Dynamical Behavior of a Stochastic Delayed One-Predator and Two-Mutualistic-Prey Model with Markovian Switching and Different Functional Responses
We propose a stochastic delayed one-predator and two-mutualistic-prey model perturbed by white noise and telegraph noise. By the -matrix analysis and Lyapunov functions, sufficient conditions of stochastic permanence and extinction are established, respectively. These conditions are all dependent on the subsystems’ parameters and the stationary probability distribution of the Markov chain. We also investigate another asymptotic property and finally give two examples and numerical simulations to illustrate main results.
Mutualism plays a key part in ecology, and researchers have proposed many mathematical models to describe the mutualistic interaction [1–6]. In particular, motivated by Holling type II functional response , Wright  established the Holling type II mutualistic model:For the biological meaning of parameters in the above model, we refer to [5, 6, 8].
Besides, the predator-prey interaction is extremely common in the natural world, and many researchers have paid attention to the predator-prey model. Predator-prey models with Holling types I, II, III, and IV responses were investigated in [9–12]. The Beddington-DeAngelis, Crowley-Martin, and ratio-dependent functional responses were also further considered in [13–15]. But limited work is available on predator-prey model with mutualism. By considering the coexistence of antagonism, mutualism, and competition, Mougi and Kondoh  showed that interaction-type diversity generally enhanced stability of complex communities. Motivated by the above ideas, we consider the following one-predator and two-mutualistic-prey model with Holling type II and Beddington-DeAngelis responses:where species , are two mutualistic preys and is the predator. Furthermore, it is more realistic and reasonable that the future state of population dynamics is determined by not only the present states but also the past [17, 18]. Up to now, there have been many works considering the effect of time delay [13, 14, 19, 20]. Then, taking time delay on mutualistic interaction and predation into account, model (2) can be modified as the following model:where denotes the time delay, and we drop from , , and and do that throughout this paper.
However, it is not enough to only consider certain factors. The biological system is more or less affected by stochastic fluctuations. One of these general fluctuations is white noise. Recently, many authors have studied lots of stochastic models with white noise, for example, [12, 19, 21]. They mostly put the effect of white noise on the intrinsic birth rate and death rate. In this paper, we assume that white noise affects the intrinsic birth rate and intraspecific competition rate; that is, where , denoting white noise, are independent standard Brownian motions and denotes the intensity of white noise.
Besides white noise, the biological system is inevitably affected by another environment noise, that is, telegraph noise. This noise can be represented by switching among two or more regimes of environment, which are distinguished by factors such as rain falls and nutrition [22, 23]. Suppose is a Markov chain controlling the switching among regimes and taking values in a finite state space . Then, taking white noise and telegraph noise into consideration and on the basis of model (3), we finally developed the following stochastic delayed one-predator and two-mutualistic-prey model with Markovian switching and different functional responses:
with the initial data where , , and all of the parameters are positive. In regime , system (5) obeysTherefore, (7) is regarded as the subsystem of system (5). In this paper, our main aim is to reveal how two kinds of environment noise, that is, white noise and telegraph noise, affect permanence and extinction of system (5).
The stochastic differential equations controlled by a continuous Markov chain have been applied to the population models with telegraph noise. Li et al.  investigated the logistic population system without intraspecific competition incorporating white and telegraph noise and mainly researched stochastic permanence and extinction. A two-dimensional stochastic predator-prey model with Markovian switching was developed by Ouyang and Li , and they explored permanence and asymptotical behavior. However, for the stochastic predator-prey model with Markovian switching, most of previous works focused on two-dimensional systems. And to the best of our knowledge, there is no work about -dimensional stochastic delayed predator-prey models with Markovian switching, two mutualistic preys, and different functional responses till now.
We arrange the rest of this paper as follows. In Section 2, we prepare some notations and consider the existence and uniqueness of the solution of system (5). By the -matrix analysis and Lyapunov functions, we study stochastically ultimate boundedness and stochastic permanence, and the sufficient condition of stochastic permanence is given in Section 3. Section 4 gives the sample Lyapunov exponent and hence shows the sufficient condition of extinction. We study another asymptotic property in Section 5. In Section 6, we give two examples and make numerical simulations to illustrate main results and reveal the dynamical behavior. In Section 7, we give conclusions and the future direction.
Throughout this paper, let be a complete probability space with the filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). Denote by the positive cone in . Let denote the space of continuous function with the norm . Denote by a solution of system (5) and its norm is defined by . Let , , denote , , , respectively. Let the initial data of system (5) be
Assume that is a right-continuous Markov chain taking values in the finite state space with the generator defined by where . Here is the transition rate from to and if , whileWe also assume that the Markov chain is independent of the Brownian motion and is irreducible (e.g., ), which indicates that the overall system (5) can switch from any regime to any another regime. Under this assumption, the Markov chain has a unique stationary probability distribution , depending on the equationsubject to
For convenience and simplicity, we give the following notations:
In order to study the dynamic behavior, we must firstly guarantee that there exists a unique, positive, and global solution.
3. Stochastic Permanence
In this section, we will consider stochastic permanence and firstly study stochastically ultimate boundedness.
Definition 2 (e.g., see ). System (5) is said to be stochastically permanent if for any there are two positive constants and such that for any initial data (8) the solution of system (5) satisfies
Proof. Define a function By means of the generalized Itô formula (e.g., see ), we get Note that , , where is a positive constant. Then applying the generalized Itô formula to , we get Integrating both sides of from to , taking the expectation, and taking the limit superior, we obtain Note that , and thus we have
Theorem 4. System (5) is stochastically ultimately bounded.
Next we will investigate stochastic permanence. Based on the above conclusion, we only need to prove another inequality about stochastic permanence. And one of the main methods in this section is the -matrix analysis which was introduced by  and used in [15, 24].
Now we give notations, the classical result, and some assumptions. Let be a vector or matrix. Denote by that all elements of are positive. Set
Lemma 5 (e.g., see ). If , then the following statements are equivalent:(i)is a nonsingular -matrix.(ii) is semipositive; that is, there exists in such that .
Assumption (A1). For some , , .
Assumption (A2). , where
Assumption (A3). For some
The proof of stochastic permanence is rather long and technical. To make it more understandable, we divide the proof into several lemmas.
Lemma 6. Assumptions (A1) and (A2) imply that there exists a constant such that the matrixis a nonsingular -matrix, where
Proof. Define By the generalized Itô formula, we have where Define again By the generalized Itô formula, we get For given , by Lemma 5, there exists a vector such that ; that is,Define the third function By the generalized Itô formula, we haveComputing and substituting (30) into (29), we obtain Under (28), there is a sufficiently small constant such that ; that is,Applying the generalized Itô formula to and noticing (23), we obtain By (32), it is obvious that , where Thus, Noting that , we have
Theorem 8. Under Assumptions (A1) and (A2), system (5) is stochastically permanent.
On the basis of the above theorem, we directly give the following corollary about subsystems permanence.
Corollary 9. Under Assumption (A3), subsystem (7) is stochastically permanent.
In this section, we will discuss the sample Lyapunov exponent of system (5) and hence get the sufficient condition for three species to be extinct.
Proof. By the generalized Itô formula, we have Integrating from to on both sides of the above inequality, we obtain where is real-valued continuous local martingale and its quadratic form is defined by Let be arbitrary. By the exponential martingale inequality (e.g., see ), for each , Noting that the series converges and by the Borel-Cantelli lemma (e.g., see ), there exists with such that, for any , there is an integer such thatfor all and . Substituting (43) into (39) and noting that , we get for all and . Then for any , if and , we obtain Taking the limit superior on both sides of the above inequality and by the strong law of large numbers and the ergodic property of Markov chain (e.g., see ), we finally obtain By the above same methods and procedures, we have The proof is completed.
On the basis of the above theorem, we directly give the following corollary about subsystem’s extinction.
Corollary 11. For subsystem (7), if the solution satisfies , , and , then
5. Asymptotic Property
In this section, we will consider another asymptotic property of system (5).
Proof. Define Let be arbitrary. Applying the generalized Itô formula to , we get where and are defined in Section 2.is real-valued continuous local martingale and its quadratic form is defined by Let and be arbitrary. By the exponential martingale inequality, for each , Noting that the series converges and by the Borel-Cantelli lemma, there exists with such that, for any , there is an integer such thatfor all and . Note that and choose three constants such thatSubstituting (55)–(57) into (51) and noting that , we obtain for all and . It is obvious that there is a positive constant such that Then for all and . Thus, for any , if and , we have This implies Let , , and , so we get Then we can directly get the desired conclusion.
6. Examples and Numerical Simulations
In this section, we will give two examples and make some numerical simulations to support main results. By the method mentioned in , the discrete form of system (5) can be given by where , are Gaussian random variables that follow and denotes the integer part of . For the procedure of generating the discrete Markov chain , please refer to .
Throughout this section, we assume that , , , , , and let the initial data be , , , , , where . Please note that in this section all of numerical results are computed by MATLAB and displayed in rational format. Therefore there may be an error for some numerical results.
Example 1. Let be a right-continuous Markov chain taking values in . System (5) may be regarded as the result of the following two subsystems: