一类带有时滞效应的修正的Leslie-Gower捕食者–食饵模型动力学分析
Dynamics Analysis of a Modified Leslie-Gower Predator-Prey Model with Time Delay
DOI: 10.12677/AAM.2021.104131, PDF,   
作者: 苏婧婧：温州大学，浙江 温州
关键词: 时滞；Allee效应；食饵避难；Hopf分岔；Time Delay； Allee Effect； Prey Refuge； Hopf Bifurcation
摘要: 本文构建了一类带有时滞效应的修正的Leslie-Gower捕食者–食饵模型，旨在研究时滞效应对捕食者–食饵模型动力学性质的影响。运用时滞微分方程理论，分析了模型中由时滞导致的Hopf分岔的存在性。借助于中心流形定理和规范型理论进一步推导Hopf分岔的方向以及周期解的稳定性。此外，数值模拟揭示了种群的密度与平衡点的稳定性对Allee效应、食饵避难以及时滞的变化具有一定的敏感性。
Abstract: This paper constructs a modified Leslie-Gower predator-prey model with Allee effect and prey refuge. The aim of this paper is to study the effect of time delay on the population dynamics. By using delay differential equation theory, we demonstrate the existence of Hopf bifurcation which is induced by time delay. According to center manifold theorem and normal form theory, the direction of Hopf bifurcation and stability of periodic solution are discussed. Additionally, numerical simulations reveal that the population density and the stability of positive equilibrium are sensitive for variations of time delay, Allee effect and prey refuge.
文章引用：苏婧婧. 一类带有时滞效应的修正的Leslie-Gower捕食者–食饵模型动力学分析[J]. 应用数学进展, 2021, 10(4): 1207-1221. https://doi.org/10.12677/AAM.2021.104131

参考文献

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[32] Freedman, H.I. and Mathsen, R.M. (1993) Persistence in Predator-Prey Systems with Ratio-Dependent Predator Influence. Bulletin of Mathematical Biology, 55, 817-827. [Google Scholar] [CrossRef] 
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[34] 潘陈蓉, 陈松林. 具有广义Holling III型功能反应的Leslie捕食-食饵系统的平衡点分类和稳定性分析[J]. 应用数学进展, 2019, 8(10): 1625-1631.
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[36] Leslie, P.H. and Gower, J.C. (1960) The Properties of a Stochastic Model for the Predator-Prey Type of Interaction between Two Species. Biometrica, 47, 219-234. [Google Scholar] [CrossRef] 
[37] Li, Y. and Xiao, D.M. (2007) Bifurcation of a Predator-Prey System of Holling and Leslie Types. Chaos Solitons Fractals, 34, 606-620. [Google Scholar] [CrossRef] 
[38] Liu, M. and Wang, K. (2013) Dynamics of a Leslie-Gower Holling-Type II Predator-Prey System with Lévy Jumps. Nonlinear Analysis: Theory, Methods and Applications, 85, 204-213. [Google Scholar] [CrossRef] 
[39] Huang, J.C., Ruan, S.G. and Song, J. (2014) Bifurcations in a Predator-Prey System of Leslie Type with Generalized Hollingtype III Functional Response. Journal of Differential Equations, 257, 1721-1752. [Google Scholar] [CrossRef] 
[40] Aziz-Alaoui, M.A. and Okiye, M.D. (2003) Boundedness and Global Stability for a Predator Prey Model with Modified Leslie Gower and Holling Type II Schemes. Applied Mathematics Letters, 16, 1069-1075. [Google Scholar] [CrossRef] 
[41] Gupta, R.P. and Chandra, P. (2013) Bifurcation Analysis of Modified Leslie-Gower Predator-Prey Model with Michaelis-Menten Type Prey Harvesting. Journal of Mathematical Analysis and Applications, 398, 278-295. [Google Scholar] [CrossRef] 
[42] Pal, P.J. and Mandal, P.K. (2014) Bifurcation Analysis of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response and Strong Allee Effect. Mathematical and Computers in Simulation, 97, 123-146. [Google Scholar] [CrossRef] 
[43] Peng, F. and Yun, K. (2015) Dynamics of a Modified Leslie-Gower Model with Double Allee Effects. Nonlinear Dynamics, 80, 1051-1062. [Google Scholar] [CrossRef] 
[44] Song, X.Y. and Li, Y.F. (2008) Dynamic Behaviors of the Periodic Predator-Prey Model with Modified Leslie-GowerHolling-Type II Schemes and Impulsive Effect. Nonlinear Analysis: Real World Applications, 9, 64-79. [Google Scholar] [CrossRef] 
[45] Ji, C.Y., Jiang, D.Q. and Shi, N.Z. (2009) Analysis of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type IIschemes with Stochastic Perturbation. SIAM Journal on Applied Mathematics, 359, 482-498. [Google Scholar] [CrossRef] 
[46] Nie, L.N., Teng, Z.D., Hu, L., et al. (2010) Qualitative Analysis of a Modified Leslie-Gower and Holling-Type II Predator-Prey Model with State Dependent Impulsive Effects. Nonlinear Analysis: Real World Applications, 11, 1364-1373. [Google Scholar] [CrossRef] 
[47] Tian, Y.L. and Weng, P.X. (2011) Stability Analysis of Diffusive Predator-Prey Model with Modified Leslie-Gower and Holling-Type III Schemes. Applied Mathematics and Computation, 218, 3733-3745. [Google Scholar] [CrossRef] 
[48] Nindjin, A.F., Aziz-Alaoui, M.A. and Cadivel, M. (2006) Analysis of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Time Delay. Nonlinear Analysis: Real World Applications, 7, 1104-1118. [Google Scholar] [CrossRef] 
[49] Wright, E.M. (1945) On a Sequence Defined by a Nonlinear Recurrence Formula. Journal of the London Mathematical Society, 20, 68-73. [Google Scholar] [CrossRef] 
[50] May, R.M. (1973) Time Delay versus Stability in Population Models with Two and Three Trophic Levels. Ecology, 54, 315-325. [Google Scholar] [CrossRef] 
[51] Yafia, R., Adnani, F.E. and Alaoui, H.T. (2008) Limit Cycle and Numerical Simulations for Small and Large Delays in a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes. Nonlinear Analysis: Real World Applications, 9, 2055-2067. [Google Scholar] [CrossRef] 
[52] Celik, C. (2008) The Stability and Hopf Bifurcation for a Predator-Prey System with Time Delay. Chaos, Solitons and Fractals, 37, 87-99. [Google Scholar] [CrossRef] 
[53] Shu, H.Y., Hu, X., Wang, L., et al. (2015) Delay Induced Stability Switch, Multitype Bistability and Chaos in an Intraguild Predation Multitype Bistability and Chaos in an Intraguild Predation Model. Journal of Mathematical Biology, 71, 1269-1298. [Google Scholar] [CrossRef] [PubMed]
[54] González-Olivares, E., Mena-Lorca, J., Rojas-Palma, A., et al. (2011) Dynamical Complexities in the Leslie-Gower Predator-Prey Model as Consequences of the Allee Effect on Prey. Applied Mathematical Modelling, 35, 366-381. [Google Scholar] [CrossRef] 
[55] Cai, Y.L., Zhao, C.D., Wang, W.W., et al. (2015) Dynamics of Leslie-Gower Predator-Prey Model with Additive Allee Effect. Applied Mathematical Modelling, 39, 2096-2106. [Google Scholar] [CrossRef] 
[56] Aguirre, P., González-Olivares, E. and Sáez, E. (2009) Three Limit Cycles in a Leslie-Gower Predator-Prey Model with Additive Allee Effect. SIAM Journal on Applied Mathematics, 69, 1244-1262. [Google Scholar] [CrossRef] 
[57] Aguirre, P., González-Olivares, E. and Sáez, E. (2009) Two Limit Cycles in a Leslie-Gower Predator-Prey Model with Additive Allee Effect. Nonlinear Analysis: Real World Applications, 10, 1401-1416. [Google Scholar] [CrossRef] 
[58] Wang, Y. and Wang, J.Z. (2012) Influence of Prey Refuge on Predator-Prey Dynamics. Nonlinear Dynamics, 67, 191-201. [Google Scholar] [CrossRef] 
[59] McNair, J.N. (1986) The Effect of Refuge on Predator-Prey Interactions: A Reconsideration. Theoretical Population Biology, 29, 38-63. [Google Scholar] [CrossRef] [PubMed]
[60] Kar, T.K. (2005) Stability Analysis of a Prey-Predator Model Incorporating a Prey Refuge. Communications in Nonlinear Science and Numerical Simulations, 10, 681-691. [Google Scholar] [CrossRef] 

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