# Dynamics of a Stochastic Intraguild Predation Model
## 3 Stochastic Persistence and Stochastic Extinction
: It follows from [PERSON]'s formula that  
\[d\ln R = (r-\alpha_{r}^{2}/2-a_{rr}R(t)-a_{rn}N(t)-a_{rp}P(t))dt+\alpha_{r }dB_{r}(t),\] \[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]0 \[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]1  
By integrating from \(0\) to \(t\) on both sides of the above equation and dividing by \(t\), we have  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]2 \[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]3 \[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]4  
(i) It follows from the first equality of Equation (21) that  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]5  
By Lemma 1, we have  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]6  
since \(2r<\alpha_{r}^{2}\) holds. Substituting Equation (23) into the second equality of Equation (21) yields  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]7for sufficiently large \(t\) and sufficiently small \(\varepsilon\) such that \(-d_{n}-\alpha_{n}^{2}/2+\varepsilon<0\). Applying Lemma 1 to Equation (24), we get  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]8  
Similarly, in view of the third equality of Equations (21), (23), (25) and Lemma 1, we can conclude that \(\lim_{t\to+\infty}P(t)=0\;a.s.\) This implies that (i) of Theorem 8 holds. (ii) It follows from Equation (22) and Lemma 1 that  
\[\lim_{t\rightarrow+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds=\frac{r-\alpha_{r}^{2} /2}{a_{rr}}\ \ a.s. \tag{16}\]9  
Combining the second equality of Equation (21) with Equation (26) gives  
\[ \begin{split}\frac{1}{t}\ln\frac{N(t)}{N(0)}&\leq -d_{n}-\frac{\alpha_{n}^{2}}{2}+e_{rn}a_{rn}[R]^{*}+\varepsilon-a_{nn}[N(t)]+ \frac{\alpha_{n}B_{n}(t)}{t}\\ &\leq\frac{\delta_{5}-\bar{\delta}_{3}}{a_{rr}}+\varepsilon-a_{nn }[N(t)]+\frac{\alpha_{n}B_{n}(t)}{t}\end{split}  \tag{27}\]  
for sufficiently large \(t\). Then  
\[\lim_{t\to+\infty}N(t)=0\;\;a.s. \tag{28}\]  
if \(\delta_{5}/\bar{\delta}_{3}<1\) and \(\varepsilon\) is sufficiently small such that \(\delta_{5}-\bar{\delta}_{3}+a_{rr}\varepsilon<0\). It follows from the third equality of Equation (21), (26), (28) and Lemma 1 that  
\[\lim_{t\to+\infty}P(t)=0\;\;a.s. \tag{29}\]  
since \(\delta_{6}/\bar{\delta}_{4}<1\). From Equation (28) and (29) and Lemma 1, we obtain  
\[\frac{1}{t}\ln\frac{R(t)}{R(0)}\geq r-\frac{\alpha_{r}^{2}}{2}-a_{rr}[R(t)]- \varepsilon+\frac{\alpha_{r}B_{r}(t)}{t}\]  
for sufficiently large \(t\) and  
\[[R]_{*}\geq\frac{r-\alpha_{r}^{2}/2}{a_{rr}}\;\;a.s. \tag{30}\]  
Combining Equation (26) with Equation (30) implies that (ii) holds. (iii) Let  
\[\mu_{1}=(a_{m}a_{pp}+a_{rp}e_{np}a_{np})/(a_{nn}a_{pp}+e_{np}a_{np}^{2}),\;\mu_ {2}=-(a_{rn}a_{np}-a_{rp}a_{nm})/(a_{mn}a_{pp}+e_{np}a_{np}^{2}).\]  
A direct calculation gives \(a_{mn}\mu_{1}-e_{np}a_{np}\mu_{2}-a_{rn}=0\) and \(a_{np}\mu_{1}+a_{pp}\mu_{2}-a_{rp}=0\). Multiplying both sides of three equalities of Equation (21) by \(-1\), \(\mu_{1}\) and \(\mu_{2}\), respectively, and then adding these three equalities, we have  
\[ \begin{split}\frac{1}{t}\ln\frac{R(t)}{R(0)}=&\frac {\mu_{1}}{t}\ln\frac{N(t)}{N(0)}+\frac{\mu_{2}}{t}\ln\frac{P(t)}{P(0)}+\frac {L_{1}-M_{1}}{a_{nn}a_{pp}+e_{np}a_{np}^{2}}-\frac{L}{a_{nn}a_{pp}+e_{np}a_{np} ^{2}}[R(t)]\\ &+\frac{\alpha_{r}B_{1}(t)-\mu_{1}\alpha_{n}B_{2}(t)-\mu_{2} \alpha_{p}B_{3}(t)}{t}.\end{split}