Abstract
Considering the impacts of white noise, Holling-type II functional response, and regime switching, we formulate a stochastic predator-prey model in this paper. By constructing some suitable functionals, we establish the sufficient criteria of the stationary distribution and stochastic permanence. By numerical simulations, we illustrate the results and analyze the influence of regime switching on the dynamics.
1. Introduction
Functional responses are very important in the predator-prey system, which is the amount of prey catch per predator per unit of time and has significant effect on the dynamical properties. Usually there are two kinds of functional response: prey dependent (such as Holling II and Holling IV, see [1–3]) and predator dependent (such as Hassell–Varley, Beddington–DeAngelis, and Crowley–Martin, see [4, 5]). Recently, a number of researchers have devoted their efforts to the predator-prey system with functional response and obtained some nice results [1–7].
For the ecological system, the growth rate of population is inevitably affected by environmental white noise, which almost exists everywhere in real world [8–10]. May reveals that due to stochastic fluctuations in environmental conditions, all the natural parameters exhibit a certain amount of random perturbations, and hence, random disturbance is introduced in many mathematical models to reveal the effect of white noise [10–15]. Besides the white noise, the growth of species also suffers from fluctuating environments such as hurricanes and earthquakes, which is described by colorful noise in mathematical modelling [16–18]. The colorful noise may take several values and switch among different regimes of environments. The switching is memoryless, and the waiting time for the next switching follows an exponential distribution. That is, in mathematical sense, it is a Markovian process. Actually, when the environments fluctuate frequently, colorful noise may bring great influence to population dynamics and even change the permanence and extinction of species, so the impacts of colorful noise on population dynamics have attracted many researchers, see, e.g., [19–22].
Motivated by above discussion, in this article, we formulate a stochastic model with Holling-type II functional response and colorful noise. By stochastic analysis, we aim to study the stability in distribution and stochastic permanence of the system.
The rest of this paper is structured as follows. Section 2 begins with our model and some notations. Section 3 is devoted to the stability in distribution of the above system. Section 4 focuses on the stochastic permanence. Some examples are given to illustrate our main results in Section 5. Finally, a brief conclusion and discussion are given to end the paper in Section 6.
2. The Model and Notations
Hsu and Huang [6] proposed the following predator-prey model with Holling-type II functional response:where and represent the birth rate of prey and death rate of predator, respectively; and are intraspecific competition rate between species; is the capture rate, and is the conversion rate of food; denotes the density of white noise; is the Holling-type II functional response. and are independent standard Brownian motions defined on the probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null set). In view of the impact of regime switching (colorful noise) analyzed before, system (1) turns to the following:
The regime switching is a Markovian chain in a finite state space . The generator of is defined as withwhere is the transition rate from the th stage to the th stage and if while . It is often assumed that every sample of is a right continuous step function and irreducible with a finite simple jumps in any finite subinterval of . It obeys a unique stationary distribution satisfying and . The detailed switching mechanism of the hybrid system is referred to [19, 23].
Let , then system (2) is equivalent to the following model:
For the later discuss, we introduce some notations about the Itô’s integral for stochastic differential equations with Markovian switching [19, 22]. Letwhere are measurable functions. Let . Define the operator as follows:where , , and .
The generalized Itô’s formula is defined as
Lemma 1 (see [21]). If the following conditions hold.(i)For .(ii)For each and any holds with for all .(iii)There exists a bounded open subset with a regular boundary (i.e., smooth) such that, for any , there exists a nonnegative function satisfying is twice continuously differentiable and for some ,
Then, (5) is ergodic and positive recurrent; that is, there exists a unique stationary density , for any Borel measurable function with , we have
About the existence and uniqueness of positive solutions and the moment boundedness of (2), we have the following two lemmas. The proofs of them are very standard and are omitted here. Readers may refer to [3, 21].
Lemma 2. There is a unique positive solution for system (2) on with initial value , and the solution will remain in with probability 1.
Lemma 3. For any initial value and any , there exists a constant such that the solution for system (2) satisfying for all .
For simplicity, we give some notations as follows:
3. Stationary Distribution
In this section, we discuss the stationary distribution of (2).
Theorem 1. For any initial value and any , the solution of (2) is ergodic and has a unique stationary distribution in if the following condition holds:
Proof. According to the equivalent property of (2) and (4), we only need to prove it for (4). Define , where , and then we havewhereOn the other hand,Set , and similarly we haveDefine , and thenwhere . Let , where . It is easy to observe thatandDefine a bounded closed set as follows:where is a sufficiently small number, and then the set contains the following four domains:Take sufficiently small enough such thatwhere are defined later. Next, we verify for all .
Case 1. If , namely, , then . By (17), (18), and (21), we have
Case 2. If , namely, , then , and similarly we have
Case 3. If , then we derive from (17) and (22) thatwhere .
Case 4. If , similarly, from (17) and (22) we havewhere .
Consequently, we deduce that on all . Obviously, the other condition of Lemma 1 holds too, so we conclude from Lemma 1 that system (4) is ergodic and has a unique stationary distribution in ; that is, system (2) is ergodic and has a unique stationary distribution in . This completes the proof.
For (2), if the state Markovian chain takes value in space , namely, there is no switching, then (2) turns to the following subsystem:For (27), from Theorem 1, we can easily obtain the following conclusion.
Corollary 1. For any initial value , the solution of (27) is ergodic and has a unique stationary distribution in if the following condition holds:
Remark 1. It is clear that, for any positive integer . That is, Theorem 1 shows that switching system (2) has stationary distribution only under the condition that every subsystem of (2) has stationary distribution. If there exists no switching, Corollary 1 gives the sufficient condition of stationary distribution of (27), which is accordant with Theorem 1 of [3].
4. Stochastic Permanence
For (2), if we consider the birth rate instead of the death rate of predator, then (2) turns to the following model:where is the birth rate of species and other parameters are the same as before. Now, we consider the stochastic permanence of (29).
Definition 1. (see [16])System (29) is stochastically permanent if for every and any , there is a pair of constants and such that for any initial data , the solution of (29) has the property thatwhere represents the probability of events.
Assumption 1. For some .
Lemma 4. Under Assumption 1, if , then there exists such that for any is a nonsingular M-matrix, where .
Remark 2. The proof is rather standard. Readers may refer to the details in [24] or [21, 23].
Theorem 2. For any initial value , system (29) is stochastically permanent under conditions of Lemma 3.
Proof. The proof is motivated by [22]. Let be a matrix or vector, and denote by all the elements of are positive. Under the hypotheses, Lemma 2 shows is a nonsingular M-matrix, and then by M-matrix theory (see Theorem 2.1 [22]), there exists such that , that is, . So, there exists a constant such thatDefine functional , where , then for above , we compute as follows:where , andBy Ito’s formula, we havewhere . Integrating from 0 to and taking expectation giveHence,Let , thenSince , then we deduce that , and hence,Therefore, holds. By Lemma 3, using Chebyshev’s inequality again, it is clear that for some constant . Therefore, (29) is stochastically permanent by Definition 1. The proof is completed.
Obviously, if there is no switching, we can similarly obtain the following corollary.
Corollary 2. For any initial value , the subsystem of (29) is stochastically permanent if .
Remark 3. Theorem 2 reveals that when some subsystems of (2) are no stochastic permanent, if we give a suitable switching, then switching system (2) may be stochastic permanent, which implies the switching has very important influence to the dynamics of (2). By simulation, we can verify it directly, see Figure 1.
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5. Examples and Simulations
In this section, some examples are given to illustrate our theoretical results and reveal the effects of regime switching and stochastic factors [25]. For simplicity, we assume that the continuous-time discrete state Markovian chain takes value in the space , then system (2) reduces to the following subsystems:
We let ; . By Corollary 1, we know that (39) and (40) both have stationary distribution, see Figure 2.
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Suppose the distribution of is (see Figure 3). It is easy to verify that . Theorem 1 implies that (2) has stationary distribution, see Figure 4.
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If then Theorem 2 shows that (39) and (40) are stochastic permanence, see Figure 5.
If , and , then the switching system is not stochastic permanence, but if we take , then the system is stochastic permanence, which implies the switching is very important to make (39) and (40) be permanent, see Figure 1.
6. Conclusions and Discussion
In this paper, we study a stochastic predator-prey system with regime switching and Holling-type II functional responses. Theorems 1 and 2 give the sufficient conditions of stationary distribution and the stochastic permanence of (2). Finally, some examples are given to verify the main results. Our numerical examples reveal that switching and random factors bring much influence to the dynamics of this system.
By comparison analysis, we give Remarks 1 and 2 to show that our main results improve or generalize the corresponding results in [3]. The main method applied in this paper is by constructing some suitable functionals instead of stochastic analysis techniques to study the stationary distribution, which is less applied for switching system. In the process of our analysis, Holling-type II functional response brings some difficulties and we apply inequality techniques to overcome them.
As Arditi and Ginzburg [23] pointed out that the predator-dependent functional response can provide better description in some cases, then how to deal with predator-dependent functional response such as Beddington–DeAngelis type? Furthermore, studies have shown that the mental state of the adolescent prey can be mediated by fear induced from predators and the alternation causes deleterious outcomes on their adult’s survival [24] and then how fear will impact our system? All these are interesting for us to study in the future.
Data Availability
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.