# A Simple Predator-Prey Population Model with Rich Dynamics
## 4 Numerical Simulation  
In this section, we present numerical examples for system Equation (3). **Example 4.1**: (**Example 3.1**: continued) The parameters values \(r,K,a,c,d\) and \(m\) are the same as in Example 3.1. We obtain \(\hat{n}>1\), \(h_{1}=0.0081\), and \(h_{2}=0.0072\). A forward bifurcation diagram is given in Figure 1.  
Selecting \(P_{0}=0.25\), we get \(h=0.0075>h_{2}\). The equilibrium \(E^{*}(0.2,0.24)\) exists, but \(E_{1}\) and \(E_{2}\) do not exist (Theorem 2\(b)\)). Its phase portrait is given in Figure 4, which shows that the unique positive equilibrium \(E^{*}\) is globally asymptotically stable. If we choose \(P_{0}=0.2\), then \(h=0.006<h_{2}\). Equilibrium \(E_{1}(0.16044,0.27165)\) exists, but \(E_{2}\) and \(E^{*}\) do not exist (Theorem 2 _b_)). Its phase portrait is given in Figure 5. The unique positive equilibrium \(E_{1}\) is globally asymptotically stable in \(D\). The equilibrium \(E^{*}\) in Figure 4 corresponds to some point on the curve of \(P^{*}\) in Figure 1 (\(h>h_{2}\)), and the equilibrium \(E_{1}\) in Figure 5 corresponds to some point on the curve of \(P_{1}\) in Figure 1 (\(h<h_{2}\)). **Example 4.2**. : (**Example 3.2**. continued) Choosing the same parameters values as in Example 3.2, we have \(h_{1}=0.00378\) and \(\hat{P}=0.34367\). A backward bifurcation diagram is given in Figure 2.  
If we choose \(P_{0}=0.1\), then \(P_{0}<\hat{P}\). Equilibria \(E_{1}(0.2,0.5)\) and \(E_{2}(0.325,0.1875)\) exist, but equilibrium \(E^{*}\) does not exist. Its phase portrait is illustrated in Figure 6. It shows that equilibria \(E_{0}\) and \(E_{1}\) are asymptotically stable. The equilibria \(E_{1}\) and \(E_{2}\) in Figure 6 correspond to some points on the curves of \(P_{1}\) and \(P_{2}\), respectively, in Figure 2 (\(h<h_{1}\)). Figure 4: The phase portrait of model Equation (3) when \(E^{*}\) is globally asymptotically stable and \(E_{0}\) is unstable. Figure 5: The phase portrait of model Equation (3) when \(E_{1}\) is locally asymptotically stable and \(E_{0}\) is unstable. **Example 4.3 (I). ** (**Example 3.3. ** continued) We identically select parameter values as in Example 3.3, and set \(P_{0}=0.1\). Thus we have \(\hat{n}>1\), \(h_{1}=0.0032\), \(h_{2}=0.0024\), \(\hat{P}=0.16\), and \(h=0.003\). The above parameter values satisfy conditions \(a)\) and \(c)\) in Theorem 2, and condition \(a)\) in Theorem 6. Obviously, equilibria \(E_{0}(0.25,0)\), \(E^{*}(0.2,0.08)\), \(E_{1}(0.125,0.2)\), and \(E_{2}(0.175,0.12)\) all exist. The phase portrait of model Equation (3) is shown in Figure 7. Equilibria \(E^{*}\) and \(E_{1}\) are asymptotically stable, and \(E_{0}\) and \(E_{2}\) are unstable. So model Equation (3) has bistable positive equilibria \(E^{*}\) and \(E_{1}\). It follows from Figure 7 that the stable region \(\Gamma^{s}\) of the saddle point \(E_{2}\) divides the positive invariant set into two regions. The attractive basin for the stable equilibrium \(E_{1}\) is the region above \(\Gamma^{s}\) and the region below \(\Gamma^{s}\) is the basin of attraction for the stable equilibrium \(E^{*}\). The equilibria \(E_{1}\) and \(E_{2}\) in Figure 7 correspond to some points on the curves of \(P_{1}\) and \(P_{2}\), respectively, and \(E^{*}\) corresponds to some point on the curve of \(P^{*}\) in Figure 3 (\(h_{2}<h<h_{1}\)). **Example 4.3 (II). ** We set parameter values \(r=0.03,K=0.25,a=0.5,c=0.4,d=0.01,m=0.03\), and \(P_{0}=0.0133\). Thus we have \(\hat{n}=1.25\), \(uh_{1}=0.00048\), \(h_{2}=0.00036\), \(\hat{P}=0.024\), and \(h=0.000399\). These parameter values satisfy conditions \(a)\) and \(c)\) in Theorem 2 and the condition \(c)\) in Theorem 6. Thus, equilibria \(E_{0}(0.25,0)\), \(E^{*}(0.2,0.012)\), \(E_{1}(0.1092,0.0338)\), and \(E_{2}(0.1908,0.0142)\) all exist. The phase portrait of model Equation (3) is shown in Figure 8. Equilibria \(E^{*}\) and \(E_{1}\) are stable, and \(E_{0}\) and \(E_{2}\) are unstable. So model Equation (3) has bistable positive equilibria \(E^{*}\) and \(E_{1}\).