×

Dynamical properties of a stochastic predator-prey model with functional response. (English) Zbl 1456.92123

Summary: A stochastic prey-predator model with functional response is investigated in this paper. A complete threshold analysis of coexistence and extinction is obtained. Moreover, we point out that the stochastic predator-prey model undergoes a stochastic Hopf bifurcation from the viewpoint of numerical simulations. Some numerical simulations are carried out to support our results.

MSC:

92D25 Population dynamics (general)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. R. Beddington,Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 1975, 44, 331-340.
[2] Y. Chen, Z. Liu and M. Haque,Analysis of a Leslie-Gower-type prey-predator model with periodic impulsive perturbations, Commun. Nonlinear Sci., 2009, 14, 3412-3423. · Zbl 1221.34032
[3] S. Chen, J. Wei and J. Yu,Stationary patterns of a diffusive predator-prey model with Crowley-Martin functional response, Nonliear Anal.-Real., 2018, 39, 33-57. · Zbl 1379.35113
[4] C. Chiarella, X. He, D. Wang and M. Zheng,The stochastic bifurcation behaviour of speculative financial markets, Physica A., 2008, 387, 3837-3846.
[5] P. H. Crowley and E. K. Martin,Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 1989, 8, 211-221.
[6] D. L. DeAngelis, R. A. Goldsten and R. V. O’Neill,A model for trophic interaction, Ecology., 1975, 56, 881-892.
[7] T. S. Doan, M. Engel, J. S. W. Lamb and M. Rasmussen,Hopf bifurcation with additive noise, Nonlinearity., 2018, 31(10), 4567-4601. · Zbl 1396.37062
[8] N. Du, D. H. Nguyen and G. Yin,Conditions for permanence and ergodicity of certain stochastic predator-prey models, J. Appl. Probab., 2016, 53, 187- šC202. · Zbl 1338.34091
[9] T. C. Gard,Persistence in stochastic food web models, B. Math. Biol., 1984, 46, 357-370. · Zbl 0533.92028
[10] T. C. Gard,Stability for multispecies population models in random environments, Nonlinear Anal., 1986, 10, 1411-1419. · Zbl 0598.92017
[11] M. P. Hassell and G. C. Varley,New inductive population model for intersect parasites and its bearing on biological control, Nature. 1969, 223, 1133-1137.
[12] A. Hening and D. H. Nguyen,Coexistence and extinction for stochastic Kolmogorov systems, Ann. Appl. Probab., 2018, 28, 1893-1942. · Zbl 1410.60094
[13] A. Hening and D. H. Nguyen,Persistence in stochastic Lotka-Volterra food chains with intraspecific competition, Bull. Math. Biol., 2018, 80, 2527-2560. · Zbl 1400.92435
[14] Z. Huang, Q. Yang and J. Cao,Stochastic stability and bifurcation for the chronic state in Marchuk’s model with noise, Appl. Math. Model., 2011, 35, 5842-5855. · Zbl 1228.93086
[15] D. Huang, H. Wang, J. Feng and Z. Zhu,Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics, Chaos Solit. Fract., 2006, 27, 1072- 1079. · Zbl 1134.34312
[16] C. Ji, D. Jiang and N. Shi,Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 2009, 359, 482-498. · Zbl 1190.34064
[17] W. Li, W. Xu, J. Zhao and Y. Jin,Stochastic stability and bifurcation in a macroeconomic model, Chaos Solit. Fract., 2007, 31, 702-711. · Zbl 1133.91484
[18] X. Liu, S. Zhong, B. Tian and F. Zheng,Asymptotic of a stochastic predatorprey model with Crowley-Martin functional response, J. Appl. Math. Comput., 2013, 43, 479-490. · Zbl 1325.92073
[19] J. Lv, H. Liu and X. Zou,Stationary distribution and persistence of a stochastic predator-prey model with a functional response, J. Appl. Anal. Comput., 2019, 9(1), 1-11. · Zbl 1465.92096
[20] A. P. Maiti, B. Dubey and J. Tushar,A delayed prey-predator model with Crowley-Martin-type functional response including prey refuge, Math. Method Appl. Sci., 2017, 40, 5792-5809. · Zbl 1383.37073
[21] R. M. May,Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1973.
[22] G. Pang, F. Wang and L. Chen,Extinction and permanence in delayed stagestructure predator-prey system with impulsive effects, Chaos Soliton. Frac., 2009, 39, 2216-2224. · Zbl 1197.34158
[23] K. R. Schenk-Hopp´e,Stochastic hopf bifurcation: an example, Int. J. Non-Lin Mech., 1996, 31, 685-692. · Zbl 0900.70357
[24] H. Shi and S. Ruan,Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 2015, 80, 1534-1568. · Zbl 1327.35376
[25] X. Shi, X. Zho and X. Song,Analysis of a stage-structured predator-prey model with Crowley-Martin function, J. Appl. Math. Comput., 2011, 36, 459-472. · Zbl 1222.34054
[26] J. P. Tripathi, S. Tyagi and S. Abbas,Global analysis of a delayed density dependent predator-prey model with Crowley-Martin functional response, Commun. Nonlinear Sci., 2016, 30, 45-69. · Zbl 1489.92125
[27] C. Wang, Z. Xiong, R. He and H. Yin,Dynamical behaviors of stochastic delayed one-predator and two-competing-prey systems with Holling type IV and Crowley-Martin type functinal responses, Discrete Dyn. Nat. Soc., 2016, 7676101, 1-16. · Zbl 1368.92159
[28] Y. Zhang, S. Gao, K. Fan and Y. Dai,On the dynamics of a stochastic ratiodependent predator-prey model with a specific functional reponse, J. Comput. Math. Appl., 2015, 48, 441-460. · Zbl 1321.34072
[29] Z. Sun, J. Lv and X. Zou,Dynamical analysis on two stochastic single-species models, Appl. Math. Lett., 2020. DOI: 10.1016/j.aml.2019.07.013. · Zbl 1423.92223
[30] X. Zou, Y. Zheng, L. Zhang and J. Lv,Survivability and stochastic bifurcations for a stochastic Holling type II predator-prey model, Commun. Nonlinear Sci., 2020. DOI: 10.1016/j.cnsns.2019.105136. · Zbl 1453.92282
[31] X. Zou, J. Lv and Y. Wu,A note on a stochastic Holling-II predator-prey model with a prey refuge, J. Franklin Inst., 2020, 357(7), 4486-4502. · Zbl 1437.92108
[32] J. Zhou,Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses, Commun. Pur. Appl. Anal., 2015, 14, 1127-1145. · Zbl 1312.35015
[33] X. Zhou and J. Cui,Global stability of the viral dynamics with crowley-martin functional response, Bull. Korean Math. Soc., 2011, 48, 555-574. · Zbl 1364.34076
[34] X.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.