几类种群动力学模型持久性和绝灭性的研究
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摘要
本文研究四类种群动力学模型:
第一部分,研究一类食饵种群具有阶段结构、捕食者的捕食率为非线性的捕食-食饵模型.首先得到了保证系统强持续生存的充分条件,接着通过构造适当的Lyapunov函数,得到了非负边界平衡点的全局渐近稳定性即捕食者种群走向绝灭的充分条件.
第二部分,研究一类具有阶段结构和Ⅳ类功能性反应的周期捕食系统,在一定的条件下,得到了保证该生态系统永久持续生存的充分必要条件.
第三部分,研究一类离散的具有单调功能性反应的捕食系统.运用差分方程的比较原理,得到了保证该系统永久持续生存的充分条件和捕食者种群走向绝灭的充分条件.
第四部分,研究一类离散的具有时滞和反馈控制的非线性n-种群竞争系统.利用差分方程的比较原理和运用一定的精细的分析技巧,得到了保证该系统永久持续生存的充分条件.
In this paper, we consider four kinds of population dynamic systems:
Firstly, we study a predator-prey system with stage-structure and functional response, we obtain sufficient condition for the strong persistence of the proposed ecological system; by constructing a suitable Lyapunov function, some sufficient conditions which guarantee nonnegative verge equilibrium to be global asympototics stability are obtained.
Secondly, we study a periodic Holling type-IV predator-prey system with stage structure for prey. Under certain assumptions, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.
Thirdly, we study a delayed discrete ratio-dependent predator-prey model with monotonic functional responses. By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system; also, we obtain sufficient condition for extinction of the predator.
Finally, we study a discrete nonlinear n-species competition system with time delays and feedback controls. With the help of the comparison theorem of difference equation and some subtly analysis, sufficient conditions are obtained for the permanence of the system.
第一部分,研究一类食饵种群具有阶段结构、捕食者的捕食率为非线性的捕食-食饵模型.首先得到了保证系统强持续生存的充分条件,接着通过构造适当的Lyapunov函数,得到了非负边界平衡点的全局渐近稳定性即捕食者种群走向绝灭的充分条件.
第二部分,研究一类具有阶段结构和Ⅳ类功能性反应的周期捕食系统,在一定的条件下,得到了保证该生态系统永久持续生存的充分必要条件.
第三部分,研究一类离散的具有单调功能性反应的捕食系统.运用差分方程的比较原理,得到了保证该系统永久持续生存的充分条件和捕食者种群走向绝灭的充分条件.
第四部分,研究一类离散的具有时滞和反馈控制的非线性n-种群竞争系统.利用差分方程的比较原理和运用一定的精细的分析技巧,得到了保证该系统永久持续生存的充分条件.
In this paper, we consider four kinds of population dynamic systems:
Firstly, we study a predator-prey system with stage-structure and functional response, we obtain sufficient condition for the strong persistence of the proposed ecological system; by constructing a suitable Lyapunov function, some sufficient conditions which guarantee nonnegative verge equilibrium to be global asympototics stability are obtained.
Secondly, we study a periodic Holling type-IV predator-prey system with stage structure for prey. Under certain assumptions, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.
Thirdly, we study a delayed discrete ratio-dependent predator-prey model with monotonic functional responses. By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system; also, we obtain sufficient condition for extinction of the predator.
Finally, we study a discrete nonlinear n-species competition system with time delays and feedback controls. With the help of the comparison theorem of difference equation and some subtly analysis, sufficient conditions are obtained for the permanence of the system.
引文
[1]Andrews,J.F.A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates.Biotechnol.Bioeng,10:707-723(1968).
[2]W.Sokol,W.,Howell,J.A.Kinetics of phenol oxidation by washed cells.Biotechnol.Bioeng.,23:2039-2049(1980)
[3]J.A.Cui,X.Y.Song.Permanence of a predator-prey system with stage structure,Discret.Contin.Dyn.Syst.,Ser.B 4(3)(2004):547-554.
[4]J.A.Cui,Y.Takeuchib.A predator-prey system with a stage structure for the prey,Math.Comput.Modelling 44(2006):1126-1132.
[5]X.Zhang,L.Chen,A.U.Neumann,The stage-structured predator-prey model and optimal harvesting policy,Math.Biosci.168(2000):201-210.
[6]J.Cui,L.Chen,W.Wang.The effect of dispersal on population growth with stagestructure,Comput.Math.Appl.39(2000):91-102.
[7]X.-Q.Zhao.The qualitative analysis of N-species Lotka-Volterra periodic competition systems,Math.Comput.Modelling 15(1991):3-8.
[8]X.-Q.Zhao.Dynamical Systems in Population Biology,Springer-Verlag,New York,2003.
[9]H.Huo,W.Li.Positive periodic solutions of a class of delay differential system with feedback control,Appl.Math.Comput.148(1)(2004):35-46.
10]L.Wang,M.Q.Wang.Ordinary Difference Equation,Xinjiang University Press,China,1991(in Chinese).
[11]Y.H.Fan,W.T.Li.Permanence in delayed ratio-dependent predator-prey models with monotonic functional responses,Nonlinear Anal.8(2007):424-434.
[12]Y.H.Fan,W.T.Li.Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response,J.Math.Anal.Appl.299(2)(2004):357-374.
[13]M.Fan,K.Wang.Periodic solutions of a discrete time nonautonomous ratiodependent predator-prey system,Math.Comput.Model.35(9-10)(2002):951-961.
[14]F.D.Chen,Permanence of a single species discrete model with feedback control and delay,Appl.Math.Lett.20(2007):729-733.
[15]J.M.Zhang,J.Wang.Periodic solutions for discrete predator-prey systems with the Beddington-DeAngelis functional response,Appl.Math.Lett.19(2006):1361-1366.
[16]C.Y.Xu,M.J.Wang.Permanence for a delayed discrete three-level food-chain model with Beddington-DeAngelis functional response,Appl.Math.Comput.187(2007):1109-1119.
[17]F.D.Chen.Permanence of a discrete n-species food-chain system with time delays,Appl.Math.Comput.185(2007):719-726.
[18]Y.Takeuchi.Global Dynamical Properties of Lotka-Volterra Systems,World Scientific Press,1996.
[19]K.Gopalsamy,P.X.Weng.Feedback regulation of logistic growth,Int.J.Math.Sci.16(1)(1993):177-192.
[20]F.Yang,D.Q.Jiang.Existence and global attractivity of positive periodic solution of a Logistic growth system with feedback control and deviating arguments,Ann.Diff.Eqs.17(4)(2001):337-384.
[21]F.D.Chen.Global asymptotic stability in n-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control,Appl.Math.Comput.170(2)(2005):1452-1468.
[22] F.Q. Yin, Y.K. Li. Positive periodic solutions of a single species model with feedback regulation and distributed time delay, Appl. Math. Comput. 153 (2004): 475-484.
[23] M. Fan, Patrica J.Y. Wong, Ravi P. Agarwal. Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Math. Sinica 19 (4) (2003): 801-822.
[24] H. Huo, W. Li. Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (1) (2004): 35-46.
[25] L. Wang and M. Q. Wang, Ordinary Difference Equation (in Chinese), Xinjiang University Press, China, 1991.
[26] F. Montes de Oca, M.L. Zeeman. Extinction in nonautonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc. 124 (1996): 3677-3687.
[27] H. Huo, W. Li. Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (1) (2004): 35-46.
[28] M.L. Zeeman. Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc. 123 (1995): 87-96.
[29] Agarwal, R. P.. Difference Equations and Inequalities: Theory, Methods and Applications, Monographs and Textbooks in Pure and Applied Mathematics, No. 228 Marcel Dekker, New York, 2000.
[30] J.D. Murry, Mathematical Biology, Springer-Verlag, New York, 1989.
[31] Crone, E. E., Delayed density dependence and the stability of interacting populations and subpopulations, Theoret. Population Biol. 51, 67-76 (1997) (in Chinese).
[32] W.D.Wang, Z.Y.Lu. Global stability of discrete models of Lotka-Volterra type, Nonlinear Anal.35(1999): 1019-1030.
[33] Turchin, P., Chaos and stability in rodent population dynamics: Evidence from nonlinear time series analysis, Oikos. 68, 167-182 (1993).
[34] F.D. Chen et al., Positive periodic solutions of a class of non-autonomous single species population model with delays and feedback control, Acta Math. Sinica 21 (6) (2005): 1319-1336.
[35] F.D. Chen. Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput. 162 (3) (2005): 1279-1302.
[36] Wang, W. D. and Lu, Z. Y., Global stability of discrete population models with delays and fluctuating environment, J. Math. Anal. Appl. 264, 147-167 (2001).
[37] F.D. Chen. Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems, Appl. Math. Comput. 182 (1)(2006): 3-12.
[38] J. Chattopadhyay. Effect of toxic substances on a two-species competitive system, Ecological Modelling, 84 (1996): 287-289.
[39] F.D. Chen, X.D. Xie, J.L. Shi. Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. Comput. Appl. Math. 194 (2) (2006): 368-387.
[40] F.D. Chen. Permanence of a delayed nonautonomous Gilpin-Ayala competition model, Appl. Math. Comput. 179 (1) (2006): 55-66.
[41] F.D. Chen. Some new results on the permanence and extinction of nonautonomous Gilpin-Ayala type competition model with delays, Nonlinear Anal. RWA 7 (5) (2006): 1205-1222.
[42] F.D. Chen. Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model, Nonlinear Anal. RWA 7 (4) (2006): 895-915.
[43]F.D.Chen,J.L.Shi.Periodicity in a logistic type system with several delays,Comput.Math.Appl.48(1-2)(2004):35-44.
[44]F.D.Chen.Permanence in nonautonomous multi-species predator-prey system with feedback controls,Appl.Math.Comput.173(2)(2006):694-709.
[45]F.D.Chen.Positive periodic solutions of neutral Lotka-Volterra system with feedback control,Appl.Math.Comput.162(3)(2005):1279-1302.
[46]F.D.Chen.On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay,J.Comput.Appl.Math.180(1)(2005):33-49.
[47]M.Fan,K.Wang.Global periodic solutions of a generalized n-species Gilpin-Ayala competition model,Comput.Math.Appl.40(7-8)(2000):1141-1151.
[48]L.Chen.Mathematical Models and Methods in Ecology,Science Press,Beijing,1988.
[49]X.Liao,et al.,Permanence and global stability in a discrete n-species competition system with feedback controls,Nonlinear Anal.:RealWorld Appl.(2007),doi:10.1016/j.nonrwa.2007.05.001.
[50]F.D.Chen.Permanence of a discrete N-species cooperation system with time delays and feedback controls,Appl.Math.Comput.186(2007):23-29.
[51]陈兰荪,刘平舟,肖藻.种群生态系统的持续生存.生物数学学报,1988,3(1):18-32.
[52]Erbe L H,Zhang B G.Oscillation of discrete analogues of delay equations.Diferential and Integral Equations,1989,2(3):300-309.
[53]Patula W.Growth and oscillation properties of second order linear diference equations.SIAM Journal on Mathematical Analysis,1979,10:55-61.
[54]Elaydi S N,Zhang S.Stability and periodicity of diference equations with finite delay.Funkcialaj Ekvacioj,1994,37:401-413.
[55] AgarwalR P. Diference Equations and Inequalities: Theory Method and Applications. New York: Marcel Dekker, 1992,32 -102.
[56] Huang L H ,Yu J S. Convergencein neutral delay diference equation. Diferential Equations and Dynamical Systems,19 99,7(1):9-20.
[57] W. G. Kelley and A. C. Peterson. Difference Equations with Applications. New York: Acad. Press, 1991.
[58] S. Mohamad and K. Gopalsamy. Continuous and discrete Halanay-type inequalities. Bull.Aus.Math.Sot., 2000, 61: 371-385.
[59] Z. J. Liu and L. S. Chen. Positive periodic solution of a general discrete non-autonomous difference system of plankton allelopathy with delays. Journal of Computational and Ap- plied Mathematics, 2006, 197(2): 446-456.
[60] Z. J. Liu and L. S. Chen. Periodic solution of a two-species competitive system with toxicant and birth pulse. Chaos. Solitons and Fractals, in press.
[61] K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Population Dynamics (Kluwer Academic Publishers, Dordrecht, 1992).
[62] G. Fan, Y. Li, M. Qin. The existence of positive periodic solutions for periodic feedback control systems with delays, ZAMM · Z. Angew. Math. Mech. 84, No. 6, 425-430 (2004)
[63] R.P. Agarwal. Difference Equations and Inequalities: Theory, Method and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker, New York, 2000.
[64] R.P. Agarwal, P.J.Y. Wong. Advance Topics in Difference Equations, Kluwer Publisher, Dordrecht, 1997.
[65]H.I.Freedman.Deterministic Mathematics Models in Population Ecology,Marcel Dekker,New York,1980.
[66]J.D.Murry.Mathematical Biology,Springer-Verlag,New York,1989.
[67]K.Gopalsamy.Stability and Oscillations in Delay Differential Equations of Population Dynamics,Kluwer Academic Publishers,Boston,1992.
[68]M.Fan,Patrica J.Y.Wong,Ravi P.Agarwal.Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments,Acta Math.Sinica 19(4)(2003):801-822.
[69]X.T.Yang.Uniform persistence and periodic solutions for a discrete predator-prey system with delays,J.Math.Anal.Appl.316(1)(2006):161-177.
[70]Smith H.Monotone Semiflows Generated by Functional Differential Equations[J].J Diff Equa,1987,66:420-442.
[71]叶彦谦.极限环论[M].第2版.上海:上海科技出版社,1984.
[2]W.Sokol,W.,Howell,J.A.Kinetics of phenol oxidation by washed cells.Biotechnol.Bioeng.,23:2039-2049(1980)
[3]J.A.Cui,X.Y.Song.Permanence of a predator-prey system with stage structure,Discret.Contin.Dyn.Syst.,Ser.B 4(3)(2004):547-554.
[4]J.A.Cui,Y.Takeuchib.A predator-prey system with a stage structure for the prey,Math.Comput.Modelling 44(2006):1126-1132.
[5]X.Zhang,L.Chen,A.U.Neumann,The stage-structured predator-prey model and optimal harvesting policy,Math.Biosci.168(2000):201-210.
[6]J.Cui,L.Chen,W.Wang.The effect of dispersal on population growth with stagestructure,Comput.Math.Appl.39(2000):91-102.
[7]X.-Q.Zhao.The qualitative analysis of N-species Lotka-Volterra periodic competition systems,Math.Comput.Modelling 15(1991):3-8.
[8]X.-Q.Zhao.Dynamical Systems in Population Biology,Springer-Verlag,New York,2003.
[9]H.Huo,W.Li.Positive periodic solutions of a class of delay differential system with feedback control,Appl.Math.Comput.148(1)(2004):35-46.
10]L.Wang,M.Q.Wang.Ordinary Difference Equation,Xinjiang University Press,China,1991(in Chinese).
[11]Y.H.Fan,W.T.Li.Permanence in delayed ratio-dependent predator-prey models with monotonic functional responses,Nonlinear Anal.8(2007):424-434.
[12]Y.H.Fan,W.T.Li.Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response,J.Math.Anal.Appl.299(2)(2004):357-374.
[13]M.Fan,K.Wang.Periodic solutions of a discrete time nonautonomous ratiodependent predator-prey system,Math.Comput.Model.35(9-10)(2002):951-961.
[14]F.D.Chen,Permanence of a single species discrete model with feedback control and delay,Appl.Math.Lett.20(2007):729-733.
[15]J.M.Zhang,J.Wang.Periodic solutions for discrete predator-prey systems with the Beddington-DeAngelis functional response,Appl.Math.Lett.19(2006):1361-1366.
[16]C.Y.Xu,M.J.Wang.Permanence for a delayed discrete three-level food-chain model with Beddington-DeAngelis functional response,Appl.Math.Comput.187(2007):1109-1119.
[17]F.D.Chen.Permanence of a discrete n-species food-chain system with time delays,Appl.Math.Comput.185(2007):719-726.
[18]Y.Takeuchi.Global Dynamical Properties of Lotka-Volterra Systems,World Scientific Press,1996.
[19]K.Gopalsamy,P.X.Weng.Feedback regulation of logistic growth,Int.J.Math.Sci.16(1)(1993):177-192.
[20]F.Yang,D.Q.Jiang.Existence and global attractivity of positive periodic solution of a Logistic growth system with feedback control and deviating arguments,Ann.Diff.Eqs.17(4)(2001):337-384.
[21]F.D.Chen.Global asymptotic stability in n-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control,Appl.Math.Comput.170(2)(2005):1452-1468.
[22] F.Q. Yin, Y.K. Li. Positive periodic solutions of a single species model with feedback regulation and distributed time delay, Appl. Math. Comput. 153 (2004): 475-484.
[23] M. Fan, Patrica J.Y. Wong, Ravi P. Agarwal. Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Math. Sinica 19 (4) (2003): 801-822.
[24] H. Huo, W. Li. Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (1) (2004): 35-46.
[25] L. Wang and M. Q. Wang, Ordinary Difference Equation (in Chinese), Xinjiang University Press, China, 1991.
[26] F. Montes de Oca, M.L. Zeeman. Extinction in nonautonomous competitive Lotka-Volterra systems, Proc. Amer. Math. Soc. 124 (1996): 3677-3687.
[27] H. Huo, W. Li. Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (1) (2004): 35-46.
[28] M.L. Zeeman. Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc. 123 (1995): 87-96.
[29] Agarwal, R. P.. Difference Equations and Inequalities: Theory, Methods and Applications, Monographs and Textbooks in Pure and Applied Mathematics, No. 228 Marcel Dekker, New York, 2000.
[30] J.D. Murry, Mathematical Biology, Springer-Verlag, New York, 1989.
[31] Crone, E. E., Delayed density dependence and the stability of interacting populations and subpopulations, Theoret. Population Biol. 51, 67-76 (1997) (in Chinese).
[32] W.D.Wang, Z.Y.Lu. Global stability of discrete models of Lotka-Volterra type, Nonlinear Anal.35(1999): 1019-1030.
[33] Turchin, P., Chaos and stability in rodent population dynamics: Evidence from nonlinear time series analysis, Oikos. 68, 167-182 (1993).
[34] F.D. Chen et al., Positive periodic solutions of a class of non-autonomous single species population model with delays and feedback control, Acta Math. Sinica 21 (6) (2005): 1319-1336.
[35] F.D. Chen. Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput. 162 (3) (2005): 1279-1302.
[36] Wang, W. D. and Lu, Z. Y., Global stability of discrete population models with delays and fluctuating environment, J. Math. Anal. Appl. 264, 147-167 (2001).
[37] F.D. Chen. Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems, Appl. Math. Comput. 182 (1)(2006): 3-12.
[38] J. Chattopadhyay. Effect of toxic substances on a two-species competitive system, Ecological Modelling, 84 (1996): 287-289.
[39] F.D. Chen, X.D. Xie, J.L. Shi. Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. Comput. Appl. Math. 194 (2) (2006): 368-387.
[40] F.D. Chen. Permanence of a delayed nonautonomous Gilpin-Ayala competition model, Appl. Math. Comput. 179 (1) (2006): 55-66.
[41] F.D. Chen. Some new results on the permanence and extinction of nonautonomous Gilpin-Ayala type competition model with delays, Nonlinear Anal. RWA 7 (5) (2006): 1205-1222.
[42] F.D. Chen. Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model, Nonlinear Anal. RWA 7 (4) (2006): 895-915.
[43]F.D.Chen,J.L.Shi.Periodicity in a logistic type system with several delays,Comput.Math.Appl.48(1-2)(2004):35-44.
[44]F.D.Chen.Permanence in nonautonomous multi-species predator-prey system with feedback controls,Appl.Math.Comput.173(2)(2006):694-709.
[45]F.D.Chen.Positive periodic solutions of neutral Lotka-Volterra system with feedback control,Appl.Math.Comput.162(3)(2005):1279-1302.
[46]F.D.Chen.On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay,J.Comput.Appl.Math.180(1)(2005):33-49.
[47]M.Fan,K.Wang.Global periodic solutions of a generalized n-species Gilpin-Ayala competition model,Comput.Math.Appl.40(7-8)(2000):1141-1151.
[48]L.Chen.Mathematical Models and Methods in Ecology,Science Press,Beijing,1988.
[49]X.Liao,et al.,Permanence and global stability in a discrete n-species competition system with feedback controls,Nonlinear Anal.:RealWorld Appl.(2007),doi:10.1016/j.nonrwa.2007.05.001.
[50]F.D.Chen.Permanence of a discrete N-species cooperation system with time delays and feedback controls,Appl.Math.Comput.186(2007):23-29.
[51]陈兰荪,刘平舟,肖藻.种群生态系统的持续生存.生物数学学报,1988,3(1):18-32.
[52]Erbe L H,Zhang B G.Oscillation of discrete analogues of delay equations.Diferential and Integral Equations,1989,2(3):300-309.
[53]Patula W.Growth and oscillation properties of second order linear diference equations.SIAM Journal on Mathematical Analysis,1979,10:55-61.
[54]Elaydi S N,Zhang S.Stability and periodicity of diference equations with finite delay.Funkcialaj Ekvacioj,1994,37:401-413.
[55] AgarwalR P. Diference Equations and Inequalities: Theory Method and Applications. New York: Marcel Dekker, 1992,32 -102.
[56] Huang L H ,Yu J S. Convergencein neutral delay diference equation. Diferential Equations and Dynamical Systems,19 99,7(1):9-20.
[57] W. G. Kelley and A. C. Peterson. Difference Equations with Applications. New York: Acad. Press, 1991.
[58] S. Mohamad and K. Gopalsamy. Continuous and discrete Halanay-type inequalities. Bull.Aus.Math.Sot., 2000, 61: 371-385.
[59] Z. J. Liu and L. S. Chen. Positive periodic solution of a general discrete non-autonomous difference system of plankton allelopathy with delays. Journal of Computational and Ap- plied Mathematics, 2006, 197(2): 446-456.
[60] Z. J. Liu and L. S. Chen. Periodic solution of a two-species competitive system with toxicant and birth pulse. Chaos. Solitons and Fractals, in press.
[61] K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Population Dynamics (Kluwer Academic Publishers, Dordrecht, 1992).
[62] G. Fan, Y. Li, M. Qin. The existence of positive periodic solutions for periodic feedback control systems with delays, ZAMM · Z. Angew. Math. Mech. 84, No. 6, 425-430 (2004)
[63] R.P. Agarwal. Difference Equations and Inequalities: Theory, Method and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker, New York, 2000.
[64] R.P. Agarwal, P.J.Y. Wong. Advance Topics in Difference Equations, Kluwer Publisher, Dordrecht, 1997.
[65]H.I.Freedman.Deterministic Mathematics Models in Population Ecology,Marcel Dekker,New York,1980.
[66]J.D.Murry.Mathematical Biology,Springer-Verlag,New York,1989.
[67]K.Gopalsamy.Stability and Oscillations in Delay Differential Equations of Population Dynamics,Kluwer Academic Publishers,Boston,1992.
[68]M.Fan,Patrica J.Y.Wong,Ravi P.Agarwal.Periodicity and stability in periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments,Acta Math.Sinica 19(4)(2003):801-822.
[69]X.T.Yang.Uniform persistence and periodic solutions for a discrete predator-prey system with delays,J.Math.Anal.Appl.316(1)(2006):161-177.
[70]Smith H.Monotone Semiflows Generated by Functional Differential Equations[J].J Diff Equa,1987,66:420-442.
[71]叶彦谦.极限环论[M].第2版.上海:上海科技出版社,1984.