# Dynamics of a Stochastic Intraguild Predation Model
## 4 Stationary Distribution and Ergodicity  
In this section, we establish the stationary distribution of the stochastic IGP model (3) and show that it has the ergodic property. It is clear that the diffusion matrix of (3) is \(\Lambda(x)=\operatorname{diag}(\alpha_{r}^{2}R^{2},\alpha_{n}^{2}N^{2},\alpha _{p}^{2}P^{2})\). Let  
\[ \begin{split}\lambda_{1}&=a_{rr}-(a_{rn}+e_{rn}a_{ rn}+a_{rp}+e_{rp}a_{rp})/2,\\ \lambda_{2}&=a_{nn}-(a_{np}+e_{np}a_{np}+a_{rn}+e_{ rn}a_{rn})/2,\\ \lambda_{3}&=a_{pp}-(a_{np}+e_{np}a_{np}+a_{rp}+e_{ rp}a_{rp})/2.\end{split}  \tag{66}\]  
**Theorem 11**: _If \(\lambda_{i}>0\), \(i=1,2,3\) and \((R^{*},N^{*},P^{*})\) is the positive equilibrium point of the deterministic model (1) with_  
\[(\alpha_{r}^{2}R^{*}+\alpha_{n}^{2}N^{*}+\alpha_{p}^{2}P^{*})/2<\min\{\lambda_ {1}(R^{*})^{2},\lambda_{2}(N^{*})^{2},\lambda_{3}(P^{*})^{2}\}, \tag{67}\]_then there is a stationary distribution \(\mu(\cdot)\) for (3) and it has the ergodic property_  
\[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{1}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1, \tag{68}\] \[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}N(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{2}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1,\] \[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}P(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{3}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1.\]  
Proof. : To obtain the conclusion, we need to show that (K\({}_{1}\)) and (K\({}_{2}\)) hold. It follows from (67) that the ellipsoid  
\[-\lambda_{1}(R-R^{*})^{2}-\lambda_{2}(N-N^{*})^{2}-\lambda_{3}(P-P^{*})^{2}+ \frac{\alpha_{r}^{2}R^{*}}{2}+\frac{\alpha_{n}^{2}N^{*}}{2}+\frac{\alpha_{p}^ {2}P^{*}}{2}=0 \tag{69}\]  
lies entirely in \(\mathbb{R}^{3}_{+}\). Let \(U\) be a neighborhood of the ellipsoid with \(\overline{U}\subseteq\mathbb{R}^{3}_{+}\). It is not difficult to show that there exists a \(\rho>0\) such that  
\[\sum_{i,j=1}^{3}a_{ij}(x)\omega_{i}\omega_{j}=\alpha_{r}^{2}R^{2}\omega_{1}^{2 }+\alpha_{n}^{2}N^{2}\omega_{2}^{2}+\alpha_{p}^{2}P^{2}\omega_{3}^{2}\geq\rho |\omega|^{2} \tag{70}\]  
for \(x\in\overline{U}\) and \(\omega\in\mathbb{R}^{3}\). This implies that (K\({}_{1}\)) holds. Let  
\[V(R,N,P)=R-R^{*}-R^{*}\ln\frac{R}{R^{*}}+N-N^{*}-N^{*}\ln\frac{N}{N^{*}}+P-P^{* }-P^{*}\ln\frac{P}{P^{*}}. \tag{71}\]  
Then,  
\[dV(R,N,P) =LV(R,N,P)dt \tag{72}\] \[\quad+(R-R^{*})\alpha_{r}dB_{r}(t)+(N-N^{*})\alpha_{n}dB_{n}(t)+( P-P^{*})\alpha_{p}dB_{p}(t),\]  
where  
\[LV(R,N,P)= (R-R^{*})[r-a_{rr}R-a_{rn}N-a_{rp}P]+\alpha_{r}^{2}R^{*}/2 \tag{73}\] \[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{1}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1, \tag{68}\]0 \[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{1}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1, \tag{68}\]1  
Since \((R^{*},N^{*},P^{*})\) is the positive equilibrium point of (1), we have  
\[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{1}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1, \tag{68}\]2 \[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{1}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1, \tag{68}\]3 \[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{1}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1, \tag{68}\]4 \[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{1}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1, \tag{68}\]5 \[P\Big{\{}\lim_{t\to+\infty}\frac{1}{t}\int_{0}^{t}R(s)ds =\int_{\mathbb{R}^{3}_{+}}\omega_{1}\mu(d\omega_{1},d\omega_{2},d \omega_{3})\Big{\}} =1, \tag{68}\]6  
Then, for any \(x\in\mathbb{R}^{3}_{+}\setminus U\), we get \(LV(x)<0\), which means that (K\({}_{2}\)) holds. It follows from Lemma 6 that (3) has a stationary distribution \(\mu(\cdot)\), and it is ergodic.