A Stochastic Holling-Type II Predator-Prey Model with Stage Structure and Refuge for Prey

Shi, Wanying; Huang, Youlin; Wei, Chunjin; Zhang, Shuwen

doi:https://doi.org/10.1155/2021/9479012

Advances in Mathematical Physics

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9479012 | https://doi.org/10.1155/2021/9479012

A Stochastic Holling-Type II Predator-Prey Model with Stage Structure and Refuge for Prey

Wanying Shi,¹Youlin Huang,¹Chunjin Wei,¹and Shuwen Zhang¹

Academic Editor: Giorgio Kaniadakis

Received21 Apr 2021

Accepted14 Jun 2021

Published10 Jul 2021

Abstract

In this paper, we study a stochastic Holling-type II predator-prey model with stage structure and refuge for prey. Firstly, the existence and uniqueness of the global positive solution of the system are proved. Secondly, the stochastically ultimate boundedness of the solution is discussed. Next, sufficient conditions for the existence and uniqueness of ergodic stationary distribution of the positive solution are established by constructing a suitable stochastic Lyapunov function. Then, sufficient conditions for the extinction of predator population in two cases and that of prey population in one case are obtained. Finally, some numerical simulations are presented to verify our results.

1. Introduction

Population ecology is one of the most important research fields in biomathematics. There are three basic relationships among different species living in the same natural environment: (i) competitive relationship [1, 2], for example, crops and weeds compete for fertilizer, sunlight, and other resources for their own growth; (ii) reciprocal relationship [3, 4], such as rhinoceros birds feed on the parasites from rhinoceroses. For rhinoceroses, rhinoceros birds can get rid of the parasites and warn them when they are in danger; (iii) predation relationship [5, 6], as the saying goes, “big fish eat small fish, small fish eat shrimps.” Among the relationships of population interaction, the predator-prey system has been extensively studied. Lotka and Volterra put forward the predator-prey model in 1925 and 1926, respectively. Since then, many scholars have devoted themselves to the study of various predator-prey models and obtained abundant research results [7–10].

In all the above predator-prey systems, it is generally assumed that predators in the same population have identical predation ability, and the prey in the same population has identical viability and fertility. However, in the real world, the life of many creatures should be divided into two stages: immature and mature. There are recognizable morphological and behavioral differences that may exist between these stages. For example, the immature creatures have no fertility and predation ability, and their survival ability and defense ability are relatively weak. On the contrary, the mature creatures not only have reproductive ability and predator ability but also have strong survival ability and defensive ability. These will have a great impact on the dynamic behavior of the population. Thus, it is more practical to study the predator-prey model with stage structure. In view of this, many scholars have studied assorted predator-prey systems with stage structure and given corresponding dynamic analysis [11–17]. In particular, the following predator-prey model with stage structure for prey has been considered in [16]. where , , , , , , , , , and are positive constants and , and represent the size of the immature prey population, mature prey population, and predator population, respectively. And , and indicate the mortality rates of immature prey population, mature prey population, and predator population, respectively. The parameter denotes the birth rate of the immature prey population; is the transformation rate of the mature prey population. and denote the intraspecific competition coefficient of the immature prey population and predator population, respectively. is the digestion constant. Predators only prey on immature prey.

Functional response plays an important role in describing the predator-prey model, which refers to the response of predators’ predation rate to the density of prey, that is, the predation effect of predators on prey. Model (1) uses bilinear functional response to describe the interaction between predator and prey. However, many researchers [18–20] have pointed out that nonlinear functional response can more accurately describe the trend of population density. Specifically, the growth rate of predators described by Holling-type II functional response is increasing with the increase of the number of prey [21]. When the survival space is far from saturated and the resources are sufficient, the growth rate of predators is relatively fast, but when the survival space and resources are limited, the growth rate changes less significantly and finally approaches a fixed value. Holling-type II functional response is according to some practical phenomena and takes into account the influence of density constraint. Therefore, it has been discussed by many scholars and extensively applied [13, 22–26]. In addition, it is well known that in many cases, prey can avoid predators through refuges. For example, crabs on the beach hide under sand or stones to avoid seabirds, thus increasing their survival chance. In recent years, different predator-prey systems with refuges have attracted much attention [6, 27–30]. In [27], Qi and Meng studied the threshold behavior of a stochastic predator-prey system with prey refuge and fear effect. They concluded that the survival rate of prey can be improved by increasing the strength of refuge. In [29], Ghosh et al. have studied a prey-predator model incorporating prey refuge and additional food for predators and found that slightly higher refuge is beneficial to the coexistence of species, but strong refuge will lead to the extinction of the predator population.

Inspired by the above motivations, we consider the following Holling-type II predator-prey model with stage structure and refuge for prey: where denotes the influence of predators on prey, represents the half-saturation constant, denotes the refuge rate of immature prey, is the number of immature prey that predators can capture, and is the capture rate of predators.

On the other hand, in nature, the biological population is inevitably affected by environmental noise. May [31] pointed out that due to the continuous fluctuation of the environment, the birth rate, mortality, environmental capacity, and other parameters in the model will show random fluctuations in varying degrees. However, the deterministic model does not consider the impact of environmental disturbance, so to some extent, it cannot accurately predict the dynamic behavior of the population. In order to describe the environmental noise better, many scholars consider the environmental noise as white noise, for instance, [22–27, 32, 33]. Considering the sensitivity of mortality to environmental noise, in this paper, we assume that the mortality rates , and are disturbed by environmental noise, i.e., where are the intensities of the white noise and represent independent standard Brownian motions which are defined on the complete probability space with a filtration satisfying the usual conditions (that means it is increasing and right continuous while contains all -null sets). Then, the stochastic model corresponding to the deterministic model (2) is derived in the following form:

Based on the analysis above, the paper focuses on the analysis of some dynamic behaviors of the stochastic Holling-type II predator-prey model with stage structure and refuge for prey. The major contributions are presented as follows. First, for the first time, the Holling-type II functional response is applied to establish a predator-prey model with stage structure and refuge for prey. Second, the existence and uniqueness as well as the stochastically ultimate boundedness of positive solution of the stochastic model are analyzed. Third, some conditions for the existence and uniqueness of ergodic stationary distribution of the stochastic model are established. Finally, some extinction conditions of the stochastic model are studied.

Moreover, in [26], Xu et al. studied a stochastic two-predator one-prey model with modified Leslie-Gower and Holling-type II schemes. The authors proved the existence and uniqueness of global positive solution of their stochastic model by using comparison theorem. In this paper, we consider it by constructing a Lyapunov function. Besides, in [4], Liu et al. discussed a mutualism system in random environments. They obtained the existence and uniqueness of a stable stationary distribution by means of Markov semigroup theory and Fokker-Planck equation. In this paper, we consider the existence and uniqueness of an ergodic stationary distribution of the stochastic model (4) by the stochastic Lyapunov function method.

The rest of this paper is arranged as follows. Some useful definitions and lemmas are given in Section 2. The existence and uniqueness of the global positive solution are discussed in Section 3. In Section 4, we obtain sufficient conditions for stochastically ultimate bounded of the prey and predator. In Section 5, the sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solution are established. In Section 6, we establish sufficient conditions for the extinction of the predator and prey in two cases. The first case is the extinction of the predator, and another case is that both the prey and the predator are extinct. To verify our results, some numerical simulations are presented in Section 7. Finally, the conclusion is given in Section 8.

2. Preliminaries

In this section, some definitions and lemmas are given to prepare for further work.

Lemma 1 ( formula) (see [34]). Let be a -dimensional process on with the stochastic differential where and Let Then, is again an process with the stochastic differential given by where

Definition 2 (see [35]). With respect to system (4), the solution is said to be stochastically ultimate bounded, if for , there is a positive constant such that for any initial data , the solution has the property that where

Definition 3 (see [36]). The transition probability function is said to be time-homogeneous (and the corresponding Markov process is called time-homogeneous) if the function is independent of , where and , denotes the -algebra of Borel sets in
Let be a regular time-homogeneous Markov process in described by the stochastic differential equation The diffusion matrix of the process is defined as follows:

Lemma 4 (see [36]). The Markov process has a unique ergodic stationary distribution if there exists a bounded open domain with regular boundary , having the following properties:
: the diffusion matrix is strictly positive definite for all .
: there exists a nonnegative -function such that is negative for any .

3. Existence and Uniqueness of the Global Positive Solution

The existence and uniqueness of global positive solution are the premises of studying the properties of population dynamics. From [34], for any given initial data , if the coefficients of SDE model (4) satisfy the linear growth condition and the local Lipschitz condition, then there exists a unique global positive solution . But the coefficients of model (4) only satisfy the local Lipschitz condition; then, the solution of the system will explode in a finite time. In this section, we will prove that SDE model (4) has a unique global positive solution.

Theorem 5. For any given initial data , there is a unique solution to system (4) on and the solution will remain in with probability 1.

Proof. Since the coefficients of system (4) satisfy the local Lipschitz condition, then for any given initial value , there is a unique local solution on (see [34]), where denotes the explosion time. To show this solution is global, we only need to prove Let be sufficiently large such that . For any integer , we define the following stopping time: We let ( denotes the empty set). From the definition of stopping time, it is easy to know that is increasing as . Set ; hence, If we can show that , then and for all . Thus, to complete the proof, we only need to prove If this statement is false, for any , there exists a constant such that Therefore, there is an integer , for all , we have Now, we define a -function : where is a positive constant that will be determined later. Since thus, is a nonnegative function. Applying formula to , we have where is defined by Choose , then where is a positive constant. The rest of the proof is similar to [37] and hence is omitted here. The proof is completed.

4. Stochastically Ultimate Boundedness

Theorem 5 shows that the solution of system (4) remains in the positive cone . However, this nonexplosion property in a population dynamical system is often not good enough. Hence, we need to consider another important asymptotic property: stochastically ultimate boundedness, which means that the solution will be ultimately bounded with large probability.

Theorem 6. If the conditions , , and hold, then, the solution of system (4) is stochastically ultimately bounded for any initial data .

Proof. For , define By Itô formula, we have where Let According to the conditions, we can find that function has an upper bound. We assume that its upper bound is as follows: Let and noticing , thus, According to (19), we can obtain By formula, we have Integrating both sides of (24) from 0 to and then taking expectations, we get Hence, we have namely, For any , let . By Chebyshev’s inequality, we can obtain Then, The proof of Theorem 6 is completed.□

5. Existence of Ergodic Stationary Distribution

In this section, we will study the sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions of system (4). The existence of an ergodic stationary distribution means that all species can coexist for a long time and are stochastic weakly persistent [17].

Theorem 7. Assume that and , then for any initial , system (4) has a unique ergodic stationary distribution.

Proof. According to Lemma 4, we first prove that the condition is true. The diffusion matrix of system (4) is given by Clearly, the matrix is strictly positive definite for all Hence, the condition in Lemma 4 holds.
Next, we verify that the condition holds. Let . By formula, we have Define , then we have where Define where is a sufficiently large number. Then, we obtain Define where . Let then where From the assumptions that has an upper bound. Let Then, Define a -function , where is a sufficiently small number satisfying the following condition: is a positive constant. Since is continuous and tends to as approaches the boundary of Hence, it must have a lower bound. Now, we assume that it gets this lower bound at point Let By (32), (34), (39), and (41), we have Define a bounded open set as follows: is a sufficiently small number. In the set , we can choose sufficiently small such that the following conditions hold: where is a positive constant which will be determined later. We can divide into six domains: Clearly, . Next, we will prove that for any .
For any , due to , by (43) and (45), we have For any , since , by (43) and (45), we can get From (43), we have For any , by (46) and (50), we obtain For any , by (46) and (50), we have For any , by (46) and (50), we obtain For any , by (46) and (50), we get Therefore, there is a sufficiently small , s.t. for any . So the condition in Lemma 4 is satisfied. Thus, system (4) has a unique stationary distribution and it has the ergodic property. We have completed the proof.□

6. Stochastic Extinction

In this section, we will establish sufficient conditions for the extinction of the prey and predator populations.

Theorem 8. Let be the solution of system (4) with an initial value . If and , then the predator population will die out exponentially with probability one, i.e.,

Proof. Define where is a positive constant that will be determined later. By formula, we have Let , then By formula, And hence, Integrating (61) from 0 to and then dividing by on both sides, we get Similar to [38], we can get and . Then, Thus, Then, we obtain

Remark 9. Theorem 8 shows that the larger noise intensity or high mortality of predators as well as high refuge rate of immature prey can lead to the extinction of the predator population. However, under these conditions, we cannot judge whether the immature and mature prey populations are extinct or not.

Theorem 10. Assume that is the solution of system (4) with initial data , if , then, all the populations will die out exponentially with probability one, i.e.,

Proof. Let . By formula, we have Rewriting as the following form Let the matrix it is clear that Thus, is negative definite. Define as the maximum eigenvalue of matrix , then we have Thus, we obtain Integrating (72) from 0 to and then dividing by on both sides, we get By the strong law of large number of the martingale, we have According to the nonnegativity of and , we can obtain that which implies that . In other words, the prey population goes to extinction exponentially with probability one. Because of , thus, and a set such that and By Itô formula, we have Integrating (77) from 0 to and then dividing by on both sides, we get Then, we obtain Let , then , which implies that .□

Remark 11. According to Theorem 10, we can find that the extinction of immature prey population leads to the extinction of predator population. This is true, since model (4) shows that the predator population has no additional food source.

7. Numerical Simulations

For the sake of confirming our theoretical results established in the previous sections, we numerically simulate stochastic system (4). We use Milstein’s higher order method mentioned in [39] to give numerical simulations. The discretization transformation of system (4) is given as follows: where the time increment are the intensities of the white noise, and denote mutually independent Gaussian random variables which follow distribution

First, we select the parameter values as follows: (i) , , , , , , , , , , Here, we consider the strengths of white noise as , , and . By simple calculation, we have that , Thus, under these circumstances, Theorem 7 holds. Then, we can obtain that system (4) has a unique ergodic stationary distribution. Figure 1 illustrates this. And from Figure 1, we can directly see that immature prey population, mature prey population, and predator population coexist for a long time and are stochastic persistent. In addition, Figure 1 shows that the trajectories of the solution for the perturbed system (4) will show random fluctuations (red lines) when the undisturbed system are smooth curves (blue lines). These results indicate that the population density does fluctuate more or less due to the influence of white noise.

Next, we choose the parameter values as (ii) , , , , , , , , , , , , , We can calculate that , Therefore, the conditions of Theorem 8 are satisfied. By Theorem 8, we get that the predator population goes extinct exponentially with probability one. Figures 2(a)–2(c) show the simulation results. In this case, we can see that the predator population is extinct, but the prey population is persistent. In addition, Theorem 8 shows that large noise intensity will lead to the extinction of the predator population. Therefore, in order to illustrate the effect of noise intensity on predator population, we choose and , respectively, and other parameters are taken as (ii). The simulation result is shown in Figure 2(d). From Figure 2(d), we can see that when the noise intensity is large, that is, when (see the cyan line of Figure 2(d)), the predator population is extinct, which can be directly proved by the conclusion of Theorem 8.

(a)

(b)

(c)

(d)

In order to illustrate the conclusion of Theorem 10, we choose the noise intensities as , , and , respectively. And other parameters are taken as (i). By simple calculation, we have that . Thus, the condition of Theorem 10 holds. By Theorem 10, we obtain that the prey and predator populations die out exponentially with probability one. The numerical simulation results are shown in Figure 3.

Finally, we use the numerical simulation method to analyze the effect of prey refuge on system (4). The refuge rate of immature prey varies from 0 to 1. means that the immature prey population has no refuge, that is to say, the immature prey population is completely exposed to the predatory environment of the predator population. On the contrary, indicates that the immature prey is totally protected by refuge. Now, we take and , respectively, and other parameters are as (i). The numerical simulation results are shown in Figure 4. Figure 4 shows that when , the immature prey population, mature prey population, and predator population are persistent, that is, the predator population and the prey population coexist, which can be verified by Theorem 7. But when , the predator population becomes extinct, because the predator population has no additional food source. Theorem 8 also proves this result.

(a)

(b)

(c)

8. Conclusion

In this paper, a stochastic Holling-type II predator-prey model with stage structure and refuge for prey is analyzed. Firstly, the existence and uniqueness of the global positive solution for system (4) have been proved. Due to its nonexplosive property, we obtain the conditions for the stochastically ultimate boundedness of positive solution. Next, the sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solution are established. Then, we obtain sufficient conditions for the extinction of predator population in two cases and that of prey population in one case. The theoretical results are proved by numerical simulations.

The following conclusions are verified by numerical simulations. Figure 1 shows that the existence of ergodic stationary distribution can allow all populations are coexistent and stochastic persistent in a long time. In Figure 2, it is shown that the large noise intensity will lead to the extinction of predator population, as shown in Figures 2(c) and 2(d), but the prey population can survive, as shown in Figures 2(a) and 2(b). Figure 3 shows that the extinction of immature prey population will lead to the extinction of predator population. This is due to the fact that predators do not have additional food sources. Finally, it can be seen from Figure 4(c) that if the immature prey population is completely protected by refuge, it will lead predator population towards extinction.

In addition, based on our results, from the perspective of ecological protection, we can provide additional food to the predators to ensure that the prey population and predator population maintain a high-density level. On the one hand, it can reduce the predation rate of the predator population to the prey [30]; on the other hand, it can ensure that the predator population will not be completely extinct when the refuge rate of the prey population is high.

Data Availability

Please contact the author for data requests.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

SZ and CW suggested the model, helped in result interpretation and manuscript evaluation, and supervised the development of work. WS and YH helped to evaluate, revise, and edit the manuscript. All authors read and approved the final manuscript.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (No. 11971405), Fujian Provincial Natural Science of China (No. 2018J01418), and National Natural Science Foundation Breeding Program of Jimei University (No. ZP2020064).

References

S. Yu, “Extinction for a discrete competition system with feedback controls,” Advances in Difference Equations, vol. 2017, no. 1, 10 pages, 2017.
View at: Google Scholar
G. P. Samanta, “A stochastic two species competition model: nonequilibrium fluctuation and stability,” International Journal of Stochastic Analysis, vol. 2011, Article ID 489386, 7 pages, 2011.
View at: Publisher Site | Google Scholar
Y. Wang and D. Deangelis, “A mutualism-parasitism system modeling host and parasite with mutualism at low density,” Mathematical Biosciences and Engineering, vol. 9, no. 2, pp. 431–444, 2012.
View at: Publisher Site | Google Scholar
G. Liu, H. Qi, Z. Chang, and X. Meng, “Asymptotic stability of a stochastic May mutualism system,” Computers and Mathematics with Applications, vol. 79, no. 3, pp. 735–745, 2020.
View at: Publisher Site | Google Scholar
J. Tripathi, S. Bugalia, T. Vandana, and K. Yun, “A predator-prey model with Crowley-Martin functional response: a nonautonomous study,” Natural Resource Modeling, vol. 33, no. 4, article e12287, 2020.
View at: Google Scholar
M. Onana, B. Mewoli, and J. Tewa, “Hopf bifurcation analysis in a delayed Leslie-Gower predator-prey model incorporating additional food for predators, refuge and threshold harvesting of preys,” Nonlinear Dynamics, vol. 100, no. 7, pp. 1–22, 2020.
View at: Google Scholar
G. Zhang, W. Wang, and X. Wang, “Coexistence states for a diffusive one-prey and two-predators model with B-D functional response,” Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 931–948, 2012.
View at: Publisher Site | Google Scholar
K. Sarkar and S. Khajanchi, “Impact of fear effect on the growth of prey in a predator-prey interaction model,” Ecological Complexity, vol. 42, article 100826, 2020.
View at: Publisher Site | Google Scholar
Z. Wang, M. Deng, and M. Liu, “Stationary distribution of a stochastic ratio-dependent predator-prey system with regime-switching,” Chaos Solitons and Fractals, vol. 142, article 110462, 2020.
View at: Google Scholar
F. Bian, W. Zhao, Y. Song, and R. Yue, “Dynamical analysis of a class of prey-predator model with beddington-deangelis functional response, stochastic perturbation, and impulsive toxicant input,” Complexity, vol. 2017, Article ID 3742197, 18 pages, 2017.
View at: Publisher Site | Google Scholar
B. Ghosh, O. L. Zhdanova, B. Barman, and E. Frisman, “Dynamics of stage-structure predator-prey systems under density-dependent effect and mortality,” Ecological Complexity, vol. 41, p. 100812, 2020.
View at: Google Scholar
C. Xu, G. Ren, and Y. Yu, “Extinction analysis of stochastic predator-prey system with stage structure and Crowley-Martin functional response,” Entropy, vol. 21, no. 3, p. 252, 2019.
View at: Publisher Site | Google Scholar
G. P. Neverova, O. L. Zhdanova, B. Ghosh, and E. Frisman, “Dynamics of a discrete-time stage-structured predator-prey system with Holling type II response function,” Nonlinear Dynamics, vol. 98, no. 1, pp. 427–446, 2019.
View at: Publisher Site | Google Scholar
X. Zhao and Z. Zeng, “Stationary distribution of a stochastic predator-prey system with stage structure for prey,” Physica A: Statistical Mechanics and its Applications, vol. 545, p. 123318, 2020.
View at: Publisher Site | Google Scholar
Y. Lu, K. Pawelek, and S. Liu, “A stage-structured predator-prey model with predation over juvenile prey,” Applied Mathematics and Computation, vol. 297, pp. 115–130, 2017.
View at: Publisher Site | Google Scholar
X. A. Zhang, L. S. Chen, and A. U. Neumann, “The stage-structured predator-prey model and optimal harvesting policy,” Mathematical Biosciences, vol. 168, no. 2, pp. 201–210, 2000.
View at: Publisher Site | Google Scholar
Q. Liu, D. Jiang, A. Alsaedi, and B. Ahmad, “Dynamical behavior of a stochastic predator-prey model with stage structure for prey,” Stochastic Analysis and Applications, vol. 35, no. 4, pp. 1–21, 2020.
View at: Google Scholar
M. Liu and K. Wang, “Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1114–1121, 2011.
View at: Publisher Site | Google Scholar
G. Gilioli, S. Pasquali, and F. Ruggeri, “Nonlinear functional response parameter estimation in a stochastic predator-prey model,” Mathematical Biosciences and Engineering, vol. 9, no. 1, pp. 75–96, 2017.
View at: Publisher Site | Google Scholar
X. Huang, X. Kong, and W. Yang, “Permanence of periodic predator-prey system with general nonlinear functional response and stage structure for both predator and prey,” Abstract and Applied Analysis, vol. 2009, Article ID 481712, 8 pages, 2009.
View at: Publisher Site | Google Scholar
C. S. Holling, “The functional response of predator to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 97, pp. 5–60, 1965.
View at: Publisher Site | Google Scholar
Z. Li and D. Jiang, “Periodic solution for a stochastic non-autonomous predator-prey model with Holling II functional response,” Acta Applicandae Mathematicae, vol. 161, no. 1, pp. 89–105, 2019.
View at: Publisher Site | Google Scholar
M. Liu, X. Chen, and M. Deng, “Persistence and extinction of a modified Leslie-Gower Holling-type II stochastic predator-prey model with impulsive toxicant input in polluted environments,” Nonlinear Analysis: Hybrid Systems, vol. 27, pp. 177–190, 2018.
View at: Publisher Site | Google Scholar
M. Sambath, K. Balachandran, and Y. K. Ma, “Analysis of stochastic predator-prey model with disease in the prey and Holling type II functional response,” Advances in Mathematical Physics, vol. 2020, Article ID 3632091, 17 pages, 2020.
View at: Publisher Site | Google Scholar
M. Liu, “Dynamics of a stochastic regime-switching predator–prey model with modified Leslie–Gower Holling-type II schemes and prey harvesting,” Nonlinear Dynamics, vol. 96, no. 1, pp. 417–442, 2019.
View at: Publisher Site | Google Scholar
Y. Xu, M. Liu, and Y. Yang, “Analysis of a stochastic two-predators one-prey system with modified Leslie-Gower and Holling-type II schemes,” Journal of Applied Analysis & Computation, vol. 7, no. 2, pp. 713–727, 2017.
View at: Publisher Site | Google Scholar
H. Qi and X. Meng, “Threshold behavior of a stochastic predator-prey system with prey refuge and fear effect,” Applied Mathematics Letters, vol. 113, article 106846, 2021.
View at: Publisher Site | Google Scholar
F. Wei and Q. Fu, “Globally asymptotic stability of a predator-prey model with stage structure incorporating prey refuge,” International Journal of Biomathematics, vol. 9, no. 4, pp. 155–168, 2016.
View at: Google Scholar
J. Ghosh, B. Sahoo, and S. Poria, “Prey-predator dynamics with prey refuge providing additional food to predator,” Chaos Solitons and Fractals, vol. 96, pp. 110–119, 2017.
View at: Publisher Site | Google Scholar
A. Das and G. P. Samanta, “A prey-predator model with refuge for prey and additional food for predator in a fluctuating environment,” Physica A: Statistical Mechanics and its Applications, vol. 538, article 122844, 2019.
View at: Google Scholar
R. M. May, Stability and Complexity in Model Ecosystems, Princeton university press, 2001.
P. Das and S. Sengupta, “Dynamics of two-prey one-predator non-autonomous type-III stochastic model with effect of climate change and harvesting,” Nonlinear Dynamics, vol. 97, no. 4, pp. 2777–2798, 2019.
View at: Google Scholar
Z. Cao, W. Feng, X. Wen, and L. Zu, “Stationary distribution of a stochastic predator-prey model with distributed delay and higher order perturbations,” Physica A: Statistical Mechanics and its Applications, vol. 521, pp. 467–475, 2019.
View at: Publisher Site | Google Scholar
X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997.
X. Li, A. Gray, D. Jiang, and X. Mao, “Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 11–28, 2011.
View at: Publisher Site | Google Scholar
R. Khasminskii, Stochastic Stability of Differential Equations, Springer Science & Business Media, The Netherlands, 1980.
G. Lan, Y. Fu, C. Wei, and S. Zhang, “Dynamical analysis of a ratio-dependent predator–prey model with Holling III type functional response and nonlinear harvesting in a random environment,” Advances in Difference Equations, vol. 2018, no. 1, 25 pages, 2018.
View at: Publisher Site | Google Scholar
Y. Zhao and D. Jiang, “The threshold of a stochastic SIS epidemic model with vaccination,” Applied Mathematics and Computation, vol. 243, pp. 718–727, 2014.
View at: Publisher Site | Google Scholar
D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Wanying Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

583

Downloads

668

Citations