# Dynamics of a Stochastic Intraguild Predation Model
## 5 Conclusions  
In this section, we first focus on the stochastic food chains model and the stochastic exploitative competition model. In the model (3), if we let \(a_{rp}=0\) or \(a_{np}=0\), then we get the stochastic food chains model  
\[ \begin{split} dR(t)&=R(t)(r-a_{rr}R(t)-a_{rn}N(t)) dt+\alpha_{r}R(t)dB_{r}(t),\\ dN(t)&=N(t)(-d_{n}+e_{m}a_{rn}R(t)-a_{nn}N(t)-a_{ np}P(t))dt+\alpha_{n}N(t)dB_{n}(t),\\ dP(t)&=P(t)(-d_{p}+e_{np}a_{np}N(t)-a_{pp}P(t)) dt+\alpha_{p}P(t)dB_{p}(t),\end{split}  \tag{77}\]  
and the stochastic exploitative competition model  
\[ \begin{split} dR(t)&=R(t)(r-a_{rr}R(t)-a_{rn}N(t)- a_{rp}P(t))dt+\alpha_{r}R(t)dB_{r}(t),\\ dN(t)&=N(t)(-d_{n}+e_{m}a_{rn}R(t)-a_{nn}N(t)) dt+\alpha_{n}N(t)dB_{n}(t),\\ dP(t)&=P(t)(-d_{p}+e_{rp}a_{rp}R(t)-a_{pp}P(t)) dt+\alpha_{p}P(t)dB_{p}(t).\end{split}  \tag{78}\]  
In view of the stochastic IGP model (3), Theorems 8, 10, 11 reduce the corresponding results of models (77) and (78), that is, we get the stochastic persistence and stochastic extinction, stationary distribution and ergodicity, and globally asymptotically stability of the positive solution for the stochastic food chains model (77), and the stochastic exploitative competition model (78), in the case of \(a_{rp}=0\) or \(a_{np}=0\). In this paper, we have developed a stochastic IGP model (3) describing the interactions among a top predator (IG predator \(P\)), an intermediate consumer (IG prey \(N\)), and a shared prey (\(R\)) under the influence of environmental noise. We have analyzed the dynamic properties for the stochastic IGP model (3) and the deterministic IGP model (1). As applications, we show that our results may be extended to two well-known biological systems: food chains and exploitative competition. Comparing the stochastic IGP model (3) with the deterministic IGP model (1) (see Theorems 8, 10, 11 and Table 1), we obtain the following conclusions:  
* In the deterministic model (1), the total extinction of three populations is impossible since \(E_{0}\) is unstable. However, this situation is possible for the stochastic model (3) when the noise intensity \(a_{r}\) is large enough (see Figure 1a);
* The existence of the shared prey with the extinction of both IG prey and IG predators is a possible outcome of the stochastic model (3) (see Figure 1b). There is also evidence that the noise is a harmful factor for the shared prey population (see \(E_{r}\) of Table 1 and (ii) of Theorem 8);
* The existence of both the shared prey and IG prey with the extinction of IG predators, and the existence of both the shared prey and IG predators with the extinction of IG prey are both possible outcomes of the stochastic model (3) with different sets of parameters (see Figure 1c,d). Here, it is worth noting that the noise has a negative effect for IG prey and IG predators, and may also have a positive effect for the shared prey if the values of \(a_{n}\) and \(a_{p}\) grow larger (see (iii) and (iv) of Theorem 8). This also implies that stochastic fluctuation of \(N\) or \(P\) would help \(R\) to grow larger;
* This study suggests that the shared prey, IG prey and IG predators can coexist together for the stochastic model (3), which implies that it is possible for the coexistence of three species under the influence of environmental noise (see Figure 1e). There is recognition that the noise may be favorable to three-species coexistence if \(M_{i}<0\), \(i=1,2,3\) (see (v) of Theorem 8). In addition, we also prove that three-species is stable coexistence for the influence of environmental noise (see Theorem 10 and Figure 11);
* The study of Theorem 11 suggests that the time average of the population size of model (3) with the development of time is equal to the stationary distribution in space. **Acknowledgments:** The authors are grateful to the anonymous referees for carefully reading the manuscript and for important suggestions and comments, which led to the improvement of their manuscript.