# Dynamics of a Stochastic Intraguild Predation Model
## 3 Stochastic Persistence and Stochastic Extinction  
To illuminate the effect of the stochastic perturbations for population and compare the stochastic IGP model (3) with the deterministic IGP model (1), we first explore the existence and local stability of boundary and positive equilibria for model (1). The summary of conditions for the existence and local stability of equilibria are listed in Table 1.  
Now, we analyze stochastic persistence and stochastic extinction of model (3). **Definition 7** (see [32]). : Species \(x(t)\) is said to be persistent in the mean if \([x]_{*}>0\). Let  
\[ \begin{split}\delta_{1}&=\alpha_{r}^{2}a_{nn}/2-\alpha_ {n}^{2}a_{rn}/2,\quad\delta_{2}=\alpha_{r}^{2}a_{pp}/2-\alpha_{p}^{2}a_{rp}/2, \\ \delta_{3}&=\alpha_{n}^{2}a_{rr}/2+\alpha_{r}^{2}e_{ rn}a_{rn}/2,\;\delta_{4}=\alpha_{p}^{2}a_{rr}/2+\alpha_{r}^{2}e_{rp}a_{rp}/2. \end{split}  \tag{14}\]  
A direct calculation gives  
\[ \begin{split} 2r/\alpha_{r}^{2}-\delta_{5}/\tilde{\delta}_{3}& =a_{rr}(r\alpha_{n}^{2}+d_{n}\alpha_{r}^{2})/(\alpha_{r}^{2}\tilde{ \delta}_{3})>0,\\ 2r/\alpha_{r}^{2}-\delta_{6}/\tilde{\delta}_{4}& =a_{rr}(ra_{p}^{2}+d_{p}\alpha_{r}^{2})/(\alpha_{r}^{2}\tilde{ \delta}_{4})>0,\\ \delta_{4}/\tilde{\delta}_{1}-2r/\alpha_{r}^{2}& =a_{rn}(ra_{n}^{2}+d_{n}\alpha_{r}^{2})/(\tilde{\delta}_{1}\alpha_{r}^{2})>0, \\ \delta_{3}/\tilde{\delta}_{2}-2r/\alpha_{r}^{2}& =a_{rp}(ra_{p}^{2}+d_{p}\alpha_{p}^{2})/(\tilde{\delta}_{2}\alpha_{r}^{2})>0. \end{split}  \tag{15}\]  
**Theorem 8**. : _The following five cases hold:_  
(i) _If_ \(2r<\alpha_{r}^{2}\)_, then all the populations are extinction a.s._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Equilibria** & **Existence** & **Local Stability** \\ \hline \(E_{0}(0,0,0)\) & Always & Never \\ \(E_{r}(r/a_{rr},0,0)\) & Always & \(\delta_{5}<0,\delta_{6}<0\) \\ \(E_{rn}(\delta_{4}/\delta_{1},\delta_{5}/\delta_{1},0)\) & \(\delta_{5}>0\) & \(L_{3}<0\) \\ \(E_{rp}(\delta_{3}/\delta_{2},0,\delta_{6}/\delta_{2})\) & \(\delta_{6}>0\) & \(L_{2}<0\) \\ \(E_{mp}(L_{1}/L,L_{2}/L,L_{3}/L)\) & \(L>0,L_{i}>0,i=1,2,3\) & \(\delta_{1}\delta_{2}+\delta_{7}\delta_{8}>0\) \\ \hline _Here,_ \(\delta_{1}=a_{rr}a_{nn}+e_{rn}a_{rn}^{2};\delta_{2}=a_{rr}a_{pp}+e_{rp}a_{rp}^{2} ;\delta_{3}=ra_{pp}+a_{rp}d_{p};\delta_{4}=ra_{nn}+a_{rn}d_{n};\\ \delta_{5}=-a_{rr}d_{n}+re_{rn}a_{rn};\delta_{6}=-a_{rr}d_{p}+re_{rp}a_{rp}; \delta_{7}=-a_{rr}e_{np}a_{np}+a_{rn}e_{rp}a_{rp};\delta_{8}=a_{rr}a_{np}+a_{rp} e_{rn}a_{rn}.\end{split}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Existence and local stability of equilibria for model (1). _If \(2r/\alpha_{r}^{2}>1>\max\{\delta_{5}/\tilde{\delta}_{3},\delta_{6}/\tilde{\delta}_{4}\}\), then \(N(t)\) and \(P(t)\) are extinction a.s. and_  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]  
_If \(L>0\), \(\delta_{5}/\tilde{\delta}_{3}>1\) and \(L_{3}<M_{3}\), then \(P(t)\) is extinction a.s. and_  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{ \delta_{4}-\tilde{\delta}_{1}}{\delta_{1}}\ \ a.s., \tag{17}\] \[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}N(s)ds=\frac{ \delta_{5}-\tilde{\delta}_{3}}{\delta_{1}}\ \ a.s.\]  
_If \(L>0\), \(\delta_{6}/\tilde{\delta}_{4}>1\) and \(L_{2}<M_{2}\), then \(N(t)\) is extinction a.s. and_  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{ \delta_{3}-\tilde{\delta}_{2}}{\delta_{2}}\ \ a.s., \tag{18}\] \[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}P(s)ds=\frac{ \delta_{6}-\tilde{\delta}_{4}}{\delta_{2}}\ \ a.s.\]  
_If \(L>0\), \(L_{i}>M_{i}\), \(i=1,2,3\), then_  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{ L_{1}-M_{1}}{L}\ \ a.s., \tag{19}\] \[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}N(s)ds=\frac{ L_{2}-M_{2}}{L}\ \ a.s.,\] \[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}P(s)ds=\frac{ L_{3}-M_{3}}{L}\ \ a.s.\]  
Proof.