具有Ⅳ类功能反应的离散非自治捕食系统永久持续生存性和周期解的存在性
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摘要
本文主要讨论了具有Ⅳ类功能反应的离散非自治捕食模型和时滞影响的离散非自治捕食系统分别获得了系统(1)和系统(2)的永久持续生存性,利用Brouwer不动点定理和重合度理论的连续性定理,讨论了系统(1)和系统(2)周期解的存在性,并且在一些特定假设下,讨论了系统(1)任意正解的全局稳定性,以及系统(1)周期解的稳定性.
This paper discusses discrete non-autonomous predator-prey system with IV type functional response and delayed discrete non-autonomous predator-prey system
The author obtain the permanence of the system(1)and(2),with the help of Brouwer fixed point theorem and continuation theorem in coincidence degree theory,we derive sufficient and realistic that guarantee the existence of at least one positive periodic solution for discrete predator-prey system(1) and(2),we also obtain the existence and stability of any positive solution and periodic solution under some additional conditions.
This paper discusses discrete non-autonomous predator-prey system with IV type functional response and delayed discrete non-autonomous predator-prey system
The author obtain the permanence of the system(1)and(2),with the help of Brouwer fixed point theorem and continuation theorem in coincidence degree theory,we derive sufficient and realistic that guarantee the existence of at least one positive periodic solution for discrete predator-prey system(1) and(2),we also obtain the existence and stability of any positive solution and periodic solution under some additional conditions.
引文
[1]郑祖麻,泛函微分方程理论,合肥,安徽教育出版社,1996.
[2]郭大钧,孙经先,刘兆理,非线性常微分方程泛函方法,济南,山东科学技术出版社,2005.
[3]Saber Elaydi,An Introuction to Difference Equation,3th,Springer,2005.
[4]傅希林,闫宝强,刘衍胜,脉冲微分系统引论,北京,科学出版社,2004.
[5]Hui-Hsiung Kuo,Introduction to Stochastic Intergration,Springer,2006.
[6]E.Allen,Modelling with Ito Stochastic Differential Equations,Springer,2007.
[7]马知恩,种群生态学的数学建模与研究,合肥,安徽教育出版社,1996.
[8]陈兰荪,陈键,非线性生物动力系统,北京,科学出版社,1993.
[9]马知恩,周义仓,王稳地,靳祯,传染病动力学的数学建模与研究,北京,科学出版社,2004.
[10]蒋达清,张宝学,王德辉,史宁中,具有随机扰动的Logistic方程正解的存在唯一性,全局吸引性及其参数的极大似然估计,中国科学A辑,2007,37(6)：742-750.
[11]D Jiang,N Shi,X Li,Global stability and Stochastic Permanence of a Non-autonomous Logistic Equation with Random Perturbation,Math. Anal. Appl, doi:10.1016/j.jmau.2007.08.0141-10.
[12]W S C Gurney, S P Blythe,R M Nisbet,Nicholson's Blowflies Revisited Nature,1980, 287:17-21.
[13]C Feng, W Ge,Almost Periodic Solutions for a class of Population Equations with Infinite Delays,Acta. Mathematicae Applicatae Sinica,2003,26(3):402-407.
[14]E M Elabbasy, S H Saker,Periodic solutions and oscillation of discrete non-linear delay population dynamics model with external force, IMA Journal of Applied Mathematics, 2005,70:753-767.
[15]S B Hsu,Mathematical Modelling In Biological Science,11-12.
[16]C S Holling,The functional response of predator to prey density and role in mimicry and population regulation,Men. Entomology.Soc. Can,1965,45:3-60.
[17]L Wang,W Li,Existence and global stability of positive periodic solutions of a predator-prey system with delays,Appl. Math. Comput,2003,146:167-185.
[18]H Huo,W Li,S Cheng, Periodic solutions of two-species diffusive models with con-tinuous time delays,Demonstntio Mathematica,2002,35(2):433-466.
[19]J Jia,Persistence and periodic solution for the nonautonomous predator-prey system with type III functional response,J. Biomath,2001,16(1):59-62.
[20]赵育林,陈海波,具功能性反应函数kxθ/a+kxθ的捕食-食饵系统的定性分析,数学的实践与认识,2007,37(10)：118-121.
[21]S B Hsu,T W Hwang,Global analysis of the Michaelis-Menten type ratio-dependent predator-prey system,Math. Biol,2001,42:189-506.
[22]J F Andrews,A mathematical model for the continuous culture of microorganisms utilizing inhibitory substratesl,Biotechnol. Bioeng,1968,10:707-723.
[23]W Sokol,J A Howell,Kinetics of phenol oxidation by washed cells,Biotechnol. Bioeng, 1980,23:2039-2049.
[24]S Ruan,D Xiao,Global analysis in a predator-prey system with nonmonotonic func-tional response,SIAM. Appl. Math,2001,61(4):1445-1472.
[25]J Huang, D Xiao,Analyses of Bifurcations and Stability in a Predator-prey System with Holling Type-IV Functional Response,Acta Mathematicae Applicatae Sinica, English Series,2004,20(1):167-178.
[26]J Huang, Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-ⅣFunctional Response,Acta Mathematicae Applicatae Sinica, English Series, 2005,21(1):157-176.
[27]S Zhang,D Tan,L Chen,Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations,Chaos, Solitons and Fractals,2006,27:980-990.
[28]Y Dong, E Zhou,An Application of Coincidence Degree Continuation Theorem in Existence of Solutions of Impulsive Differential Equations,Math.Anal. Appl,1996, 197:875-889.
[29]Y Dong,Periodic solutions for second order impulsive differential systems,Nonlinear Analysis,Theory,Methods and Applications,1996,27(7):811-820.
[30]K Wang,S Lu,On the existence of periodic solutions for a kind high-order neutral functional differential equation,Math.Anal. Appl,2006,4:121-126.
[31]B Liu,L Huang,Existence and uniqueness of periodic solutions for a kind of sec-ond order neutral functional differential equations,Nonlinear Analysis,Real World Application,2007,18:222-229.
[32]S Lu,W Ge,Periodic solutions for a kind of second order differential equation with multiple deviating arguments,Applied Mathematics and Computation,2003,146: 195-209.
[33]Z Liu,L Liao,Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays,Math. Anal. Appl,2004,290:247-262.
[34]Z Zhang,Zhi Wang, Periodic Solutions of a Class of Second Order Functional Differ-ential Equations,Acta Mathematica Sinica,English Series,2005,21(1):95-108.
[35]W Li,H Huo,Existence and global attractivity of positive periodic solutions of func-tional differential equations with impulses,Nonlinear Analysis,2004,59:857-877.
[36]S Zhou,W Li,G Wang,Persistence and global stability of positive periodic solutions of three species food chains with omnivory, Math.Anal. Appl,2006,324:397-408.
[37]H Huo, W Li,Existence and global stability of periodic solutions of a discrete predator-prey system with delays,Appl. Math. Comput,2004,153:337-351.
[38]D Jiang,W Zuo,Donal O'regan,Ravip Agarwal,Optimal existence conditions for second order periodic solutions of delay differential equations with upper and lower solutions in the reverse order,International Journal of Computer Mathematics,2004, 81(6):707-717.
[39]D L Bennett,S A Goukley, Periodic oscillations in a model of the glucose-insulin interaction with delay and periodic forcing,Dynamical Systems,2004,19(2):109-125.
[40]P Amster,P De Napoli,M S Mariani,Periodic solution of a resonant third-order equation,Nonlinear Analysis,2005,6:399-410.
[41]R P Agarwal,Difference Equations and Inequalities:Theory, Mathods and Applica-tions, Monographys and textbooks in pure and applied mathematics,No.229,Marcel Dekker,New York,2000.
[42]J D Murray, Mathematical Biology,Springer-Verlag,New York,1989.Springer, Berlin, New York, Vol.840,1993.
[43]J Zhang,H Fang,Multiple periodic solutions for a discrete time model of plankton allelopathy,Advances in Difference Equations,2006,Article ID 90479:1-14.
[44]F Chen,Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,Appl. Math, Comput,2006,182:3-12.
[45]余长春,李培峦,曾宪武,具有Ⅳ类功能反应的非自治捕食系统的持续性和周期解,数学杂志,2006,26(6)：619-622.
[46]X Yang,Uniform persistence and periodic solutions for a discrete predator-prey sys-tem with delays,Math.Anal.Appl,2006,316:161-177.
[47]R E Gaines, J L Mawhin,Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics,vol.568,Springer,Berlin,1977.
[48]M Fan,K Wang,Periodic solutions of a discrete time non-autonomous rati-dependent predator-prey system,Math.Comput. Model,2002,35:951-961.
[49]R Y Zhang,Z C Wang, Y Chen,J Wu,Periodic solutions of a singe species discrete population model with periodic harvest/stock,Gomput. Math.Appl,2000,39:77-90.
[2]郭大钧,孙经先,刘兆理,非线性常微分方程泛函方法,济南,山东科学技术出版社,2005.
[3]Saber Elaydi,An Introuction to Difference Equation,3th,Springer,2005.
[4]傅希林,闫宝强,刘衍胜,脉冲微分系统引论,北京,科学出版社,2004.
[5]Hui-Hsiung Kuo,Introduction to Stochastic Intergration,Springer,2006.
[6]E.Allen,Modelling with Ito Stochastic Differential Equations,Springer,2007.
[7]马知恩,种群生态学的数学建模与研究,合肥,安徽教育出版社,1996.
[8]陈兰荪,陈键,非线性生物动力系统,北京,科学出版社,1993.
[9]马知恩,周义仓,王稳地,靳祯,传染病动力学的数学建模与研究,北京,科学出版社,2004.
[10]蒋达清,张宝学,王德辉,史宁中,具有随机扰动的Logistic方程正解的存在唯一性,全局吸引性及其参数的极大似然估计,中国科学A辑,2007,37(6)：742-750.
[11]D Jiang,N Shi,X Li,Global stability and Stochastic Permanence of a Non-autonomous Logistic Equation with Random Perturbation,Math. Anal. Appl, doi:10.1016/j.jmau.2007.08.0141-10.
[12]W S C Gurney, S P Blythe,R M Nisbet,Nicholson's Blowflies Revisited Nature,1980, 287:17-21.
[13]C Feng, W Ge,Almost Periodic Solutions for a class of Population Equations with Infinite Delays,Acta. Mathematicae Applicatae Sinica,2003,26(3):402-407.
[14]E M Elabbasy, S H Saker,Periodic solutions and oscillation of discrete non-linear delay population dynamics model with external force, IMA Journal of Applied Mathematics, 2005,70:753-767.
[15]S B Hsu,Mathematical Modelling In Biological Science,11-12.
[16]C S Holling,The functional response of predator to prey density and role in mimicry and population regulation,Men. Entomology.Soc. Can,1965,45:3-60.
[17]L Wang,W Li,Existence and global stability of positive periodic solutions of a predator-prey system with delays,Appl. Math. Comput,2003,146:167-185.
[18]H Huo,W Li,S Cheng, Periodic solutions of two-species diffusive models with con-tinuous time delays,Demonstntio Mathematica,2002,35(2):433-466.
[19]J Jia,Persistence and periodic solution for the nonautonomous predator-prey system with type III functional response,J. Biomath,2001,16(1):59-62.
[20]赵育林,陈海波,具功能性反应函数kxθ/a+kxθ的捕食-食饵系统的定性分析,数学的实践与认识,2007,37(10)：118-121.
[21]S B Hsu,T W Hwang,Global analysis of the Michaelis-Menten type ratio-dependent predator-prey system,Math. Biol,2001,42:189-506.
[22]J F Andrews,A mathematical model for the continuous culture of microorganisms utilizing inhibitory substratesl,Biotechnol. Bioeng,1968,10:707-723.
[23]W Sokol,J A Howell,Kinetics of phenol oxidation by washed cells,Biotechnol. Bioeng, 1980,23:2039-2049.
[24]S Ruan,D Xiao,Global analysis in a predator-prey system with nonmonotonic func-tional response,SIAM. Appl. Math,2001,61(4):1445-1472.
[25]J Huang, D Xiao,Analyses of Bifurcations and Stability in a Predator-prey System with Holling Type-IV Functional Response,Acta Mathematicae Applicatae Sinica, English Series,2004,20(1):167-178.
[26]J Huang, Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-ⅣFunctional Response,Acta Mathematicae Applicatae Sinica, English Series, 2005,21(1):157-176.
[27]S Zhang,D Tan,L Chen,Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations,Chaos, Solitons and Fractals,2006,27:980-990.
[28]Y Dong, E Zhou,An Application of Coincidence Degree Continuation Theorem in Existence of Solutions of Impulsive Differential Equations,Math.Anal. Appl,1996, 197:875-889.
[29]Y Dong,Periodic solutions for second order impulsive differential systems,Nonlinear Analysis,Theory,Methods and Applications,1996,27(7):811-820.
[30]K Wang,S Lu,On the existence of periodic solutions for a kind high-order neutral functional differential equation,Math.Anal. Appl,2006,4:121-126.
[31]B Liu,L Huang,Existence and uniqueness of periodic solutions for a kind of sec-ond order neutral functional differential equations,Nonlinear Analysis,Real World Application,2007,18:222-229.
[32]S Lu,W Ge,Periodic solutions for a kind of second order differential equation with multiple deviating arguments,Applied Mathematics and Computation,2003,146: 195-209.
[33]Z Liu,L Liao,Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays,Math. Anal. Appl,2004,290:247-262.
[34]Z Zhang,Zhi Wang, Periodic Solutions of a Class of Second Order Functional Differ-ential Equations,Acta Mathematica Sinica,English Series,2005,21(1):95-108.
[35]W Li,H Huo,Existence and global attractivity of positive periodic solutions of func-tional differential equations with impulses,Nonlinear Analysis,2004,59:857-877.
[36]S Zhou,W Li,G Wang,Persistence and global stability of positive periodic solutions of three species food chains with omnivory, Math.Anal. Appl,2006,324:397-408.
[37]H Huo, W Li,Existence and global stability of periodic solutions of a discrete predator-prey system with delays,Appl. Math. Comput,2004,153:337-351.
[38]D Jiang,W Zuo,Donal O'regan,Ravip Agarwal,Optimal existence conditions for second order periodic solutions of delay differential equations with upper and lower solutions in the reverse order,International Journal of Computer Mathematics,2004, 81(6):707-717.
[39]D L Bennett,S A Goukley, Periodic oscillations in a model of the glucose-insulin interaction with delay and periodic forcing,Dynamical Systems,2004,19(2):109-125.
[40]P Amster,P De Napoli,M S Mariani,Periodic solution of a resonant third-order equation,Nonlinear Analysis,2005,6:399-410.
[41]R P Agarwal,Difference Equations and Inequalities:Theory, Mathods and Applica-tions, Monographys and textbooks in pure and applied mathematics,No.229,Marcel Dekker,New York,2000.
[42]J D Murray, Mathematical Biology,Springer-Verlag,New York,1989.Springer, Berlin, New York, Vol.840,1993.
[43]J Zhang,H Fang,Multiple periodic solutions for a discrete time model of plankton allelopathy,Advances in Difference Equations,2006,Article ID 90479:1-14.
[44]F Chen,Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,Appl. Math, Comput,2006,182:3-12.
[45]余长春,李培峦,曾宪武,具有Ⅳ类功能反应的非自治捕食系统的持续性和周期解,数学杂志,2006,26(6)：619-622.
[46]X Yang,Uniform persistence and periodic solutions for a discrete predator-prey sys-tem with delays,Math.Anal.Appl,2006,316:161-177.
[47]R E Gaines, J L Mawhin,Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics,vol.568,Springer,Berlin,1977.
[48]M Fan,K Wang,Periodic solutions of a discrete time non-autonomous rati-dependent predator-prey system,Math.Comput. Model,2002,35:951-961.
[49]R Y Zhang,Z C Wang, Y Chen,J Wu,Periodic solutions of a singe species discrete population model with periodic harvest/stock,Gomput. Math.Appl,2000,39:77-90.