Dynamics of a Stochastic Three-Species Food Web Model with Omnivory and Ratio-Dependent Functional Response

is paper is concerned with a stochastic three-species food web model with omnivory which is dened as feeding on more than one trophic level. e model involves a prey, an intermediate predator, and an omnivorous top predator. First, by the stochastic comparison theorem, we show that there is a unique global positive solution to the model. Next, we investigate the asymptotic pathwise behavior of the model. en, we conclude that the model is persistent in mean and extinct and discuss the stochastic persistence of themodel. Further, by constructing a suitable Lyapunov function, we establish sucient conditions for the existence of an ergodic stationary distribution to the model. en, we present the application of the main results in some special models. Finally, we introduce some numerical simulations to support the main results obtained. e results in this paper generalize and improve the previous related results.


Introduction
e dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [1].During the past one hundred years, there have been many investigations on predator-prey models.To the best of our knowledge, in the predator-prey interaction, the functional response plays an important role in the population dynamics, and most of the predator-prey models with the functional responses only depend on the prey.However, laboratory experiments show that the ratio-dependent response function is more reasonable in characterizing the relationship between predators and their preys [2].Arditi and Ginzburg [3] rst proposed a ratio-dependent functional response of form (αx)/ (x + βy).Kuang and Beretta [4] investigated the following ratio-dependent type predator-prey model: where x 1 (t) and x 2 (t) represent population sizes of prey and predator at time t, respectively.r, a 1 , and d 2 stand for the prey intrinsic growth rate, the intraspeci c competition rate of the prey, and the predator death rate, respectively; α 12 , β 12 , and e 12 represent the encounter rate, half capturing saturation constant, and conversion rate, respectively, that predator x 2 preys on prey x 1 .Long-term ecological research studies show that threespecies predator-prey models are fundamental building blocks of large scale ecosystems.However, it was only in the 1970s that some scholars began to study the dynamics of three-species predator-prey systems [5].In particular, Hsu et al. [6] have classified all three-species predator-prey models into five types: two predators competing for one prey, one predator acting on two preys, food chain, food chain with omnivory, and food chain with cycle.Food chain architecture and strengths of species interactions are important determinants of trophic dynamics (see [7]).It is well known that tritrophic food chain model consists of one prey, one intermediate predator, and one top predator.Note that omnivory is a widespread mechanism in interacting populations.In [6], the authors investigated the following threespecies predator-prey food chain model with an omnivory top predator: where x 1 , x 2 , and x 3 denote the number of prey, intermediate predator, and omnivorous top predator, respectively, r 1 is the growth rate of prey, r i is the death rate of species x i (i � 2, 3), a 11 is the intraspecific competition rate of prey, a 12 , a 13 , and a 23 are the capture rates, and a 21 , a 31 , and a 32 denote the efficiency of food conversion.Model (2) describes that the intermediate predator preys on only the prey and the omnivorous top predator preys on both the prey and the intermediate predator.is is a general part of marine or terrestrial food web ecological systems.Based on model (2), Namba et al. [8] considered the intraspecific competition of the intermediate predator and the intraspecific competition of the top predator.Moreover, the authors demonstrated the stabilizing role of intraspecific competition among intermediate and top predators when the growth rate of prey species is adequate to support both the predator species.Furthermore, Sen et al. [9] investigated the following three-species Lotka-Volterra model with intraguild predation and mixed functional responses: where x 1 , x 2 , and x 3 denote the number of prey, intermediate predator, and omnivorous top predator, respectively.Obviously, in [9], the authors considered Holling type-II functional response between the intermediate predator and top predator and other functional responses were assumed to be linear.All meanings of the parameters are exact to or similar as those for (2) except the following.
Here, a ii is the intraspecific competition rate of species x i (i � 2, 3) and β is the reciprocal of the half-saturation constant.
Note that the three-species food web models ( 2) and ( 3) with the functional responses only depend on prey density.However, in fact, the predator has to search and compete for food and the ratio-dependent function of the prey and the predator is more suitable to substitute for the model with complicated interaction between the prey and predator.
en, the ratio-dependent type three-species food web model with omnivory is expressed in the form: where x 1 (t) stands for the total number of prey at time As mentioned above, we notice that population models (1)-( 4) are described by the deterministic model.is is valid only at the macroscopic scale, that is, the stochastic effects can be neglected or averaged out, in view of the law of large numbers.However, in the real world, populations are actually subject to the environmental fluctuations.Generally speaking, such fluctuations could be modeled by a colored noise.It has been noted that if the colored noise is not strongly correlated, 2 Complexity then we can approximate the colored noise by a white noise _ w(t), and the approximation works quite well (see [10]).It turns out that the white noise _ w(t) is formally regarded as the derivative of a Brownian motion w(t), i.e., _ w(t) � dw(t)/dt (see [11]).As a result, the study of stochastic ecological dynamics model has already become one of the important subjects in biological mathematics.
After taking the effect of randomly fluctuating environment into account, many researchers introduced stochastic environmental variation described by the Brownian motion into parameters in the deterministic model to establish the stochastic population model (see [12][13][14][15]).Liu and Bai [12] considered the optimal harvesting problem of a stochastic logistic model with time delay.In [13][14][15], the authors investigated the dynamics of stochastic predatorprey models.Ji et al. [13] discussed a stochastic predatorprey model with modified Leslie-Gowerand Holling-type II schemes.Jovanović and Krstić [14] investigated the extinction of a stochastic predator-prey model with the Allee effect on the prey.Liu and Jiang [15] considered the periodic solution and stationary distribution of stochastic predatorprey models with higher-order perturbation.In [16], considering that fluctuations in the environment would manifest themselves mainly as fluctuations in the intrinsic growth rate of the prey population and in the death rate of the predator population (see [17]), Ji et al. supposed parameters r and d 2 in model (1) were perturbed with where w 1 (t) and w 2 (t) are mutually independent Brownian motions and σ 2 i represents the intensity of white noise _ w i (t) (i � 1, 2).Moreover, they investigated the long time behavior of the following stochastic ratio-dependent preypredator model: Based on (6), Wu et al. [18] considered the corresponding nonautonomous stochastic ratio-dependent model.Lv et al. [19] introduced the intraspecific competition of the predator population, denoted by a 2 , into model (6).Nguyen and Ta [20] considered a corresponding nonautonomous stochastic ratio-dependent prey-predator model, in which the white noise makes the effect on both the growth rates of species and the intraspecific competition coefficient of the species.
For the study of stochastic three-species models, consult [21][22][23][24][25][26] and the references therein.Geng et al. [21] investigated the stability of a stochastic one-predator-two-prey population model with time delay, while Liu et al. [22] studied the stability of a stochastic two-predator one-prey population model with time delay.In [23,24], the authors discussed the dynamical behaviors of stochastic tri-trophic food-chain models.Li et al. [23] investigated the persistence and nonpersistence of a stochastic food-chain model, while Liu and Bai [24] considered the optimal harvesting problem of a stochastic three species food-chain model.Furthermore, in [25,26], the stochastic three-species food-chain models with omnivory are discussed.Qiu and Deng [25] investigated the stationary distribution and global asymptotic stability of a stochastic food-web model with omnivory and linear functional response, while R. Liu and G. Liu [26] discussed the persistence in mean and extinction of a stochastic food-web model with intraguild predation and mixed functional responses.In [26], the authors considered Holling type-II functional response between the intermediate predator and the top predator and other functional responses were assumed to be linear.
To the best of our knowledge, so far there is no investigation on the dynamics of the stochastic three-species food web model with omnivory and ratio-dependent functional response.e purpose of this paper is to make some contribution in this direction.Recall that parameters r, d 2 , and d 3 in model (4) represent the intrinsic growth rate of the prey population, the death rate of the intermediate predator, and the death rate of the omnivorous top predator, respectively.As done in [16], in this paper, we may replace r, d 2 , and d 3 in model (4), respectively, by where _ w i (t) is the white noise and σ 2 i is the intensity of white noise _ w i (t) (i � 1, 2, 3).en, the stochastic three-species food web model with omnivory and ratio-dependent functional response took the following form: Complexity with (x 1 (0), x 2 (0), x 3 (0)) � (x 10 , x 20 , x 30 . All meanings of the parameters are exact to or similar as those for (4) except the following.Here, w � w 1 (t), w 2 (t), w 3 (t) : t ≥ 0   represents the threedimensional standard Brownian motion defined on a complete filtered probability space (Ω, F, F t   t ≥ 0 , P) satisfying the usual conditions.σ 2 i represents the intensity of noise w i (t) (i � 1, 2, 3).roughout this paper, unless otherwise specified, we would rather assume that a 1 > 0, a 2 ≥ 0, a 3 ≥ 0, α 13 ≥ 0, α 23 ≥ 0, β 13 > 0, β 23 > 0, e 13 > 0, and e 23 > 0.

Existence and Uniqueness of Positive Solution
In this section, we consider the existence of the positive solution for all times.Typically, conditions assuring the nonexplosion of the solution involve local Lipschitz continuity and a linear growth condition.In our case, we miss this last condition, so it is necessary to prove that the solution does not explode at a finite time.To prove the solution is positive and does not explode at a finite time, we use the stochastic comparison theorem.For simplicity, we introduce the following notations: Proof.Consider the following system: dX 3 (t) � − d 3 − a 3 e X 3 (t) + e 13 α 13 e X 1 (t) e X 1 (t) + β 13 e X 3 (t) + e 23 α 23 e X 2 (t) e X 2 (t) + β 23 e X 3 (t) − with initial value (X 1 (0), X 2 (0), X 3 (0)) � (ln x 10 , ln x 20 , ln x 30 ).Obviously, the coefficients of (10) are locally Lipschitz continuous.us, there is a unique maximal local solution (X 1 (t), X 2 (t), X 3 (t)) of (10) for t ∈ [0, τ e ), where τ e denotes the explosion time.Let x i (t) � e X i (t) (i � 1, 2, 3).Using Itô formula, it follows that (x 1 (t), x 2 (t), x 3 (t)) � (e X 1 (t) , e X 2 (t) , e X 3 (t) ) is the unique positive local solution of (8) with initial value (x 10 , x 20 , x 30 ) for t ∈ [0, τ e ).
erefore, for any initial value (x 10 , x 20 , x 30 ) ∈ R 3 + , model ( 8) has a unique global positive solution e proof is therefore complete.

Persistence in Mean and Extinction
In this section, we show that under some conditions, model ( 8) is persistent in mean and extinct.
Theorem 3. Suppose that a 2 > 0, a 3 > 0, and (8) Proof.For prey x 1 , from eorem 1, it follows that and ϕ 1 (t) is the solution of the following stochastic equation: with initial value x 10 > 0. Obviously, from Lemma 1, it follows that if is, together with (35), yields lim inf For intermediate predator x 2 , using Itô formula, it follows that (39) Hence, By the strong law of numbers of local martingales and eorem 2, we get lim inf For the omnivorous top predator x 3 , it follows from Itô formula that

Stochastic Permanence
In this section, we discuss the stochastic permanence of model (8).
e definition of stochastic permanence and stochastically ultimately boundness of model (8) were introduced in the literature [29,30] as follows.

Boundness.
In this subsection, we investigate the stochastically ultimate boundness of model (8) in two different ways.
Proof.For Φ 1 in system (11), applying Itô formula to Φ p 1 leads to 8 Complexity Taking the expectation on both sides of the above equation, we have (53) en, using the Höder inequality, it follows that From Lemma 2 and the comparison theorem, it follows that By a similar the discussion as in Φ ( From eorem 1, it follows that 0 < x i (t) ≤ Φ i (t) a.s.i � 1, 2, 3. en, for any p ≥ 0, we have Now eorem 5 follows immediately from the above analysis.e proof is complete.□ Theorem 6.For any (x 10 , x 20 , x 30 ) ∈ R 3 + , let (x 1 (t), x 2 (t), x 3 (t)) be the solution of model (8)  (60) Integrating it from 0 to t and taking expectation yields e proof is therefore complete.□

Stochastic Permanence.
In this section, we give some sufficient conditions to guarantee that model ( 8) is stochastically permanent.Denote By the Itô formula, we have be the solution of model ( 8) with any initial value (x 10 , x 20 , x 30 where Proof.First, integrating both sides of the first equation of (69) from 0 to t yields Taking the expectation on both sides of the above equation, we have with initial value E[u 1 (0)] � 1/x 10 .By a simple computation, we can get is, together with c 1 > 0, yields Next, integrating both sides of the second equation of system (69) from 0 to t yields (79) From ( 75), it follows that is, together with (79) yields At last, integrating both sides of the third equation of system (69) from 0 to t and taking the expectation, we have ( From the comparison theorem of stochastic differential equations, it follows that Now, Lemma 3 follows immediately from the above analysis.e proof is complete.8) is stochastically permanent.
e proof is therefore complete.□

Stationary Distribution and Ergodicity
In this section, we will show that there is an ergodic stationary distribution for the solution of (8).For the completeness of the paper, in this section, we list some theories about stationary distribution (see [32]).Let X(t) be a homogeneous Markov process in E d (denotes d-dimensional Euclidean space), described by the following stochastic differential equation: Complexity e diffusion matrix of the process X(t) is defined as J(X) � g(X)g T (X) � (a ij (X)).
According to Lemma 4, model ( 8) is ergodic and admits a unique stationary distribution.e proof is therefore complete.

Application of Main Results
In this section, we present the application of the main results in some special models.

Two Species Predator-Prey
en, the first two equations of (8) form the following closed two-population system: with initial value (x 10 , x 20 ) ∈ R 2 + . is is also the stochastic predator-prey model discussed in [19].From eorems 3 and 4, we have the following result.

Complexity 13
Remark 1.It is clear that Corollary 1 is consistent with eorems 7 and 8 in [19].Moreover, from eorems 3 and 4, the persistence in mean and extinction conditions of the three-species model ( 8) are more complicated.us, our work can be seen as the extension of [19].
For model (106), similar to the proof of eorem 6 (denote H � e 12 x 1 + x 2 ), we have where For model (109), we have the following results.

Numerical Simulations
In this section, we use the Milstein method (see [35]) to substantiate our main results.e numerical simulations of population dynamics are carried out for the academic tests with the arbitrary values of the vital rates and other parameters which do not correspond to some specific biological populations and exhibit only the theoretical properties of numerical solutions of the considered model.

Conclusions and Discussions
is paper is concerned with a stochastic three-species predator-prey food web model with omnivory and ratiodependent functional response.First, by the comparison theorem of stochastic differential equations, we prove the existence and uniqueness of global positive solution of the model.Next, we investigate an important asymptotic property of the solution, which is crucial to the study of the dynamic behavior of the model.en, under some conditions, we conclude that the model is persistent in mean and extinct.Moreover, we discuss the stochastic persistence of the model.Furthermore, by constructing a suitable Lyapunov function, we establish sufficient conditions for the 16 Complexity existence of an ergodic stationary distribution to the model.en, we present the application of the main results in some special models.Finally, some numerical simulations are introduced to support the main results.
In Section 4, we prove that there are two typical phenomena arising in accordance with the relative values of the parameters of the model.In eorem 3, we give the conditions on the parameters that informally can be stated by saying that the noise intensities σ 2 i (i � 1, 2, 3) are small compared to the other parameters, such that the species in model (8) are persistent in mean.From eorem 4, it follows that in the case that the noise intensities σ 2 i (i � 1, 2, 3) are large with respect to the other parameters, then the solution of model ( 8) tends to extinction almost surely.
Later, in Section 5, we discuss on the stochastic permanence of the solution.
is concept, which can be paraphrased by saying that the species in model (8) will survive forever, is one of the most important and interesting topics in the analysis of the model.From eorem 8, if the
Moreover, in Section 6, by constructing a suitable Lyapunov function, we show that there is an ergodic stationary distribution for the solution of model (8).In eorem 9, we give the conditions on the parameters that can be stated by saying that the intensity σ 2 i of white noise _ w i (t) is sufficiently small, such that the solution model ( 8) has an ergodic stationary distribution.
e results in this paper generalize and improve the previous related results.From Remark 1, we know that our work can be seen as the extension of [19].From Remark 2, we know that eorem 8 generalizes and improves eorem 4.11 in [18].The PDF of x 3 with initial value (600, 300, 100) The PDF of x 3 with initial value (700, 350, 150)