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 theM-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 andnumerical simulations to illustrate main results.


Introduction
Mutualism plays a key part in ecology, and researchers have proposed many mathematical models to describe the mutualistic interaction [1][2][3][4][5][6].In particular, motivated by Holling type II functional response [7], Wright [5] 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][10][11][12].The Beddington-DeAngelis, Crowley-Martin, and ratio-dependent functional responses were also further considered in [13][14][15].But limited work is available on predator-prey model with mutualism.By considering the coexistence of antagonism, mutualism, and competition, Mougi and Kondoh [16] showed that interaction-type diversity generally enhanced stability of complex communities.Motivated by the above ideas, we consider the following onepredator 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) where   > 0 ( = 1, 2, 3, 4) 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   (),  = 1, . . ., 6, denoting white noise, are independent standard Brownian motions and  2  ( = 1, . . ., 6) 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 {(),  ≥ 0} is a Markov chain controlling the switching among regimes and taking values in a finite state space S = {1, 2, . . ., }.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 (0) =  0 ∈ S,  = max 1≤≤4   , and all of the parameters are positive.In regime  ( ∈ S), system (5) obeys Therefore, (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. [24] 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 [15], and they explored permanence and asymptotical behavior.However, for the stochastic predator-prey model with Markovian switching, most of previous works focused on twodimensional systems.And to the best of our knowledge, there is no work about 3-dimensional stochastic delayed predatorprey 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.
Theorem 1.For any initial data (8), there is a unique positive solution of system ( 5) on  ≥ −, which remains in R 3 + × S with probability 1.
Proof.Define a function  : This proof is standard, so please refer to [13,25], and thus we omit it.

Stochastic Permanence
In this section, we will consider stochastic permanence and firstly study stochastically ultimate boundedness.
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 [18] and used in [15,24].Now we give notations, the classical result, and some assumptions.Let  be a vector or matrix.Denote by  ≫ 0 that all elements of  are positive.Set Lemma 5 (e.g., see [18]).If  ∈  × , then the following statements are equivalent: (i)  is a nonsingular -matrix.
(ii)  is semipositive; that is, there exists  ≫ 0 in R  such that  ≫ 0.
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  > 0 such that the matrix is a nonsingular -matrix, where Proof.This proof is standard, so please refer to [15,24], and thus we omit it.
Proof.By Lemma 7, Chebyshev's inequality, and Theorem 4, we can get the desired conclusion.
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.

Extinction
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.Theorem 10.For any initial data (8), the solution () of system (5) Integrating from 0 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  ∈ (0, 1) be arbitrary.By the exponential martingale inequality (e.g., see [18]), for each  ≥ 1, Noting that the series ∑ ∞ =1  −2 converges and by the Borel-Cantelli lemma (e.g., see [18]), there exists Ω 0 ⊆ Ω with P(Ω 0 ) = 1 such that, for any  ∈ Ω 0 , there is an integer  0 =  0 () such that for all 0 ≤  ≤  and  ≥  0 ().Substituting (43) into (39) and noting that  ∈ (0, 1), we get ln for all 0 ≤  ≤  and  ≥  0 ().Then for any  ∈ Ω 0 , if  ≤  ≤  + 1 and  ≥  0 (), 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 [18]), we finally obtain lim sup By the above same methods and procedures, we have lim sup The proof is completed.
On the basis of the above theorem, we directly give the following corollary about subsystem's extinction.

Asymptotic Property
In this section, we will consider another asymptotic property of system (5).

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 [26], the discrete form of system ( 5) can be given by Figure 1: (Stochastic permanence) the trajectories of the solution ((), (), ()) for subsystem (64) and its corresponding deterministic system.
switching from one to another according to the movement of Markov chain ().

Case 2. Assume that the generator of Markov chain 𝑟(𝑡) is
By solving (11), we get the unique stationary distribution ) .
Then we compute that ∑ ∈S   ( 5 ()) = (71/100) 1 − (49/600) 2 = −27/1300 < 0. Therefore, by Theorem 10, the overall system (5) is extinct.See Figures 6-8.   Figure 5: The subgraphs (a), (b), (c), and (d) denote the discrete point distribution of two subsystems and the overall system in , , , and , respectively.The blue, green, and red areas represent the overall system (5), subsystem (64), and subsystem (65), respectively.The green area which means stochastic permanence is far away from the origin while most points of the red area which means extinction lie in the origin.Under the control of Markov chain, the blue area also keeps away from the origin and means stochastic permanence in Case ) . (74)

Figure 4 :
Figure 4: The subgraphs (a) and (b) denote the trajectory and frequency of the discrete Markov chain   taking value in {1, 2}, respectively. = 1 and   = 2 mean that the overall system (5) switches to subsystems (64) and (65) in step , respectively.The change of   in (a) means the process of regime switching of system (5) between subsystems (64) and (65).And the frequency of   in (b) shows the result of regime switching-the total number of steps of the overall system (5) switching to subsystem (64) or subsystem (65).This graph shows that under the control of the Markov chain the overall system (5) mostly switches to subsystem (64) in Case 1 of Example 1.

Figure 7 :Case 1 .
Figure7: The above subgraphs have the same notations as in Figure4.The change of   in (a) means the process of regime switching of system (5) between subsystems (64) and (65).And the frequency of   in (b) shows the result of regime switching-the total number of steps of the overall system (5) switching to subsystem (64) or subsystem (65).This graph means that under the control of the Markov chain the overall system (5) mostly switches to subsystem (65) in Case 2 of Example 1.