is an interest in the use of bioeconomic models to gain insight into the scientific management of renewable resources [16]. Moreover, harvesting is an important and effective method to prevent and control the explosive growth of predators or prey when they are enough. So, generally speaking, it is reasonable and necessary for one to introduce the harvest of populations into models. Taking the above reasons into a consideration, we focus on the predator-prey model with harvest [12, 13, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28]. Normally, harvesting has several forms in predator-prey models. The most common one of these harvesting forms is a nonzero constant [16, 17, 20, 22, 23, 24, 26] or a linear harvesting rate [12, 13, 22, 25, 26, 27, 28]. In Ref. [22], a two-prey-one-predator model with predator harvested was studied. The authors are particularly interested in the stability properties of different harvest strategies.