# Dynamics of a Stochastic Intraguild Predation Model
## 2 Preliminaries  
In this section, we first introduce several important lemmas. **Lemma 1** (see [32]). : _Let \(z\in C(\Omega\times[0,+\infty),\mathbb{R}_{+})\), \([z]^{*}=\limsup_{t\to+\infty}\frac{1}{t}\int_{0}^{t}z(s)ds\) and \([z]_{*}=\liminf_{t\to+\infty}\frac{1}{t}\int_{0}^{t}z(s)ds\)._  
(i) _If there exist two positive constants_ \(T\) _and_ \(\lambda_{0}\) _such that_  
\[\ln z(t)\leq\lambda t-\lambda_{0}\int_{0}^{t}z(s)ds+\sum_{i=1}^{n}\sigma_{i}B _{i}(t) \tag{4}\]  
_for all_ \(t\geq T\)_, where_ \(B_{i}(t)\)_,_ \(1\leq i\leq n\)_, are independent standard Brownian motions and_ \(\sigma_{i}\)_,_ \(1\leq i\leq n\)_, are constants, then_ \([z]^{*}\leq\lambda/\lambda_{0}\) _a.s. if_ \(\lambda\geq 0\) _or_ \(\lim_{t\to+\infty}z(t)=0\) _a.s. if_ \(\lambda<0\)_._  
(ii) _If there exist three positive constants_ \(T\)_,_ \(\lambda\)_, and_ \(\lambda_{0}\) _such that_  
\[\ln z(t)\geq\lambda t-\lambda_{0}\int_{0}^{t}z(s)ds+\sum_{i=1}^{n}\sigma_{i}B _{i}(t) \tag{5}\]  
_for all_ \(t\geq T\)_, where_ \(B_{i}(t)\)_,_ \(1\leq i\leq n\)_, are independent standard Brownian motions and_ \(\sigma_{i}\)_,_ \(1\leq i\leq n\)_, are constants, then_ \([z]_{*}\geq\frac{\lambda}{\lambda_{0}}\) _a.s._  
Similar to Theorem 2.1, Lemma 3.1 and Lemma 3.4 in [25], we have the following lemma:  
**Lemma 2**. : _For any given initial value \((R(0),N(0),P(0))^{T}\in\mathbb{R}^{3}_{+}\) and any \(p>0\), model (3) has a unique solution \((R(t),N(t),P(t))^{T}\) on \(t\geq 0\) which will remain in \(\mathbb{R}^{3}_{+}\) with probability 1 and there is a constant \(K=K(p)\) such that_  
\[\limsup_{t\to+\infty}E(R(t)^{p})\leq K,\ \limsup_{t\to+\infty}E(N(t)^{p}) \leq K,\ \limsup_{t\to+\infty}E(P(t)^{p})\leq K. \tag{6}\]  
_Moreover, the solution \((R(t),N(t),P(t))^{T}\) of (3) has the properties that_  
\[\limsup_{t\to+\infty}\frac{\ln R(t)}{\ln t}\leq 1\ a.s.,\ \limsup_{t\to+\infty} \frac{\ln N(t)}{\ln t}\leq 1\ a.s.,\ \limsup_{t\to+\infty}\frac{\ln P(t)}{\ln t}\leq 1\ a.s. \tag{7}\]  
In order to obtain the conditions of global asymptotic stability of solutions for the stochastic model (3), we need the following two lemmas. **Lemma 3** (see [35]). : _If there exist positive constants \(\omega_{1},\omega_{2}\) and \(\kappa\) such that an n-dimensional stochastic process \(Y(t)\), \(t\geq 0\) satisfies_  
\[E|Y(t)-Y(s)|^{\omega_{1}}\leq\kappa|t-s|^{1+\omega_{2}} \tag{8}\]_for \(0\leq t,s<+\infty\), then there exists a continuous modification \(\overline{Y}(t)\) of \(Y(t)\) such that for every \(\omega\in(0,\omega_{1}/\omega_{2})\) there is a positive random variable \(h(\omega)\) such that_  
\[P\left\{\sup_{ \begin{subarray}{c}0<|t-s|<h(\omega)\\ 0\leq t<+\infty\end{subarray} }\frac{|\overline{Y}(t,\omega)-Y(s,\omega)|}{|t- s|^{\omega}}\leq\frac{2}{1-2^{-\omega}}\right\}=1, \tag{9}\]  
_which implies that almost every sample path of \(\overline{Y}(t)\) is locally but uniformly Holder continuous with exponent \(\omega\)._  
**Lemma 4** (see [36]). : _If \(g\) is a non-negative function defined on \([0,+\infty)\) such that \(g\) is integrable and is uniformly continuous, then \(\lim_{t\to+\infty}g(t)=0\)._  
To establish the existence of a stationary distribution of model (3) in Section 4, we introduce the theory of [PERSON] [34] and let \(Y(t)\) be a homogeneous Markov process in \(E^{l}\) (\(E^{l}\) is an \(l\)-dimensional Euclidean space) described by the stochastic equation  
\[dY(t)=b(Y)dt+\sum_{m=1}^{k}g_{m}(Y)dB_{m}(t). \tag{10}\]  
Let the diffusion matrix be \(\Lambda(x)=(a_{ij}(x))\), \(a_{ij}(x)=\sum_{m=1}^{k}g_{m}^{i}(x)g_{m}^{j}(x)\). **Assumption 5**.