具有阶段结构和Allee效应的捕食–食饵模型的随机动力学分析
Stochastic Dynamics Analysis of a Predator-Prey Model with Stage Structure and Allee Effect
摘要: 本文研究了一类具有阶段结构和Allee效应的捕食–食饵模型的随机动力学行为。首先，我们讨论的模型是由3个随机微分方程构成，它主要描述了食饵、未成熟捕食者和成熟捕食者之间的相互作用。其次，为了证明模型存在全局唯一正解，我们构造多个合适的李雅普诺夫函数，并建立了系统存在遍历平稳分布的条件。最后，我们建立了食饵和两个捕食者灭绝的条件。
Abstract: In this paper, the stochastic dynamics of a predator-prey model with stage structure and Allee effect is handled and established based on some suitable Lyapunov functions. First of all, our model is composed of three random differential equations, which describes the interaction between predator species and prey. Secondly, to prove the existence of a globally unique positive solution to the model, we construct several appropriate Lyapunov functions, and then establish the conditions for the existence of ergodic stationary distribution for the system. Finally, we establish the extinction conditions for the prey and two species of predator.
文章引用：冯赛提, 张天四. 具有阶段结构和Allee效应的捕食–食饵模型的随机动力学分析[J]. 应用数学进展, 2021, 10(4): 909-922. https://doi.org/10.12677/AAM.2021.104099

参考文献

[1] Lotka, A. (1925) The Elements of Physical Biology. Williams and Wilkins, Baltimore.
[2] Volterra, V. (1926) Fluctuations in the Abundance of Species Considered Mathematically. Nature, 118, 558-560. [Google Scholar] [CrossRef] 
[3] Cai, Y.L., Zhao, C.D. and Wang, W.M. (2015) Dynamics of a Leslie-Gower Predator-Prey Model with Additive Allee Effect. Applied Mathematical Modelling, 39, 2092-2106. [Google Scholar] [CrossRef] 
[4] Tian, B.D., Yang, L. and Zhong, S.M. (2015) Global Stability of a Stochastic Predator-Prey Model with Allee Effect. Applied Mathematical Modelling, 8, 1550044. [Google Scholar] [CrossRef] 
[5] Liu, Q., Jiang, D.Q., Ahmad, B. and Hayat, T. (2018) Stationary Distribution and Extinction of a Stochastic Predator-Prey Model with Additional Food and Nonlinear Perturbation. Applied Mathematics and Computation, 320, 226-239. [Google Scholar] [CrossRef] 
[6] Xu, D.S., Liu, M. and Xu, X.F. (2020) Analysis of a Stochastic Predator-Prey System with Modified Leslie-Gower and Holling-Type IV Schemes. Physica A: Statistical Mechanics and its Applications, 537, 122761. [Google Scholar] [CrossRef] 
[7] Allee, W.C. (1931) Animal Aggregations: A Study in General Sociology. The University of Chicago Press, Chicago. [Google Scholar] [CrossRef] 
[8] Andrew, M.K., Brian, D., Andrew, M.L. and John, M.D. (2009) The Evidence for Allee Effects. Population Ecology, 51, 341-354.
[9] Franck, C., Lude, B. and Joanna, G. (2008) Allee Effects in Ecology and Conservation. Oxford University Press, Oxford.
[10] Wang, J.F., Shi, J.P. and Wei, J.J. (2011) Predator-Prey System with Strong Allee Effect in Prey. Journal of Mathematical Biology, 62, 291-331. [Google Scholar] [CrossRef] [PubMed]
[11] Lin, Q. (2018) Allee Effect Increasing the Final Density of the Species Subject to the Allee Effect in a Lotka-Volterra Commensal Symbiosis Model. Advances in Difference Equations, 2018, Article No. 196. [Google Scholar] [CrossRef] 
[12] Lin, Q.F. (2018) Stability Analysis of a Single Species Logistic Model with Allee Effect and Feedback Control. Advances in Difference Equations, 2018, Article No. 190. [Google Scholar] [CrossRef] 
[13] Chen, B.G. (2018) Dynamic Behaviors of a Commensal Symbiosis Model Involving Allee Effect and One Party Can Not Survive Independently. Advances in Difference Equations, 2018, Article No. 212. [Google Scholar] [CrossRef] 
[14] Biswas, S., Pal, D., Mahapatra, G.S. and Samanta, G.P. (2020) Dynamics of a Prey-Predator System with Herd Behaviour in Both and Strong Allee Effect in Prey. Biophysics, 65, 826-835. [Google Scholar] [CrossRef] 
[15] Lai, L.Y., Zhu, Z.L. and Chen, F.D. (2020) Stability and Bifurcation in a Predator-Prey Model with the Additive Allee Effect and the Fear Effect. Mathematics, 8, 1280. [Google Scholar] [CrossRef] 
[16] Zu, J. (2013) Global Qualitative Analysis of a Predator-Prey System with Allee Effect on the Prey Species. Mathematics and Computers in Simulation, 94, 33-54. [Google Scholar] [CrossRef] 
[17] Xiao, Z.W. and Li, Z. (2019) Stability and Bifurcation in a Stage-Structured Predator-Prey Model with Allee Effect and Time Delay. International Journal of Applied Mathematics, 49, 1.
[18] Yang, K., Miao, Z., Chen, F. and Xie, X. (2016) Influence of Single Feedback Control Variable on an Autonomous Holling-II Type Cooperative System. Journal of Mathematical Analysis and Applications, 435, 874-888. [Google Scholar] [CrossRef] 
[19] Li, T., Chen, F., Chen, J. and Lin, Q. (2017) Stability of a Stage-Structured Plant-Pollinator Mutualism Model with the Beddington-DeAngelis Functional Response. Journal of Nonlinear Functional Analysis, 2017, Article ID: 50. [Google Scholar] [CrossRef] 
[20] Sai, A. and Nan, K. (2018) Sparse Grid Interpolation of Ito Stochastic Models in Epidemiology and Systems Biology. IAENG International Journal of Applied Mathematics, 48, 1.
[21] Wang, L.L. and Xie, P.L. (2017) Permanence and Extinction of Delayed Stage-Structured Predator-Prey System on Time Scales. Engineering Letters, 25, 2.
[22] Li, Z., Han, M.A. and Chen, F.D. (2012) Global Stability of Stage-Structured Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes. International Journal of Biomathematics, 5, 1250057. [Google Scholar] [CrossRef] 
[23] Lin, Y.H., Xie, X.D., Chen, F.D. and Li, T.T. (2016) Convergences of a Stage-Structured Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes. Advances in Difference Equations, 2016, Article No. 181. [Google Scholar] [CrossRef] 
[24] Liu, Q., Jiang, D.Q., Hayat, T. and Alsaedi, A. (2018) Dynamics of a Stochastic Predator-Prey Model with Stage Structure for Predator and Holling Type II Functional Response. Journal of Nonlinear Science, 28, 1151-1187. [Google Scholar] [CrossRef] 
[25] Liu, M. and Wang, K. (2011) Global Stability of Stage-Structured Predator-Prey Models with Beddington-DeAngelis Functional Response. Communications in Nonlinear Science and Numerical Simulation, 16, 3792-3797. [Google Scholar] [CrossRef] 
[26] Zhao, X. and Zeng, Z.J. (2020) Stationary Distribution and Extinction of a Stochastic Ratio-Dependent Predatorprey System with Stage Structure for the Predator. Physica A: Statistical Mechanics and its Applications, 545, 123310. [Google Scholar] [CrossRef] 
[27] Mao, X. (1997) Stochastic Differential Equations and Applications. Horwood Publishing, Chichester.
[28] Has’minskii, R.Z. (1980) Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn.
[29] Wang, H. and Liu, M. (2020) Stationary Distribution of a Stochastic Hybrid Phytoplankton-Zooplankton Model with Toxin-Producing Phytoplankton. Applied Mathematics Letters, 101, 106077. [Google Scholar] [CrossRef] 
[30] Liu, Q. and Jiang, D. (2018) Stationary Distribution and Extinction of a Stochastic Predator-Prey Model with Distributed Delay. Applied Mathematics Letters, 78, 79-87. [Google Scholar] [CrossRef] 
[31] Lu, C. and Ding, X.H. (2019) Periodic Solutions and Stationary Distribution for a Stochastic Predator-Prey System with Impulsive Perturbations. Applied Mathematics and Computation, 350, 313-322. [Google Scholar] [CrossRef] 
[32] Liu, Q., Jiang, D., Hayat, T. and Alsaedi, T. (2018) Stationary Distribution and Extinction of a Stochastic Predator-Prey Model with Herd Behavior. Journal of the Franklin Institute, 255, 8177-8193. [Google Scholar] [CrossRef] 

为你推荐