# Dynamics of a Stochastic Intraguild Predation Model
## 1 Introduction  
Interactions among species can structure biological communities by affecting the identity, number and abundance of species present. Intraguild predation (IGP) has been playing an important role in structuring ecological communities, strongly influencing the structure and function of food webs. IGP describes an interaction in which one predator species consumes another predator species with whom it also competes for shared prey [1, 2]. This suggests that IGP combines two important structuring forces in ecological communities: competition and predation. Accordingly, IGP is not only a taxonomically widespread interaction within communities which can occur at different trophic levels, but also a central force to forecast the stability of food webs and the maintenance of biodiversity. The simplest form of IGP is depicted by a simple food web model in which IGP can occur: a top predator (IG predator \(P\)), an intermediate consumer (IG prey \(N\)), and a shared prey (\(R\)). The development of IGP model can be traced back to [PERSON] and [PERSON] [1] who initially introduce a three species model with the Lotka-Volterra type to study the species coexistence of IGP and point out that it is very difficult to achieve a stable three-species steady state. After that, there are some articles to consider an IGP model with different structures and forms, such as, IGP model with the Lotka-Volterra type [3, 4, 5], the IGP model with special forms of the functional and numericalresponses [6; 7; 8], the IGP model with prey switching or adaptive prey behavior [9; 10], and the IGP model with generalist predator or time delay [11; 12; 13]. The effect of the random variation of environment is an integral part of any realistic ecosystem. Stochastic models may be more important in characterizing population dynamics in contrast to the deterministic models. In essence, random factors can lead to complete extinction of populations even if the population size is relatively large. Previous studies have explored the dynamic properties for stochastic single species models [14; 15; 16], stochastic predator-prey models [17; 18; 19; 20; 21; 22; 23], stochastic competitive models [24; 25; 26; 27], stochastic mutualism model [28; 29; 30; 31]. Specially, [PERSON] and [PERSON] [32] investigated a two-prey one-predator model with random perturbations. However, there are few studies to investigate dynamics of a stochastic IGP model. Motivated by the existing nice studies and the above considerations, we consider a following IGP model with the Lotka-Volterra type  
\[\frac{dR(t)}{dt} =R(t)(r-a_{rr}R(t)-a_{rn}N(t)-a_{rp}P(t)), \tag{1}\] \[\frac{dN(t)}{dt} =N(t)(-d_{n}+e_{rn}a_{rn}R(t)-a_{nn}N(t)-a_{np}P(t)),\] \[\frac{dP(t)}{dt} =P(t)(-d_{p}+e_{rp}a_{rp}R(t)+e_{np}a_{np}N(t)-a_{pp}P(t)),\]  
where \(R(t),N(t)\) and \(P(t)\) are the densities of the shared prey, IG prey and IG predator, respectively; \(r\) is the per capita growth rate of the shared prey and \(d_{i}(i=n,p)\) is the death rate of species \(i\); \(a_{ii}(i=r,n,p)\) is the intraspecific competition rate of species \(i\); \(a_{rp}\) and \(a_{np}\) are the predation rates of IG predator to the shared prey and IG prey; \(a_{rn}\) is the predation rate of IG prey to the shared prey; \(e_{ij}(i=r,n,j=n,p)\) is the conversion rates of resource consumption into reproduction for IG prey and IG predator. Here, \(a_{rn},a_{rp},a_{np}\) is nonnegative constants and the remaining parameters are all positive constants. In view of the fact that the per capita growth rate and the death rate exhibit random fluctuation to a greater or lesser extent (see [33]), we assume that the environmental fluctuation mainly affects the parameters \(r\), \(d_{n}\) and \(d_{p}\) and model these fluctuations by means of independent Gaussian white noises.