# A Simple Predator-Prey Population Model with Rich Dynamics
## 3 Preliminary Results
### Stability of Equilibria
Thus \(\lim_{t\to\infty}M(t)\) exists. Note that \(\frac{dM}{dt}~{}=~{}0\) if and only if \(N=N^{*}\). Substituting \(N=N^{*}\) into the first equation of Equation (3), one can directly get \(P=P^{*}\). Therefore, the maximal compact invariant set in \(\frac{dM}{dt}=0\) is the singleton \(E^{*}\). By the LaSalle invariance principle (see, for example, Theorem 5.3.1 in [PERSON] and [PERSON] [37]), positive equilibrium \(E^{*}\) is globally attracting. Further, \(E^{*}\) is globally asymptotically stable. Now we concentrate on the stability of coexistence equilibria \(E_{i}(N_{i},P_{i})\), \(i=1,2\). The Jacobian matrix of system Equation (3), at \(E_{i}(N_{i},P_{i})\), is  
\[J_{i}=\left( \begin{array}{cc}r-\frac{2 rN_{i}}{K}-aP_{i}&-aN_{i}\\ caP_{i}&caN_{i}-d\end{array} \right),\quad i=1,2 \tag{25}\]  
**Theorem 5**. : _If the coexistence equilibrium \(E_{2}(N_{2},P_{2})\) of system Equation (3) exists, it is unstable._  
Proof. : For the coexistence equilibrium \(E_{2}(N_{2},P_{2})\), \(N_{2}\) and \(P_{2}\) satisfy Equation (14). By Equations (13)-(15), after direct calculations, the determinant of matrix \(J_{2}\) is  
\[det(J_{2})=-\frac{2 car}{K}(N_{2}-\frac{d+caK}{4 ca})^{2}+\frac{2 car(d+caK)^{2}}{ 16c^{2}a^{2}K} \tag{26}\]  
Because  
\[N_{2}=\frac{crK(1+n_{0})+n_{0}\sqrt{4 crK(h_{1}-h)}}{2 crn_{0}}>\frac{crK(1+n_{0 })}{2 crn_{0}}=\frac{caK+d}{2 ca}\]  
we obtain that \(det(J_{2})<0\). Thus \(E_{2}(N_{2},P_{2})\) is a saddle point and unstable. For the coexistence equilibrium \(E_{1}(N_{1},P_{1})\), similarly, the determinant of \(J_{1}\) is  
\[det(J_{1})=-\frac{2 car}{K}(N_{1}-\frac{d+caK}{4 ca})^{2}+\frac{2 car(d+caK)^{2}}{ 16c^{2}a^{2}K} \tag{27}\]  
but since  
\[N_{1}=\frac{crK(1+n_{0})-n_{0}\sqrt{4 crK(h_{1}-h)}}{2 crn_{0}}<\frac{crK(1+n_{0 })}{2 crn_{0}}=\frac{caK+d}{2 ca}\]  
we obtain \(det(J_{1})>0\), and \(E_{1}(N_{1},P_{1})\) may be node, focus or center. By Equations (13)-(15), the sign of the trace of matrix \(J_{1}\) is determined by  
\[\varphi=r^{2}d(d-caK)+2 ca^{2}K(caK-r)h-ard\sqrt{4 crK(h_{1}-h)} \tag{28}\]  
According to Equation (28), we obtain  
1. If \(caK-r\leq 0\), then \(\varphi<0\). 2. If \(caK-r>0\), and \(h\leq\frac{r^{2}d(caK-d)}{2 ca^{2}K(caK-r)}=\frac{r^{2}d^{2}(n_{0}-1)}{2 ca^{2} K(caK-r)}=h_{3}\), then \(\varphi<0\). 3. If \(caK-r>0\), \(h>h_{3}\) and \(\eta=4 crK(h_{1}-h)-\frac{[2 ca^{2}K(caK-r)h-r^{2}d^{2}(n_{0}-1)]^{2}}{a^{2}r^{2} d^{2}}>0\), then \(\varphi<0\). Summarizing the above discussions, we have the following results on the stability of equilibrium \(E_{1}(N_{1},P_{1})\). **Theorem 6**. : _Suppose that the coexistence equilibrium \(E_{1}(N_{1},P_{1})\) exists and if one of following conditions is satisfied:_  
_(a)_ \(caK\leq r\)_;_  
_(b)_ \(caK>r\) _and_ \(h\leq h_{3}\)_;_  
_(c)_ \(caK>r\) _and_ \(h>h_{3}\) _and_ \(\eta>0\)_,_  
_then \(E_{1}(N_{1},P_{1})\) is locally asymptotically stable. It is unstable if \(caK>r,h>h_{3}\), and \(\eta<0\)._  
Combining the existence and stability results of equilibria of model Equation (3), by following similar arguments to ([31] Corollary 2.3), we present the following corollary to give conditions for bifurcation. **Corollary 1**. : _If \(\hat{n}<1\) and \(h<min\{h_{1},m\hat{P}\}\), then system Equation (3) has a backward bifurcation of positive equilibria._  
Next, we give examples to demonstrate the bifurcation of multiple equilibria. For various parameter values, model Equation (3) has a forward bifurcation from one positive equilibrium to another positive equilibrium (see Example 3.1.) and a backward bifurcation with a predator-extinct equilibrium and two positive equilibria (Example 3.2.). Note that the conditions in Theorem 2_a_) and Theorem 2_c_) guarantee the existence of three positive equilibria \(E^{*}\) and \(E_{1}\), \(E_{2}\) (Example 3.3.). **Example 3.1**. : Using the following parameter values \(r=0.1,K=0.5,a=0.25,c=0.8,d=0.01\), and \(m=0.03\), we obtain \(\hat{n}=2.5\), and \(h_{2}=0.0072\).