Dynamics of a Stochastic Intraguild Predation Model

Zejing Xing 1,2,†, Hongtao Cui 2,† and Jimin Zhang 2,*,† 1 College of Automation, Harbin Engineering University, Harbin, Heilongjiang 150001, China; xingzejing@hlju.edu.cn 2 School of Mathematical Sciences, Heilongjiang University, 74 Xuefu Street, Harbin, Heilongjiang 150080, China; hongtaocui1989@163.com * Correspondence: zhangjimin@hlju.edu.cn; Tel.: +86-0451-8661-3762 † These authors contributed equally to this work.


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 Holt and Polis [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 numerical responses [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, Liu and Wang [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 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.Let (B r (t), B n (t), B p (t)) T be a three-dimensional Brownian motion defined on a complete probability space (Ω, F , P ) and where α 2 r , α 2 n , α 2 p are the intensity of the white noise.Thus we consider the Itô's stochastic IGP model as follows: The main aim of this paper is to study the dynamics of the model (3).Theoretical studies have suggested that it is very difficult to a achieve stable three-species steady state for the deterministic IGP model.Hence, the first interesting topic of the present paper is whether we can establish a criterion for three-species coexistence under the influence of environmental noise and give the sufficient conditions for global asymptotic stability of the positive solution of model (3).Another important and interesting problem is whether there is a stationary distribution of the stochastic IGP model (3) and if it has the ergodic property.
The rest of the paper is organized as follows.In the next section, we do some necessary preparations including some notations and several important lemmas.In Section 3, we explore stochastic persistence and the extinction of model (3) for five different cases and compare them with the corresponding results of the deterministic model (1).
Then, we establish global asymptotic stability of the positive solution of the model (3).In Section 4, we prove that there is a stationary distribution of model (3), and it has the ergodic property by using the theory of Has'minskii [34].In the final section, according to the conclusions of previous sections, we first study dynamic properties of two well-known biological systems under random perturbations: food chains and exploitative competition.We state biological implications of our mathematical findings and present some figures to illustrate or complement our mathematical findings.

Preliminaries
In this section, we first introduce several important lemmas.

(i)
If there exist two positive constants T and λ 0 such that for all t ≥ T, where B i (t), 1 ≤ i ≤ n, are independent standard Brownian motions and σ i , (ii) If there exist three positive constants T, λ, and λ 0 such that for all t ≥ T, where B i (t), 1 ≤ i ≤ n, are independent standard Browniam motions and σ i , 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 ∈ R 3 + and any p > 0, model (3) has a unique solution (R(t), N(t), P(t)) T on t ≥ 0 which will remain in R 3 + with probability 1 and there is a constant Moreover, the solution (R(t), N(t), P(t)) T of (3) has the properties that 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 ω 1 , ω 2 and κ such that an n-dimensional stochastic process Y(t), t ≥ 0 satisfies for 0 ≤ t, s < +∞, then there exists a continuous modification Y(t) of Y(t) such that for every ω ∈ (0, ω 1 /ω 2 ) there is a positive random variable h(ω) such that which implies that almost every sample path of Y(t) is locally but uniformly Hölder continuous with exponent ω.
To establish the existence of a stationary distribution of model ( 3) in Section 4, we introduce the theory of Has'minskii [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 Let the diffusion matrix be Λ

x).
Assumption 5.There is a bounded domain U ⊂ E l with regular boundary Γ such that In the domain U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix Λ(x) is bounded away from zero; If x ∈ E l \ U, the mean time τ at which a path issuing from x reaches the set U is finite, and sup x∈K E x τ < +∞ for every compact subset K ⊂ E l .
It is worth noting that we can use the following two stronger conditions to verify (H 1 ) and (H 2 ) in Assumption 5: To obtain (H 1 ), we only need to show that T is uniformly elliptical in U, where [37,38] To obtain (H 2 ), we only need to prove that there exist a neighborhood U and a nonnegative C 2 -function V(x) such that for any x ∈ E l \ U, LV(x) < 0 (see [39]).

Lemma 6 ([34]). If Assumption 5 holds, then the Markov process Y(t) has a stationary distribution µ(•).
Moreover, if f (•) is a function integrable with respect to the measure µ, then In order to study dynamic properties of model (3), we do the following notations: [g(t)] =

Stochastic Persistence and Stochastic Extinction
To illuminate the effect of the stochastic perturbations for population and compare the stochastic IGP model (3) with the deterministic IGP model (1), we first explore the existence and local stability of boundary and positive equilibria for model (1).The summary of conditions for the existence and local stability of equilibria are listed in Table 1.
Table 1.Existence and local stability of equilibria for model (1).

Equilibria Existence
Local Stability

Now, we analyze stochastic persistence and stochastic extinction of model (3).
Definition 7 (see [32]).Species x(t) is said to be persistent in the mean if A direct calculation gives 2r/α 2 r − δ 5 / δ3 = a rr (rα Theorem 8.The following five cases hold: Proof.It follows from Itô's formula that By integrating from 0 to t on both sides of the above equation and dividing by t, we have (i) It follows from the first equality of Equation ( 21) that By Lemma 1, we have lim t→+∞ R(t) = 0 a.s.
(ii) It follows from Equation (22) and Lemma 1 that Combining the second equality of Equation ( 21) with Equation ( 26) gives for sufficiently large t.Then lim t→+∞ N(t) = 0 a.s.
Combining case 1 with case 2 gives lim t→+∞ P(t) = 0 a.s.The first equality of Equation (21) multiplied by e rn a rn plus the second equality of Equation ( 21 By applying the above inequality and lim t→+∞ P(t) = 0 a.s.into the first equality of Equation ( 21), we get On the other hand, for sufficiently large t, substituting Equation (41) to the second equality of Equation ( 21) gives Combining Equation (39) with Equation (43) gives It follows from Equation (43) and lim t→+∞ P(t) = 0 a.s. that Combining Equation ( 41) with (46) gets It follows from Equation ( 44) and ( 47) that (iii) holds.
(iv) Similar to the arguments of (iii), it follows from Equation ( 36) that lim t→+∞ N(t) = 0 a.s.if for sufficiently large t.By Lemma 1 and the arbitrariness of ε, we have This implies that for sufficiently large t.From Lemma 1, we get It follows from lim t→+∞ N(t) = 0 a.s. and Equation (49) that for sufficiently large t.Then, Using Equation (53), we have for sufficiently large t.Hence, It follows from Equations ( 49)-( 53) and Equation ( 55) that (iv) holds.
(v) By using Equation (37), we obtain since L 3 > M 3 .For sufficiently large t, it follows from (36) and Equation ( 56) that which means Similarly, we have and for sufficiently large t.Then, By Equation ( 33), ( 36), ( 56), ( 58) and ( 61), (v) holds.The proof of the theorem is complete.Now, we establish the sufficient criteria for global asymptotic stability of the positive solutions for the stochastic model ( 3).This stochastic model ( 3) is said to be globally asymptotically stable By the ergodic property, we have Then, R 3 + ω 1 µ(dω 1 , dω 2 , dω 3 ) ≤ K as m → +∞.By Lemma 6, the first equality of Equation (68) holds.Similarly, we can conclude that the second and third equalities of Equation (68) hold.The proof of the theorem is completed.

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 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.

•
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 α 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 α n and α 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 1f); 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.
= (a rn a pp + a rp e np a np )/(a nn a pp + e np a 2 np ), µ 2 = −(a rn a np − a rp a nn )/(a nn a pp + e np a 2 np ).
r (t) t for sufficiently large t and [R] * ≥ r ) multiplied by a rr gives − δ3 − δ 1 [N(t)] − (e rn a rn a rp + a rr a np )[P(t)] + e rn a rn α r B r (t) + a rr α n B n (t) t ≥ δ 5 − δ3 − 2ε − δ 1 [N(t)] + e rn a rn α r B r (t) + a rr α n B n (t) t (21)first equality of Equation(21)multiplied by e rp a rp plus the third equality of Equation(21)multiplied by a rr gives − δ4 − δ 2 [P(t)] − (a rn e rp a rp − a rr e np a np )[N(t)] + a rr α p B p (t) + e rp a rp α r B r (t) t ≥ δ 6 − δ4 − 2ε − δ 2 [P(t)] + a rr α p B p (t) + e rp a rp α r B r (t) t dB r (t), dN(t) = N(t)(−d n + e rn a rn R(t) − a nn N(t) − a np P(t))dt + α n N(t)dB n (t), dP(t) = P(t)(−d p + e np a np N(t) − a pp P(t))dt + α p P(t)dB p (t),