# A Simple Predator-Prey Population Model with Rich Dynamics
## 3 Preliminary Results
### Stability of Equilibria
When \(h\leq h_{2}\), a bifurcation diagram is shown in Figure 1. When the parameter \(h\) decreases, the bifurcation at \(h=h_{2}\) is forward, and model Equation (3) has a unique positive equilibrium for \(h>0\), which is similar to ([30] Example 2.4). Figure 1: The forward bifurcation diagram from \(P^{*}\) to \(P_{1}\)_versus_\(u\) for Equation (3). The line with \(P_{1}\) indicates the curve of the predator with coexistent equilibrium \(E_{1}\), and the line with \(P^{*}\) indicates the curve of the predator with coexistent equilibrium \(E^{*}\). **Example 3.2**. : Choosing \(r=0.1,K=0.4,a=0.1,c=0.8,d=0.01\), and \(m=0.03\), we obtain \(\hat{n}=0.8<1\), \(h_{1}=0.00378\), and \(\hat{P}=0.34367\). A backward bifurcation diagram is given in Figure 2, where the horizontal line denotes the curve of the predator with predator-extinct equilibrium \(E_{0}\). Two positive equilibria \(E_{1}\) and \(E_{2}\) arise simultaneously at \(h=h_{1}\) when the parameter \(h\) decreases. **Remark 3**. : In Figure 2, we consider how to set up the harvesting threshold value \(h\). We find that \(h_{1}\) is an important harvesting amount. If the harvesting threshold value exceeds \(h_{1}\), the system does not have a positive equilibrium; that is, the predator eventually tends to extinction. If the harvesting threshold value is less than \(h_{1}\), the system has two positive equilibria among which one is unstable and the other may be stable, _i.e._, the predator and prey may coexist. **Example 3.3**. : For model Equation (3), we choose \(r=0.1,K=0.25,a=0.25,c=0.8,d=0.01\), \(m=0.03\) and \(P_{0}=0.1\). Thus we obtain \(\hat{n}=1.25\), \(h_{1}=0.0032\), \(h_{2}=0.0024\), and \(\hat{P}=0.16\). A bifurcation diagram is illustrated in Figure 3, where the horizontal blue line presents the curve of the predator with the positive equilibrium \(E^{*}\). It displays that there is a bifurcation at \(h=h_{1}\) when the parameter \(h\) reduces, which produces three equilibria \(E^{*}\),\(E_{1}\) and \(E_{2}\). Figure 3: The bifurcation diagram with \(P_{1},P_{2}\) and \(P^{*}\)_versus_\(h\). The lines with \(P_{1}\) and \(P_{2}\) indicate the curves of the predator with coexistent equilibria \(E_{1}\) and \(E_{2}\), respectively, and the line with \(P^{*}\) indicates the curve of the predator with coexistent equilibrium \(E^{*}\). Figure 2: The backward bifurcation diagram of \(P_{1}\) and \(P_{2}\)_versus_\(h\) for Equation (3). The solid line with \(P_{1}\) and dotted line with \(P_{2}\) represent the curves of the predator with coexistent equilibrium \(E_{1}\) and \(E_{2}\), respectively. The existence of limit cycles plays an important role in determining the dynamical behavior of the system. For example, if there is no limit cycle in system Equation (3) and its positive equilibrium is unique and locally asymptotically stable, then it must be globally stable. Now, we explore the existence of limit cycles in system Equation (3). **Theorem 7**: _Suppose \(\hat{n}>1\) and \(h<min\{h_{1},h_{2}\}\). If \(\varphi>0\), then system Equation (3) has at least a stable limit cycle which encircles \(E_{1}\)._  
For \(\hat{n}>1\) and \(h<h_{2}\), it is known from Theorem 2\(a)\) that the equilibrium \(E^{*}\) of system Equation (3) does not exist. Furthermore, because \(\hat{n}>1\), \(h<h_{1}\), and \(h<h_{2}\), it follows from Theorem 2\(b)\) that the equilibrium \(E_{2}\) of system Equation (3) does not exist, but the equilibrium \(E_{1}\) exists. It follows from \(\varphi>0\) that \(E_{1}\) is an unstable focus or node. It is easy to see that the unstable manifold at the saddle point \(E_{0}(K,0)\) is in the first quadrant. As the set \(D\) is positively invariant for system Equation (3), and system Equation (3) does not have any equilibrium in the interior of \(D\setminus\{E_{1}\}\). It follows from the Poincare-Bendixson theorem that system Equation (3) has at least a stable limit cycle which encircles \(E_{1}\). In general, Dulac functions are only applied to smooth vector fields in the study of nonexistence of limit cycles.