# Dynamics of a Stochastic Intraguild Predation Model
## 3 Stochastic Persistence and Stochastic Extinction
Then,  
\[[P]^{*}\leq(\delta_{6}-\bar{\delta}_{4})/\delta_{2}\ \ a.s. \tag{53}\]Using Equation (53), we have  
\[ \begin{split}\frac{1}{t}\ln\frac{R(t)}{R(0)}&\geq r- \frac{\alpha_{r}^{2}}{2}-2\varepsilon-a_{rr}[R(t)]-a_{rp}[P]^{*}+\frac{\alpha_{r} B_{r}(t)}{t}\\ &\geq r-\frac{\alpha_{r}^{2}}{2}-\frac{a_{rp}(\delta_{6}-\bar{ \delta}_{4})}{\delta_{2}}-2\varepsilon-a_{rr}[R(t)]+\frac{\alpha_{r}B_{r}(t)}{t} \\ &=\frac{a_{rr}(\delta_{3}-\bar{\delta}_{2})}{\delta_{2}}-2 \varepsilon-a_{rr}[R(t)]+\frac{\alpha_{r}B_{r}(t)}{t}\end{split}  \tag{54}\]  
for sufficiently large \(t\). Hence,  
\[[R]_{*}\geq(\delta_{3}-\bar{\delta}_{2})/\delta_{2}\;\;a.s. \tag{55}\]  
It follows from Equations (49)-(53) and Equation (55) that (iv) holds. (v) By using Equation (37), we obtain  
\[[P]^{*}\leq(L_{3}-M_{3})/L \tag{56}\]  
since \(L_{3}>M_{3}\). For sufficiently large \(t\), it follows from (36) and Equation (56) that  
\[ \begin{split}\frac{1}{t}\ln\frac{R(t)}{R(0)}&\geq r- \frac{\alpha_{r}^{2}}{2}-2\varepsilon-a_{rr}[R(t)]-a_{rn}[N]^{*}-a_{rp}[P]^{*}+ \frac{\alpha_{r}B_{r}(t)}{t}\\ &\geq r-\frac{\alpha_{r}^{2}}{2}-2\varepsilon-a_{rr}[R(t)]-\frac {a_{rn}(L_{2}-M_{2})}{L}-\frac{a_{rp}(L_{3}-M_{3})}{L}+\frac{\alpha_{r}B_{r}(t )}{t}\\ &=\frac{a_{rr}(L_{1}-M_{1})}{L}-2\varepsilon-a_{rr}[R(t)]+\frac{ \alpha_{r}B_{r}(t)}{t},\end{split}  \tag{57}\]  
which means  
\[[R]_{*}\geq(L_{1}-M_{1})/L. \tag{58}\]  
Similarly, we have  
\[ \begin{split}\frac{1}{t}\ln\frac{N(t)}{N(0)}&\geq- d_{n}-\frac{\alpha_{n}^{2}}{2}-2\varepsilon+e_{rn}a_{m}[R]_{*}-a_{m}[N(t)]-a_{ np}[P]^{*}+\frac{\alpha_{n}B_{n}(t)}{t}\\ &\geq-d_{n}-\frac{\alpha_{n}^{2}}{2}-2\varepsilon+\frac{e_{rn}a_{ m}(L_{1}-M_{1})}{L}-a_{nn}[N(t)]-\frac{a_{np}(L_{3}-M_{3})}{L}+\frac{\alpha_{n}B_{n} (t)}{t}\\ &=\frac{a_{nn}(L_{2}-M_{2})}{L}-2\varepsilon-a_{nn}[N(t)]+\frac{ \alpha_{n}B_{n}(t)}{t}\end{split}  \tag{59}\]  
and  
\[ \begin{split}\frac{1}{t}\ln\frac{P(t)}{P(0)}&\geq- d_{p}-\frac{\alpha_{p}^{2}}{2}-2\varepsilon+e_{rp}a_{rp}[R]_{*}+e_{np}a_{np}[N]_{*}-a_{ pp}[P(t)]+\frac{\alpha_{p}B_{p}(t)}{t}\\ &\geq-d_{p}-\frac{\alpha_{p}^{2}}{2}-2\varepsilon+\frac{e_{rp}a_{ rp}(L_{1}-M_{1})}{L}+\frac{e_{np}a_{np}(L_{2}-M_{2})}{L}-a_{pp}[P(t)]+\frac{ \alpha_{p}B_{p}(t)}{t}\\ &=\frac{a_{pp}(L_{3}-M_{3})}{L}-2\varepsilon-a_{pp}[P(t)]+\frac{ \alpha_{p}B_{p}(t)}{t}\end{split}  \tag{60}\]  
for sufficiently large \(t\). Then,  
\[[N]_{*}\geq(L_{2}-M_{2})/L,\;\;[P]_{*}\geq(L_{3}-M_{3})/L. \tag{61}\]  
By Equation (33), (36), (56), (58) and (61), (v) holds. The proof of the theorem is complete. Now, we establish the sufficient criteria for global asymptotic stability of the positive solutions for the stochastic model (3). This stochastic model (3) is said to be globally asymptotically stable (or globally attractive) if \(\lim\limits_{t\to\infty}\max\{|R_{1}(t)-R_{2}(t)|,|N_{1}(t)-N_{2}(t)|,|P_{1}(t)-P_{ 2}(t)|\}=0\), where \((R_{i}(t),N_{i}(t),P_{i}(t))\), \(i=1,2\) are two arbitrary solutions of (3) with initial values \((R_{i}(0),N_{i}(0),P_{i}(0))\in\mathbb{R}_{+}^{3}\), \(i=1,2\). By Lemma 3, similar to arguments as those of Lemma 15 in [32], we have the following lemma. **Lemma 9**: _If \((R(t),N(t),P(t))\) is a positive solution of (3), then almost every sample path of \(R(t),N(t)\) and \(P(t)\) are uniformly continuous._  
**Theorem 10**: _If there exist positive constants \(\delta_{1},\delta_{2}\) and \(\delta_{3}\) such that_  
\[\delta_{1}a_{rr}\geq\delta_{2}e_{rn}a_{m}+\delta_{3}e_{rp}a_{rp},\ \ \delta_{2}a_{nm}\geq\delta_{1}a_{m}+\delta_{3}e_{np}a_{np},\ \ \delta_{3}a_{pp}\geq\delta_{1}a_{rp}+\delta_{2}a_{np}, \tag{62}\]  
_then (3) is globally asymptotically stable._  
We let  
\[V(t)=\delta_{1}|\ln R_{1}(t)-\ln R_{2}(t)|+\delta_{2}|\ln N_{1}(t)-\ln N_{2}(t )|+\delta_{3}|\ln P_{1}(t)-\ln P_{2}(t)| \tag{63}\]  
for \(t\geq 0\), where \((R_{i}(t),N_{i}(t),P_{i}(t))\), \(i=1,2\) are two arbitrary solutions of (3) with initial values \((R_{i}(0),N_{i}(0),P_{i}(0))\in\mathbb{R}_{+}^{3}\), \(i=1,2\). A direct calculation gives  
\[