具有避难所的Leslie-Gower捕食系统的时空分歧
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摘要
研究一类修正的具有食饵避难所的Leslie-Gower捕食-食饵模型。给出该模型非常数解的全局吸引子和持续共存性。得到该模型正平衡解的局部渐近稳定性,并通过构造Lyapunov函数得出正平衡解全局稳定的充分条件。利用分歧理论,以d为分歧参数,讨论了此模型在一维空间区域上的Hopf分歧与稳态分歧。
A modified Leslie-Gower predator-prey model with prey refuge is investigated. The global attractor and uniform persistence of the nonconstant solutions are given.The local asymptotic stability of the positive equilibrium is obtained, and by constructing a Lyapunov function, the sufficient conditions for the global asymptotic stability of the positive equilibrium are derived. By regarding d as the bifurcation parameter, the Hopf and steady state bifurcation for the model in the one-dimension space case are discussed by means of the bifurcation theory.
A modified Leslie-Gower predator-prey model with prey refuge is investigated. The global attractor and uniform persistence of the nonconstant solutions are given.The local asymptotic stability of the positive equilibrium is obtained, and by constructing a Lyapunov function, the sufficient conditions for the global asymptotic stability of the positive equilibrium are derived. By regarding d as the bifurcation parameter, the Hopf and steady state bifurcation for the model in the one-dimension space case are discussed by means of the bifurcation theory.
引文
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[7] 徐娟,李艳玲.一类捕食-食饵模型正平衡解的存在性[J].纺织高校基础科学学报,2012,25(2):149-153.XU J,LI Y L.The existence of positive steady-states solutions for a predator-prey model[J].Basis Sciences Journal of Textile Universities,2012,25(2):149-153.(in Chinese)
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[10] CAMARA B A,AZIZ-ALAOUI M A.Turing and hopf patterns formation in a predator-prey model with Leslie-Gower-type functional response[J].Dynamics of Continuous,Discrete and Impulsive Systems Series B:Applications and Algorithms,2009,16(4):479-488.
[11] KAR T K.Stability analysis of a prey-predator model incorporating a prey refuge[J].Communication in Nonlinear Science and Numerical Simulation,2005,10(6):681-691.
[12] JIA Y F.Analysis on dynamics of a population model with predator-prey-dependent functional response[J].Applied Mathematics Letters,2018,80:64-70.
[13] 叶其孝,李正元.反应扩散方程引论[M].北京:科学出版社,1990.YE Q X,LI Z Y.Introduction to reaction-diffusion equations[M].Beijing:Science Press,1990.(in Chinese)
[14] PENG R,YI F Q,ZHAO X Q.Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme[J].Journal of Differential Equations,2013,254(6):2465-2498.
[15] WANG M X,PANG P Y H.Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey-predator model[J].Applied Mathematics Letters,2008,21(11):1215-1220.
[16] ZHANG L,LIU J,BANERJEE M.Hopf and steady state bifurcation analysis in a ratio-dependent predator-prey model[J].Communication in Nonlinear Science and Numerical Simulation,2017,44:52-73.
[17] PAL P J,MANDAL P K.Bifurcation analysis of a modified Leslie-Gower predator-prey model with beddington-DeAngelis functional response and strong allee effect[J].Mathematics and Computers in Simulation,2014,97:123-146.
[2] AZIZ-ALAOUI M A,DAHER OKIYE M.Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes[J].Applied Mathematics Letters,2003,16(7):1069-1075.
[3] NINDJIN A F,AZIZ-ALAOUI M A,CADIVE M.Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay[J].Nonlinear Analysis:Real World Application,2006,7(5):1104-1118.
[4] YUAN S L,SONG Y L.Stability and Hopf bifurcations in a delayed Leslie-Gower predator-prey system[J].Journal of Mathematical Analysis Applications,2009,355(1):82-100.
[5] LOU Y,NI W M.Diffusion,self-diffusion and cross-diffusion[J].Journal of Differential Equations,1996,131(1):79-131.
[6] ZHENG S N,LIU J.Coexistence solutions for a reaction-diffusion system of unstirred chemostat model[J].Applied Mathematics and Computation,2003,145(3):590-597.
[7] 徐娟,李艳玲.一类捕食-食饵模型正平衡解的存在性[J].纺织高校基础科学学报,2012,25(2):149-153.XU J,LI Y L.The existence of positive steady-states solutions for a predator-prey model[J].Basis Sciences Journal of Textile Universities,2012,25(2):149-153.(in Chinese)
[8] GUAN X N,WANG W M,CAI Y L.Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge[J].Nonlinear Analysis:Real World Application,2011,12(4):2385-2395.
[9] ZHOU J.Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes[J].Journal of Mathematical Analysis Applications,2012,389(2):1380-1393.
[10] CAMARA B A,AZIZ-ALAOUI M A.Turing and hopf patterns formation in a predator-prey model with Leslie-Gower-type functional response[J].Dynamics of Continuous,Discrete and Impulsive Systems Series B:Applications and Algorithms,2009,16(4):479-488.
[11] KAR T K.Stability analysis of a prey-predator model incorporating a prey refuge[J].Communication in Nonlinear Science and Numerical Simulation,2005,10(6):681-691.
[12] JIA Y F.Analysis on dynamics of a population model with predator-prey-dependent functional response[J].Applied Mathematics Letters,2018,80:64-70.
[13] 叶其孝,李正元.反应扩散方程引论[M].北京:科学出版社,1990.YE Q X,LI Z Y.Introduction to reaction-diffusion equations[M].Beijing:Science Press,1990.(in Chinese)
[14] PENG R,YI F Q,ZHAO X Q.Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme[J].Journal of Differential Equations,2013,254(6):2465-2498.
[15] WANG M X,PANG P Y H.Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey-predator model[J].Applied Mathematics Letters,2008,21(11):1215-1220.
[16] ZHANG L,LIU J,BANERJEE M.Hopf and steady state bifurcation analysis in a ratio-dependent predator-prey model[J].Communication in Nonlinear Science and Numerical Simulation,2017,44:52-73.
[17] PAL P J,MANDAL P K.Bifurcation analysis of a modified Leslie-Gower predator-prey model with beddington-DeAngelis functional response and strong allee effect[J].Mathematics and Computers in Simulation,2014,97:123-146.