# A Simple Predator-Prey Population Model with Rich Dynamics
## 3 Preliminary Results
### Stability of Equilibria  
In this section, we discuss the stability of equilibria of model Equation (3). **Theorem 3**: _Equilibrium \(E_{0}(K,0)\) is locally asymptotically stable if \(\hat{n}<1\), and unstable if \(\hat{n}>1\). Moreover, \(E_{0}(K,0)\) is globally asymptotically stable in \(D\backslash\{(0,0)\}\) if \(\hat{n}<1\) and \(h>h_{1}\). If \(\hat{n}>1\), system Equation (3) is permanent._  
It is easy to obtain that the characteristic roots to the linearized equation of system Equation (3) at \(E_{0}(K,0)\) are \(\lambda_{1}=-r<0\) and \(\lambda_{2}=caK-d-m=(d+m)(\hat{n}-1)\). Thus, \(E_{0}\) is locally asymptotically stable if \(\hat{n}<1\) and unstable if \(\hat{n}>1\). Next, note that \((0,0)\) is always unstable. If \(\hat{n}<1\), by Theorems 2 and 2, there is not any other equilibrium of system Equation (3) than \(E_{0}\) in \(D\backslash\{(0,0)\}\). Since \(D\) is the invariant set of system Equation (3) and \(E_{0}\) is locally asymptotically stable, it follows from the Bendixson Theorem that every solution of system Equation (3) in \(D\) approaches \(E_{0}\) when \(t\) tends to positive infinity. Since \(E_{0}\) is unstable as \(\hat{n}>1\), following the similar arguments to [PERSON] and [PERSON] ([33] Theorem 3.1) (see also ([34] Theorems 3,4)), which is based on the uniform persistence theory introduced by [PERSON] and [PERSON] [35], we are able to conclude that system Equation (3) is permanent if \(\hat{n}>1\). The proof is complete. **Remark 2**. : It follows from [PERSON] and [PERSON] ([33] Theorem 3.1) that system Equation (5) is permanent if and only if \(\hat{n}>1\). Thus Theorem 3 suggests that if system Equation (5) is permanent, then so is system Equation (3). For the three positive equilibria \(E^{*},E_{1},E_{2}\), we have the following results. Firstly, we consider the stability of \(E^{*}\). **Theorem 4**. : _The positive equilibrium \(E^{*}(N^{*},P^{*})\) of system Equation (3) is globally asymptotically stable if \(P_{0}<\frac{r}{a}\) and \(1<\hat{n}\leq\frac{r}{r-aP_{0}}\)._  
Proof. : According to Theorem 1, positive equilibrium \(E^{*}(N^{*},P^{*})\) exists if and only if \(P_{0}<\frac{r}{a}\) and \(1<\hat{n}\leq\frac{r}{r-aP_{0}}\). The Jacobian matrix of system Equation (3) at \(E^{*}(N^{*},P^{*})\) is  
\[J^{*}=\left( \begin{array}{cc}r(1-\frac{N^{*}}{K})+rN^{*}(-\frac{1}{K})-aP^{* }&-aN^{*}\\ caP^{*}&caN^{*}-d-m\end{array} \right) \tag{23}\]  
Because \(N^{*}\) and \(P^{*}\) satisfy Equation (9), by means of Equation (9), the trace and determinant of \(J^{*}\) are simplified as  
\[ \begin{split}& tr(J^{*})=r-\frac{2 rN^{*}}{K}-aP^{*}+caN^{*}-d-m=- \frac{rN^{*}}{K}<0\\ & det(J^{*})=(r-\frac{2 rN^{*}}{K}-aP^{*})(caN^{*}-d-m)+ca^{2}N^{*}P^{*}= ca^{2}N^{*}P^{*}>0\end{split}  \tag{24}\]  
Therefore, all eigenvalues of matrix \(J^{*}\) have negative real parts when \(P_{0}<\frac{r}{a}\) and \(1<\hat{n}\leq\frac{r}{r-aP_{0}}\). It follows that \(E^{*}(N^{*},P^{*})\) is locally asymptotically stable. Then, it suffices for us to prove the global attractiveness of \(E^{*}(N^{*},P^{*})\). Inspired by the work of [PERSON] [36], we define a Lyapunov function  
\[M(t)=cN^{*}(\frac{N(t)}{N^{*}}-\ln\frac{N(t)}{N^{*}}-1)+P^{*}(\frac{P(t)}{P^{ *}}-\ln\frac{P(t)}{P^{*}}-1)\]  
where \(N^{*}=\frac{K}{\hat{n}}\), and \(P^{*}=\frac{r}{\hat{n}}(1-\frac{1}{\hat{n}})\). We know that \(\frac{N(t)}{N^{*}}-\ln\frac{N(t)}{N^{*}}-1\) and \(\frac{P(t)}{P^{*}}-\ln\frac{P(t)}{P^{*}}-1\geq 0\) for all \(N(t),P(t)>0\). (The equality holds if and only if \(N(t)=N^{*},P(t)=P^{*}\).) From the definition of \(M(t)\), we know that \(M(t)\) is well-defined and \(M(t)\geq 0\). The equality holds if and only if \(N(t)=N^{*}\) and \(P(t)=P^{*}\). Differentiating \(M(t)\) along the solutions of system Equation (3), we obtain  
\[\frac{dM(t)}{dt} =c(1-\frac{N^{*}}{N})\frac{dN}{dt}+(1-\frac{P^{*}}{P})\frac{dP}{dt}.\] \[=-\frac{cr}{K}N^{2}+\frac{2 rN(d+m)}{aK}-\frac{r(d+m)^{2}}{ca^{2}K}.\] \[=-\frac{cr}{K}(N-\frac{d+m}{ca})^{2}\leq 0\]  
It follows that \(M(t)\) is bounded and non-increasing.