Abstract
This paper is concerned with a stochastic three-species food web model with omnivory which is defined as feeding on more than one trophic level. The 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. Then, we conclude that the model is persistent in mean and extinct and discuss the stochastic persistence of the model. Further, by constructing a suitable Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Then, we present the application of the main results in some special models. Finally, we introduce some numerical simulations to support the main results obtained. The results in this paper generalize and improve the previous related results.
1. Introduction
The 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] first proposed a ratio-dependent functional response of form . Kuang and Beretta [4] investigated the following ratio-dependent type predator-prey model:where and represent population sizes of prey and predator at time t, respectively. r, , and stand for the prey intrinsic growth rate, the intraspecific competition rate of the prey, and the predator death rate, respectively; , , and represent the encounter rate, half capturing saturation constant, and conversion rate, respectively, that predator preys on prey .
Long-term ecological research studies show that three-species 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 three-species predator-prey food chain model with an omnivory top predator:where , , and denote the number of prey, intermediate predator, and omnivorous top predator, respectively, is the growth rate of prey, is the death rate of species , is the intraspecific competition rate of prey, , , and are the capture rates, and , , and 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. This 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 , , and 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, is the intraspecific competition rate of species 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. Then, the ratio-dependent type three-species food web model with omnivory is expressed in the form:where stands for the total number of prey at time t, while and represent the total number of intermediate predators and omnivorous top predators at time t, respectively. Here, r is the intrinsic growth rate of prey; represents the mortality rate of predator ; stands for the intraspecific competition rate of species ; , , and are the encounter rate, half-saturation constant, and conversion rate, respectively, that preys on ; , and stand for the same corresponding denotations that preys on ; and , , and represent the same corresponding denotations that preys on .
As mentioned above, we notice that population models (1)–(4) are described by the deterministic model. This 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, then we can approximate the colored noise by a white noise , and the approximation works quite well (see [10]). It turns out that the white noise is formally regarded as the derivative of a Brownian motion , i.e., (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 [12–15]). Liu and Bai [12] considered the optimal harvesting problem of a stochastic logistic model with time delay. In [13–15], the authors investigated the dynamics of stochastic predator-prey models. Ji et al. [13] discussed a stochastic predator-prey 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 predator-prey 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 in model (1) were perturbed withwhere and are mutually independent Brownian motions and represents the intensity of white noise . Moreover, they investigated the long time behavior of the following stochastic ratio-dependent prey-predator 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 , 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 [21–26] 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. The purpose of this paper is to make some contribution in this direction. Recall that parameters r, , and 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, , and in model (4), respectively, bywhere is the white noise and is the intensity of white noise . Then, the stochastic three-species food web model with omnivory and ratio-dependent functional response took the following form:with . All meanings of the parameters are exact to or similar as those for (4) except the following. Here, represents the three-dimensional standard Brownian motion defined on a complete filtered probability space satisfying the usual conditions. represents the intensity of noise . Throughout this paper, unless otherwise specified, we would rather assume that , , , , , , , , and .
2. 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:
Theorem 1. For any given initial value , model (8) has unique global positive solution for , that is, with probability one for .
Proof. Consider the following system:with initial value . Obviously, the coefficients of (10) are locally Lipschitz continuous. Thus, there is a unique maximal local solution of (10) for , where denotes the explosion time. Let (). Using Itô formula, it follows that is the unique positive local solution of (8) with initial value for .
Next, we show that is a global solution of (10), that is, . Consider the following two stochastic differential systems:with initial value andwith initial value .
Thanks to Lemma 4.2 in [27], systems (11) and (12) can be explicitly solved as follows:Note that the local solution is positive on . Then, from the comparison theorem of stochastic differential equations (see Theorem 3.1 in [28]), it follows that for Thus, for Since and () exist for every , it follows that . Thus, for any initial value , and (10) has a unique global solution on a.s. Note that the coefficients of (8) are local Lipschitz continuous. Therefore, for any initial value , model (8) has a unique global positive solution on a.s. The proof is therefore complete.
3. Asymptotic Behaviors
Lemma 1 (see [13]). Consider one-dimensional stochastic differential equation:where a, b, and σ are positive constants and is standard Brownian motion. For any , let be the solution of equation (16) with initial value . If , then
Theorem 2. For any , let be the solution of model (8) with initial value . If , , and , then
Proof. From Theorem 1, it follows thatNote that and are the solutions of the following stochastic equations, respectively,with initial value . Obviously, from Lemma 1, it follows that if :This, together with (19), yieldsNow, we show a.s. Note that, a.s. and is the solution of equationwith initial . Thus, from Lemma 1 and , it follows thatNote that , a.s. Thus, for any , there exists such thatBy the strong law of large numbers of local martingales (see Theorem 1.3.4 in [11]), it follows that a.s. Thus, for any , there exists such thatFrom the expression of that for any , we haveHence, from , and , it follows thatThat is a.s., for . Then, . Thus, for any ,Consequently,In addition,Thus,Similarly, if , , and , thenThe proof is therefore complete.
4. 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 , , and . Then, for any , the solution of model (8) with initial value obeys
Proof. For prey , from Theorem 1, it follows thatand is the solution of the following stochastic equation:with initial value . Obviously, from Lemma 1, it follows that if ,This, together with (35), yieldsFor intermediate predator , using Itô formula, it follows thatHence, By the strong law of numbers of local martingales and Theorem 2, we getFor the omnivorous top predator , it follows from Itô formula thatHence,By the strong law of numbers of local martingales and Theorem 2, we getThe proof is therefore complete.
Theorem 4. Suppose that , and . Then, for any initial value , model (8) is extinct exponentially with probability one.
Proof. From Itô formula, it follows thatNote that and , , and . Then,Therefore, model (8) is extinct exponentially. The proof is complete.
5. Stochastic Permanence
In this section, we discuss the stochastic permanence of model (8). The definition of stochastic permanence and stochastically ultimately boundness of model (8) were introduced in the literature [29, 30] as follows.
Definition 1 (see [29, 30]). Model (8) is called stochastically ultimate bounded, if for any , there exist three positive constants , , and such that the solution of model (8) with any initial value has the property that
Definition 2 (see [29, 30]). Model (8) is said to be stochastically permanent, if for any , there exist positive constants , , and , such that the solution of model (8) with any initial value has the property thatIt is obvious that if stochastic model (8) is stochastically permanent, its solutions must be stochastically ultimately bounded.
5.1. Boundness
In this subsection, we investigate the stochastically ultimate boundness of model (8) in two different ways.
Lemma 2 (see [31]). For any positive constants p, m, and n, the Bernoulli equationwith the initial value , has the solution
Theorem 5. For any , let be the solution of model (8) with initial value . If and , then for any ,
That is, the solution of model (8) is uniformly bounded in the pth moment.
Proof. For in system (11), applying Itô formula to leads toTaking the expectation on both sides of the above equation, we haveThen, using the Höder inequality, it follows thatFrom Lemma 2 and the comparison theorem, it follows thatNote that . Thus,By a similar the discussion as in , we also know thatFrom Theorem 1, it follows that a.s. . Then, for any , we haveNow Theorem 5 follows immediately from the above analysis. The proof is complete.
Theorem 6. For any , let be the solution of model (8) with initial value . Then,where and .
Proof. Define . By Itô formula, we haveIntegrating it from 0 to t and taking expectation yieldsThus, using the Hölder inequality yieldsIt is clear that quadratic function reaches its maximum value at . Thus, . Therefore, we haveThen, by the comparison theorem, we haveNote that the solution of model (8) is positive. Thus,The proof is therefore complete.
Theorem 7. Model (8) is stochastically ultimate bounded.
Proof. Let be solution of (8) with any initial value . For any , let , , and . Then, by Chebyshev’s inequalityHence, from Theorem 6The proof is therefore complete.
5.2. Stochastic Permanence
In this section, we give some sufficient conditions to guarantee that model (8) is stochastically permanent. Denote . Define
By the Itô formula, we havewith initial value .
Lemma 3. Let be the solution of model (8) with any initial value . If , thenwhere , , and
Proof. First, integrating both sides of the first equation of (69) from 0 to t yieldsTaking the expectation on both sides of the above equation, we haveThus,with initial value . By a simple computation, we can getThis, together with , yieldsNext, integrating both sides of the second equation of system (69) from 0 to t yieldsTaking the expectation on both sides of the above equation, we haveThus,with initial value . By a simple computation, we can getFrom (75), it follows thatThis, together with (79) yieldsAt last, integrating both sides of the third equation of system (69) from 0 to t and taking the expectation, we haveThus,with initial value . By a simple computation, we can getIt follows from (75), (81), and (84) thatFrom the comparison theorem of stochastic differential equations, it follows that Now, Lemma 3 follows immediately from the above analysis. The proof is complete.
Theorem 8. If , then model (8) is stochastically permanent.
Proof. Let be solution of (8) with initial value . For any , let , thenThus, from Lemma 3, it follows thatThis impliesLet be sufficiently small such that . From (67) and Definition 2, model (8) is stochastically permanent. The proof is therefore complete.
6. 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 be a homogeneous Markov process in (denotes d-dimensional Euclidean space), described by the following stochastic differential equation:
The diffusion matrix of the process is defined as .
Definition 3 (see [32]). Let be the probability measure induced by with initial value . That is, , for any Borel set . If there exists a probability measure such that for all and , then we say that stochastic differential equation (90) has a stationary distribution .
Lemma 4 (see [33, 34]). Assume that there exists a bounded domain with regular boundary and(i)(A1) There is a positive number M such that , , and ;(ii)(A2) There exists a nonnegative -function such that there exists a positive constant C, such that
Then, the Markov process has a unique ergodic stationary distribution . Moreover, if is a function integrable with respect to the measure μ, then
Let , , , and
Then, system (90) reduces to model (8) with diffusion matrix .
Theorem 9. If , , , , and , then for any , model (8) has a stationary distribution and the solutions have ergodic property.
Proof. Define -function : byfor . By Itô formula, we haveDefine -function : byfor . By Itô formula, we haveLet . Then,whereClearly, , , and have upper bound on . DenoteFrom , it follows thatSimilarly, from and , we haveConsequently, there exists (sufficiently small) such that whereHence, (A2) in Lemma 4 is satisfied.
Denote . Then, for any and , we havewhere . Thus, condition (A1) of Lemma 4 holds. According to Lemma 4, model (8) is ergodic and admits a unique stationary distribution. The proof is therefore complete.
7. Application of Main Results
In this section, we present the application of the main results in some special models.
7.1. Two Species Predator-Prey Model
Let . Then, the first two equations of (8) form the following closed two-population system:with initial value . This is also the stochastic predator-prey model discussed in [19]. From Theorems 3 and 4, we have the following result.
Corollary 1. Let be solution of model (106) with initial value .(i)If , , and , then(ii)If and , then (106) is extinct exponentially with probability one.
Remark 1. It is clear that Corollary 1 is consistent with Theorems 7 and 8 in [19]. Moreover, from Theorems 3 and 4, the persistence in mean and extinction conditions of the three-species model (8) are more complicated. Thus, our work can be seen as the extension of [19].
For model (106), similar to the proof of Theorem 6 (denote ), we havewhere . Furthermore, from Theorems 7–9, for model (106), we have the following result.
Corollary 2. (i)Model (106) is stochastically ultimate bounded(ii)If and , then model (106) is stochastically permanent(iii)If , , and , then for any , model (106) has a stationary distribution and the solutions have ergodic property
If we do not consider the intraspecific competition of the predator, i.e., in model (106), then model (6) is available. From Theorems 4, 7, and 8, for model (6), we have the following result.
Corollary 3. (i)If and , then model (6) is extinct exponentially with probability one(ii)Model (6) is stochastically ultimate bounded(iii)If and , then model (6) is stochastically permanent
Remark 2. From Theorem 4.11 in [18], it follows that if and , then model (6) is stochastically permanent. Obviously, if conditions of Theorem 4.11 in [18] hold, then conditions in Corollary 3 hold. On the contrary, it is not set up. Therefore, Corollary 3 generalizes and improves Theorem 4.11 in [18].
Remark 3. If and , then by Theorem 3.3 in [16], model (6) is persistent in mean, but by Corollary 3, model (6) is stochastically permanent.
7.2. Three-Species Food-Chain Model
Let . Then, we can get the following stochastic three-species food chain model:with initial value . Denote
For model (109), we have the following results.
Corollary 4. For any , let be the solution of model (109) with initial value .(i)If , , and , then(ii)If , , and , then(iii)If , , and , then model (109) is extinct exponentially with probability one.
Corollary 5. Model (109) is stochastically ultimate bounded. Furthermore, if , then model (109) is stochastically permanent.
Corollary 6. If , , , , and , then for any , model (109) has a stationary distribution and the solutions have ergodic property.
If we do not consider the intraspecific competition of the predator, i.e., in model (109), then we obtain the following stochastic three-species food chain model:with initial value . From Theorems 4, 7, and 8, for model (113), we have the following result.
Corollary 7. (i)If , , and , then model (113) is extinct exponentially with probability one(ii)Model (113) is stochastically ultimate bounded(iii)If , then model (113) is stochastically permanent
7.3. Food-Web Model without Intraspecific Competition of Predators
If we do not consider the intraspecific competition of the predator, i.e., in model (8), then we obtain the following stochastic three-species model:with initial value . For model (114), we have the following result.
Corollary 8. (i)If , , and , then model (114) is extinct exponentially with probability one(ii)Model (114) is stochastically ultimate bounded(iii)If , then model (114) is stochastically permanent
8. Numerical Simulations
In this section, we use the Milstein method (see [35]) to substantiate our main results. The 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. To illustrate the theoretical results, we take the parameter values as following with different noise intensities:
In Figure 1, we choose and get the solutions of the corresponding deterministic model.
9. Conclusions and Discussions
This paper is concerned with a stochastic three-species predator-prey food web model with omnivory and ratio-dependent 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. Then, 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 existence of an ergodic stationary distribution to the model. Then, 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 Theorem 3, we give the conditions on the parameters that informally can be stated by saying that the noise intensities (i = 1, 2, 3) are small compared to the other parameters, such that the species in model (8) are persistent in mean. From Theorem 4, it follows that in the case that the noise intensities (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. This 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 Theorem 8, if the noise intensities (i = 1, 2, 3) are small compared to the other parameters, such that (), then model (8) is stochastically permanent.
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 Theorem 9, we give the conditions on the parameters that can be stated by saying that the intensity of white noise is sufficiently small, such that the solution model (8) has an ergodic stationary distribution.
The 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 Theorem 8 generalizes and improves Theorem 4.11 in [18].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 11971279 and 11471197).