Dynamical Behaviors of Stochastic Delayed One-Predator and Two-Competing-Prey Systems with Holling Type IV and Crowley-Martin Type Functional Responses

This paper is devoted to stochastic delayed one-predator and two-competing-prey systems with two kinds of different functional responses. By establishing appropriate Lyapunov functions, the globally positive solution and stochastic boundedness are investigated. In some case, the stochastic permanence and extinction are also obtained.Moreover, sufficient conditions of the global asymptotic stability of the system are established. Finally, some numerical examples are provided to explain our conclusions.


Introduction
It is well known that predator-prey system, cooperative system, and competitive system are three kinds of important ecological systems.The dynamic relationship among species is a significant theme whether in ecology or in mathematical ecology because of its importance and universal existence with many concerned biological systems (see [1]).A lot of systems about predator-prey behaviors have been proposed (see [2][3][4]).
A main objective for ecologists is to find the relationships among species.And the consumption rate of each predator on prey is an important component of the relationships between predator and prey, that is, predator's functional response.In order to describe different situations when predators search or compete for food, many significant functional responses have been proposed, such as L-V and Holling II-IV types (see [5][6][7]).A suitable functional response is not only related to the density of prey, but also to the predator.A statistics from 19 predator-prey systems indicates that Crowley-Martin type, Beddington-DeAngelis type, and Hassell-Varley type predator-dependent functions can provide a better description in some case.In [8], the following predator-prey system bas been studied: where  is the density of prey and  is the density of predator at time .However, interaction of multiple species often occurs in nature and their relationships are much more complex than that of the two species (see [9,10]).Therefore, it is more realistic to study the multiple species predator-prey systems.Motivated by above, we consider the following systems: where  1 ,  2 and  denote two competed prey and predator densities, respectively.And the parameters  1 ,  2 , and  3 are the intrinsic growth of three species;  1 ,  2 , and  3 are the intraspecific competition rate of three species, respectively. 1 and  2 are the interspecific competition rates of two competed species,  3 and  4 are the predators' capturing rates, and  1 and  2 are the rates of conversion of nutrients into the production of predator.And all the parameters in system (2) are constants.It is very necessary to point out that  3 /(1 +  1 +  +  1 ) is a special functional response; when  =  =  = 0 it becomes a linear mass-action function response (or Holling type I functional response), when  =  = 0 it becomes a Holling type II functional response, when  = 0 it becomes a modified Holling type II functional response, and when  =  it becomes a Crowley-Martin functional response.
In the real world, population dynamics are often affected by white noise from the environment, which relate to climate, geographical distribution, geological features, human disaster, human intervention, and other environmental factors.Therefore, the flow of biological energy is a process of fluctuation.The oscillation of population biomass is directly related to the birth and death rate of random perturbation.Up to now, there have been many works considering the effect of random perturbation (see [11][12][13]).In this paper, we assume that white noise affects the intrinsic birth rate, capture rate of predator, and conversion rate of the predator population.On the other hand, the development trend of the real biological system is not only related to the status of the system, but also depends on the history of the system more or less, which is called time delays.Moreover, time delay widely exists in biological systems; for example, in the predator-prey system, the process for the conversion of prey to predators is not immediately translated into the predator population but after a certain period of time to digest the transformation.So a more realistic predator-prey model should consider the effects of time delays (see [14,15]).As a matter of fact, delay differential systems have much more complicated dynamical behaviors than the differential equations without delays.Therefore, the following stochastic delay systems are considered: 2  are the intensities of the noises,  = 1, 2, 3, 4, 5.   () are standard Brownian motions which are defined on a complete probability space (Ω, , ),  = 1, 2, 3, 4, 5. Let  = max{ 1 ,  2 ,  3 } and  = ([−,0], 3 + ) be the set of continuous functions from [−, 0] to  3  + with initial condition  ∈ ([−, 0],  3 + ) and the norm ‖‖ = sup −≤≤0 () < +∞.
Any biological system, whether it is population, biological communities, ecosystems, or the biosphere, its dynamic behavior is one of the main objects of the study, such as the resistance of ecosystem, persistence, recoverability, variability, and consistency.Therefore, we use mathematical theory and methods to study the dynamics of biological populations, which can not only protect the ecological balance but also can improve the ecological environment for human survival.
According to what we know, few current literatures are found to discuss stochastic delayed predator-prey systems with Holling type IV and Crowley-Martin type functional responses at the same time.In this paper, it is the first time to obtain the condition of global asymptotic stability of system (3).
This paper is carried out as follows.In Section 2 and in Section 3, global positive solution and stochastically ultimate boundedness of system (3) are investigated.In Sections 4 and 5, we study the stochastic permanence and extinction, respectively.In Section 6, we obtain the fact that system (3) is globally asymptotically stable.In the end, in Section 7, some numerical examples are provided to explain our findings.

Stochastically Ultimate Boundedness
Stochastically ultimate boundedness of system (3) is studied in this part.Firstly, we present a useful lemma in the following.
Lemma 2. For any initial data  ∈ ([−, 0], Also, we can obtain that By taking expectation in both sides of inequality (9), we have that By the same way, we can obtain that where This completes the proof.

Stochastic Permanence
Theorem 4. If holds, by the definition of stochastic permanence, we say that system (3) is stochastically permanent.
Let ( 1 ,  2 , ) = 1/( 1 ,  2 , ); using Itô's formula, we can obtain where If ( 16) holds, we can find a positive constant  which satisfies the following condition: Using Itô's formula, we can obtain Hence, Under the condition of (20), we can choose another positive constant  making it satisfy the following condition: Using Itô's formula, we can obtain Hence, Obviously, we can find a positive constant  which satisfies Then We can integrate above inequality and then take expectation where From Theorem 3, we have lim for any  > 0; let  = (()/) 1/ , using Chebyshev's inequality, we can obtain the conclusion.

Extinction
The extinction of system (3) will be investigated in this part.
holds; by the definition of extinction, system (3) is said to be extinct.
For stochastic permanence, we also choose which satisfies the condition (16) (see Figure 4).
In the following we choose larger noises  1 = 0.95,  2 = 0.1,  3 = 1.05,  4 = 1.5, and  5 = 0.1, which satisfy the condition of Theorem 6; then system (62) will go to extinction (see Figure 5).In the end, we discuss the global asymptotic stability.Let        which satisfy the condition of Theorem 10; then system (62) is globally asymptotically stable.By the same way, we can obtain the fact that the deterministic system of (3) is also globally stable (see Figures 6-8).

Conclusions and Discussion
In this work, stochastic delayed one-predator and twocompeting-prey systems with two different kinds of functional responses have been studied.Globally positive solution, stochastically ultimate boundedness, and the stochastic permanence and extinction for system (3) are investigated.Moreover, sufficient criteria for the global asymptotic stability of the system are established.In the end, some numerical examples are provided to explain our results.Through the study of the dynamic behavior of system (3), by comparing Theorems 4 and 6, we can obtain that if the environment noise is small, the stochastic system can maintain permanent while the system can be extinct under sufficiently large environmental noise (see Figures 4 and 5).Therefore, from Theorem 10, we can choose the appropriate parameters in a suitable environment noise intensity to make system (3) asymptotically stable.Recently, predator-prey models have been investigated extensively for their theoretical and practical significance.In the current literatures, most of this work is restricted to two-dimensional predator-prey system; few has been done on predator-prey system with interspecific competition in preys.In system (3), we study a three-dimensional hybrid system where the predator can capture two kinds of preys with different functional responses.In fact, the population models are often subjected to the influence of environmental noises inevitably.In most of predator-prey models, there is only a white noise which affects intrinsic rate of increase of predator or prey.In system (3), we consider the white noise not only has effect on the intrinsic rate of population growth but also on capture rate of predator and conversion rate of the predator population.Three time delays are introduced to make the model closer to the reality.It is interesting to point out that system (3) contains two kinds of different mathematical models; if we remove the predator, system (3) is reduced to a competitive model in [13]; if we remove any one of the two preys system (3) is reduced to a simple predator-prey system.Some meaningful questions deserve further investigation.One may investigate the stationary distribution of system (3).Moreover, it is worth considering the corresponding nonautonomous system of system (3).One may discuss dynamics behaviors contained predators with a mutual cooperation in other ecosystems.