Complex Dynamics of a Stochastic Two-Patch Predator-Prey Population Model with Ratio-Dependent Functional Responses

*is paper investigates a stochastic two-patch predator-prey model with ratio-dependent functional responses. First, the existence of a unique global positive solution is proved via the stochastic comparison theorem. *en, two different methods are used to discuss the long-time properties of the solutions pathwise. Next, sufficient conditions for extinction and persistence in mean are obtained. Moreover, stochastic persistence of the model is discussed. Furthermore, sufficient conditions for the existence of an ergodic stationary distribution are derived by a suitable Lyapunov function. Next, we apply the main results in some special models. Finally, some numerical simulations are introduced 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 been universal in mathematical ecology. In the nature world, foraging behaviour is a common phenomenon. Ecological species have the ability to adapt through learning (see [1]). An individual will adjust its behaviour by learning in response to a change of the environment in order to survive and acquire the most food. In [1], the authors studied the two-patch predator-prey population model with nonnegative initial conditions. Here, x i denotes the density of prey in patch i (i � 1, 2), and y represents the density of predators. v (0 ≤ v ≤ 1) is the proportion of time that predators stay in patch 1 on average; r i (i � 1, 2) is the intrinsic growth rate of prey in patch i; a i is the intraspecific competition coefficient of the prey in patch i; s i is the attacking rate of the predators in patch i; e i is the expected biomass of the prey converted to predators in patch i; m i is the per capita mortality rate of predators in patch i; and h i is the handling time of the predation in patch i, respectively. It is well known that the functional response between the predator and prey plays an important role in the population dynamics. In model (1), the authors assumed that an individual predator consumes the prey with functional response (x/1 + shx), which depends only on the prey. However, when predators have to search for food and, therefore, have to share or compete for food, a ratio-dependent functional response is more reasonable (see [2]). Based on the Holling-type II function, Arditi and Ginzburg [3] first proposed a ratio-dependent functional response of form (αx/x + βy). Here, α is the encounter rate with prey by a searching predator, and β is the half saturation constant for the prey. Kuang and Beretta [4] investigated the predatorprey model with ratio-dependent functional response with nonnegative initial conditions. Here, x and y represent population sizes of the prey and predator at time t, respectively. All parameters are positive constants. r and a, respectively, stand for the prey intrinsic growth rate and the intraspecific competition rate of the prey. d is the death rate of the predator population. α, β, and e, respectively, represent the encounter rate, half capturing saturation constant, and conversion rate that predator y preys on prey x. Note that population model (1) with the functional responses only depend on prey density. However, the ratiodependent functional response depends not only on the prey but also on the predator. us, the ratio-dependent function of the prey and predator is more suitable to substitute for the model. erefore, based on models (1) and (2), a two-patch predator-prey population model with ratio-dependent functional responses is expressed in the following form: where x i denotes the density of prey in patch i (i � 1, 2) and y represents the density of predators. All meanings of the parameters are exact as or similar to those for model (1) except the following. Here, α i , β i , and e i (i � 1, 2) are the encounter rate, the half-saturation constant, and the conversion rate that y preys on x i , respectively. From [5], it can be seen that stochasticity or variability plays an important role in understanding the dynamics of predator-prey populations. Note that noise in models can lead to several interesting dynamical effects, which are not anticipated by their deterministic counterpart. us, in order to simulate population dynamics, environmental fluctuations should be considered in modeling. In general, environmental fluctuations can be simulated by a colored noise. From [6], it can be seen that if the colored noise is not strongly correlated, then one can approximate the colored noise by a white noise _ w(t). In fact, 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 [7]). As a result, it is more objective to modeling stochastic population models with white noise in mathematical biology. Recently, many authors have paid their attention to stochastic prey-predator models with white noise, see [8][9][10][11][12][13][14][15] and the references therein. Reference [8] investigated the stability of a stochastic one-predator-two-prey population model with time delay, while [13] considered the stability of a stochastic two-predator one-prey population model with time delay. References [10,11,15] discussed the dynamic behaviors of stochastic population models with the Allee effect. Reference [12] is concerned with a stochastic three-species food web model with omnivory and ratio-dependent functional response.
To the best of our knowledge, so far, there is no investigation on the dynamics of the stochastic two-patch prey-predator model with ratio-dependent functional responses. e purpose of this paper is to make some contribution in this direction. As in the work of Imhof and Walcher [16], assuming that the environmental noise is proportional to the variables, we obtain the following stochastic two-patch prey-predator model: with initial value (x 1 (0 ), x 2 (0), y(0)) � (x 10 , x 20 , � y 0 ) ∈ R + 3 � (x, y, z) ∈ R 3 : x > 0, y > 0, z > 0 . All meanings of the parameters are exact as or similar to those for model (3) except the following.
Furthermore, if the intraspecific competition of the predator is considered in model (4), then one can obtain the following stochastic two-patch predator-prey model: Here, b is the interspecific competition coefficient of the predator.
In this paper, we first investigate the dynamics of the stochastic two-patch predator-prey population model (5). en, we apply the main results in the stochastic predatorprey population model (4). e rest of this paper is organized as follows. In Section 2, we first prove that model (5) has a unique global positive solution by the stochastic comparison theorem. en, we discuss the long-time properties of the solutions pathwise. Using the exponential martingale inequality and the Borel-Cantelli lemma, we show that the sample Lyapunov exponents of the solutions are nonpositive. Moreover, we prove that, under certain conditions, the sample Lyapunov exponents of the solutions are zero. In Section 3, we establish the sufficient conditions for the extinction and persistence in mean of model (5). In Section 4, we first prove the stochastic ultimate boundedness of model (5) by using two different methods. en, we show that model (5) is stochastically permanent. Moreover, in section 5, by constructing a suitable Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution to model (5). Next, in Section 6, we apply the main results to two stochastic two-species predator-prey population models and stochastic two-patch predator-prey population model (4). Section 7 contains some numerical results, which are used to demonstrate the theoretical results in this paper. Moreover, through numerical calculation, we find other dynamic properties of the model. e paper ends with a conclusion.
For simplicity, in the coming discussion, we introduce the notations

Global Positive Solution and Pathwise Estimation
In this section, we first show that model (5) has a unique positive global solution by the stochastic comparison theorem. en, we discuss the long-time properties of the solutions pathwise. Theorem 1. For any given initial value (x 10 , x 20 , y 0 ) ∈ R 3 + , model (5) has a unique global solution x 1 (t) on t ≥ 0 and the solution will remain in R 3 + with probability one.
□ From proof of eorem 3, we can get the following result with the proof being omitted. 6 Complexity

Persistence in Mean and Extinction
In this section, we show that, under some conditions, model (5) is persistent in mean and extinct. Here,

Proof.
From For the predator y(t), using the Itô formula, we obtain which implies Hence, Letting t ⟶ ∞ and by the strong law of numbers and eorem 3, we have e proof is complete.
(I) If the predator is absent, i.e., y(t) � 0 a.s. for all t ≥ 0, then the quantities of prey x 1 (t) and prey x 2 (t) satisfy the following: (II) If the prey in patch 2 is absent, i.e., x 2 (t) � 0 a.s. for all t ≥ 0, then the quantities of prey x 1 (t) and predator y(t) satisfy the following: (III) If the prey in patch 1 is absent, i.e., x 1 (t) � 0 a.s. for all t ≥ 0, then the quantities of prey x 2 (t) and predator y(t) satisfy the following: (IV) If the prey is absent, i.e., x 1 (t) � x 2 (t) � 0 a.s. for all t ≥ 0, then the predator dies with probability one

Complexity
Proof. (I) In the absence of the predator, from the Itô formula, it follows that (61) us, from Lemma 2, it follows that if Moreover, from Lemma 2, it follows that if us, (I) holds. Next, we prove (II). From eorem 3, it follows that if Moreover, in the absence of the prey in patch 2, from the Itô formula, it follows that Letting t ⟶ ∞ and by the strong law of numbers and Furthermore, from the proof of (I), if r 1 − (σ 2 1 /2) < 0, then Moreover, in the absence of the prey in patch 2, from the Itô formula, it follows that Hence, (II) holds. e proof of (III) is similar to (II) and, hence, is omitted.

Stochastic Permanence
In this section, we investigate the stochastic permanence of model (5).

Stochastically Ultimate Boundedness.
In this subsection, we first use two different ways to prove the boundedness of model (5) and then show that model (5) is stochastically ultimately bounded by Chebyshev's inequality. e definition of stochastically ultimate boundedness of model (5) was introduced in the literature [20,21] as follows.

Lemma 3 (See
for all x ∈ E 3 , where e caption of Figure 5 is unclear. Please rephrase the caption for clarity and correctness. is a function integrable with respect to the measure μ. Proof. In what follows, for the simplification, we denote x 1 (t), x 2 (t), and y(t) as x 1 , x 2 , and y, respectively. We define C 2 function V 1 : for X � (x 1 , x 2 , y) ∈ R 3 + . From the Itô formula, it follows that We define V 2 : for X � (x 1 , x 2 , y) ∈ R 3 + . Applying the Itô formula, we have 14 Complexity where Obviously, f(x 1 ), g(x 2 ), and h(y) are all functions with an upper bound on R + . us, we denote

Application of Main Results
In this section, we first apply the main results to two stochastic two-species predator-prey models. en, we present the application of the main results to stochastic two-patch predator-prey model (4).

Two-Species Predator-Prey Model.
If the predator only stays in one patch, then one can obtain the following stochastic predator-prey model (obtained by taking v � 0 or v � 1 in model (5)).

(I) By a similar discussion as in eorems 2-6, we can
obtain the following results: Here, M � (eαβ/b). is means that model (125) is persistent in mean.
(iv) If the predator is absent, i.e., y(t) � 0 a.s. for all t ≥ 0, then the quantity of prey in model (125) satisfies the following: Moreover, from Chebyshev's inequality, model (125) is stochastically ultimately bounded.
Remark 1. In [25], the authors show that model (125) has a unique global positive solution by using stopping times and contradiction. In this paper, the stochastic comparison theorem is used to prove that the model has a unique global positive solution. Reference [25] only shows that if δ 1 − (σ 2 1 /2) > 0 and δ 2 − (σ 2 2 /2) > 0, then lim t⟶∞ (ln x(t)/t) � 0 and lim t⟶∞ (ln y(t)/t) � 0 a.s. However, we also show that the sample Lyapunov exponents of the solutions are nonpositive in the absence of conditions. In [25], the authors only show that the solutions are uniformly bounded in the p-th moment. However, we give the concrete upper bound for the p-th moment. It is clear that the results of (ii) and (iii) in Corollary 2 (I) are consistent with eorems 7 and 8 in [25]. However, the result of (III) in Corollary 2 is not reflected in [25]. us, our work can be seen as the extension of [25].

Remark 2.
For the deterministic version of model (125), from [26], we can see that lim t⟶∞ y(t) � 0 holds under some special conditions, i.e., the predator dies out, but it never gets lim t⟶∞ x(t) � 0 (if lim t⟶∞ y(t) � 0, then liminf t⟶∞ x(t) ≥ (r/a) > 0). However, the result of (iii) in Corollary 2 (I) shows that great noise intensities σ 2 1 and σ 2 2 can make both the prey and predator in model (125) die out. is means that a relatively large stochastic perturbation can cause the extinction of the population.
However, Linh and Ton [26] only consider the asymptotic estimations of moments, the upper-growth rates, and exponential death rates of species in the corresponding nonautonomous stochastic model of model (125). Moreover, the results of (iv) and (v) in Corollary 2 (I) are consistent with eorems 4.3 and 4.4 in [26]. us, our paper can be regarded as the extension and supplement of [26].
Furthermore, if we do not consider the intraspecific competition of the predator, i.e., b � 0 in model (125), then one can obtain the following stochastic model: with initial value (x 0 , y 0 ) ∈ R 2 + . is is a stochastic predatorprey model discussed in [27]. Wu et al. [10] considered the corresponding nonautonomous model of stochastic model (133). By a similar discussion as in eorem 1, for any (x 0 , y 0 ) ∈ R 2 + , model (133) has a unique global positive solution (x(t), y(t)).
is means that, in the absence of the prey x 1 , the prey x 2 and the predator y will be persistent in mean (see Figure 14). (ii) We assume that σ 2 2 � 1.24 and σ 2 3 � 0.2. us, r 2 − (σ 2 2 /2) � − 0.02 < 0 and From eorem 6, lim t⟶∞ x 2 (t) � 0 and lim t⟶∞ y(t) � 0 a.s. is means that the prey x 2 and the predator y(t) will go to extinction in the absence of the prey x 1 (see Figure 15).
Example 13. We assume that σ 1 � 0.05, σ 2 � 0.05, and σ 3 � 0.02. If we take v � 0.1 (v � 0.2, v � 0.3, or v � 0.4); then, all the conditions of eorem 11 hold. us, from eorem 11, it follows that model (5) is stochastically permanent. Here, we give the numerical simulations of , and (f ) phase plan of k 3 (v, σ 2 3 ).        26 Complexity model (5) with different v (see Figure 24). As can be seen from Figure 24, with the increase of v, that is, the proportion of time that predators stay in patch 1 increases, the number of prey in patch 1 decreases, while the number of prey in patch 2 increases. is has a reasonable biological significance. However, the mortality, the encounter rate with the  prey, the half-saturation constant, and the conversion rate of the predators are different in different patches, and it is impossible to determine the number of predators with the change of v.

Conclusions and Discussion
is paper is concerned with a stochastic two-patch predator-prey model with ratio-dependent functional responses.
First, by using the comparison theorem of stochastic differential equations, we show that the model has a unique global positive solution. en, the long-time properties of the solutions are discussed pathwise. Using the exponential martingale inequality and Borel-Cantelli lemma, we show that the sample Lyapunov exponents are nonpositive. Moreover, under certain conditions, we show that the sample Lyapunov exponents are zero. Next, the sufficient conditions for the extinction and persistence in mean of the