# Dynamics of a Stochastic Intraguild Predation Model
## 3 Stochastic Persistence and Stochastic Extinction
\begin{split}\frac{1}{t}\ln\frac{R(t)}{R(0)}&\leq r -\frac{\alpha_{r}^{2}}{2}+2\varepsilon-a_{rr}[R(t)]-a_{rn}[N]_{*}+\frac{\alpha_ {r}B_{r}(t)}{t}\\ &\leq r-\frac{\alpha_{r}^{2}}{2}-\frac{a_{rn}(\delta_{5}-\delta_ {3})}{\delta_{1}}+2\varepsilon-a_{rr}[R(t)]+\frac{\alpha_{r}B_{r}(t)}{t}\\ &=\frac{a_{rr}(\delta_{4}-\delta_{1})}{\delta_{1}}+2\varepsilon-a _{rr}[R(t)]+\frac{\alpha_{r}B_{r}(t)}{t}.\end{split}  \tag{40}\]  
Then,  
\[[R]^{*}\leq(\delta_{4}-\delta_{1})/\delta_{1}\ \ a.s. \tag{41}\]  
On the other hand, for sufficiently large \(t\), substituting Equation (41) to the second equality of Equation (21) gives  
\[ \begin{split}\frac{1}{t}\ln\frac{N(t)}{N(0)}&\leq- d_{n}-\frac{\alpha_{n}^{2}}{2}+2\varepsilon+e_{rn}a_{rn}[R]^{*}-a_{nn}[N(t)]+ \frac{\alpha_{n}B_{n}(t)}{t}\\ &\leq-d_{n}-\frac{\alpha_{n}^{2}}{2}+2\varepsilon+\frac{e_{rn}a_{ rn}(\delta_{4}-\delta_{1})}{\delta_{1}}-a_{nn}[N(t)]+\frac{\alpha_{n}B_{n}(t)}{t}\\ &=\frac{a_{nn}(\delta_{5}-\delta_{3})}{\delta_{1}}+2\varepsilon-a _{nn}[N(t)]+\frac{\alpha_{n}B_{n}(t)}{t},\end{split}  \tag{42}\]  
which implies that  
\[[N]^{*}\leq(\delta_{5}-\delta_{3})/\delta_{1}\ \ a.s. \tag{43}\]  
Combining Equation (39) with Equation (43) gives  
\[[N]^{*}=(\delta_{5}-\delta_{3})/\delta_{1}\ \ a.s. \tag{44}\]  
It follows from Equation (43) and \(\lim_{t\to+\infty}P(t)=0\)\(a.s.\) that  
\[ \begin{split}\frac{1}{t}\ln\frac{R(t)}{R(0)}&\geq r -\frac{\alpha_{r}^{2}}{2}-2\varepsilon-a_{rr}[R(t)]-a_{rn}[N]^{*}+\frac{\alpha_ {r}B_{r}(t)}{t}\\ &\geq r-\frac{\alpha_{r}^{2}}{2}-\frac{a_{rn}(\delta_{5}-\delta_ {3})}{\delta_{1}}-2\varepsilon-a_{rr}[R(t)]+\frac{\alpha_{r}B_{r}(t)}{t}\\ &=\frac{a_{rr}(\delta_{4}-\delta_{1})}{\delta_{1}}-2\varepsilon- a_{rr}[R(t)]+\frac{\alpha_{r}B_{r}(t)}{t}.\end{split}  \tag{45}\]Then,  
\[[R]^{*}\geq(\delta_{4}-\bar{\delta}_{1})/\delta_{1}\ \ a.s. \tag{46}\]  
Combining Equation (41) with (46) gets  
\[[R]^{*}=(\delta_{4}-\bar{\delta}_{1})/\delta_{1}\ \ a.s. \tag{47}\]  
It follows from Equation (44) and (47) that (iii) holds. (iv) Similar to the arguments of (iii), it follows from Equation (36) that \(\lim\limits_{t\to+\infty}N(t)=0\) _a.s._ if \(L_{2}<M_{2}\). The first equality of Equation (21) multiplied by \(e_{rp}a_{rp}\) plus the third equality of Equation (21) multiplied by \(a_{rp}\) gives  
\[ \begin{split} a_{rr}\frac{1}{t}\ln\frac{P(t)}{P(0)}& =-e_{rn}a_{m}\frac{1}{t}\ln\frac{R(t)}{R(0)}+\delta_{6}-\bar{ \delta}_{4}-\delta_{2}[P(t)]-(a_{m}e_{rp}a_{rp}-a_{rp}e_{np}a_{np})[N(t)]\\ &\quad+\frac{a_{rr}a_{p}B_{p}(t)+e_{rp}a_{rp}a_{rp}B_{r}(t)}{t}\\ &\geq\delta_{6}-\bar{\delta}_{4}-2\varepsilon-\delta_{2}[P(t)]+ \frac{a_{rr}a_{p}B_{p}(t)+e_{rp}a_{rp}a_{rp}B_{r}(t)}{t}\end{split}  \tag{48}\]  
for sufficiently large \(t\). By Lemma 1 and the arbitrariness of \(\varepsilon\), we have  
\[[P]_{*}\geq(\delta_{6}-\bar{\delta}_{4})/\delta_{2}\ \ a.s. \tag{49}\]  
This implies that  
\[ \begin{split}\frac{1}{t}\ln\frac{R(t)}{R(0)}& \leq r-\frac{\alpha_{r}^{2}}{2}+2\varepsilon-a_{rr}[R(t)]-a_{rp}[P ]_{*}+\frac{\alpha_{r}B_{r}(t)}{t}\\ &\leq r-\frac{\alpha_{r}^{2}}{2}-\frac{a_{rp}(\delta_{6}-\bar{ \delta}_{4})}{\delta_{2}}+2\varepsilon-a_{rr}[R(t)]+\frac{\alpha_{r}B_{r}(t)}{ t}\\ &=\frac{a_{rr}(\delta_{3}-\bar{\delta}_{2})}{\delta_{2}}+2 \varepsilon-a_{rr}[R(t)]+\frac{\alpha_{r}B_{r}(t)}{t}\end{split}  \tag{50}\]  
for sufficiently large \(t\). From Lemma 1, we get  
\[[R]^{*}\leq(\delta_{3}-\bar{\delta}_{2})/\delta_{2}\ \ a.s. \tag{51}\]  
It follows from \(\lim\limits_{t\to+\infty}N(t)=0\) _a.s._ and Equation (49) that  
\[ \begin{split}\frac{1}{t}\ln\frac{P(t)}{P(0)}& \leq-d_{p}-\frac{\alpha_{p}^{2}}{2}+2\varepsilon+e_{rp}a_{rp}[R]^{*}-a_{ pp}[P(t)]+\frac{\alpha_{p}B_{p}(t)}{t}\\ &\leq-d_{p}-\frac{\alpha_{p}^{2}}{2}+\frac{e_{rp}a_{rp}(\delta_{ 3}-\bar{\delta}_{2})}{\delta_{2}}+2\varepsilon-a_{pp}[P(t)]+\frac{\alpha_{p}B _{p}(t)}{t}\\ &=\frac{a_{pp}(\delta_{6}-\bar{\delta}_{4})}{\delta_{2}}+2 \varepsilon-a_{pp}[P(t)]+\frac{\alpha_{p}B_{p}(t)}{t}\end{split}  \tag{52}\]  
for sufficiently large \(t\).