两种群食饵捕食系统的定性分析
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摘要
本文讨论了两种群的食饵捕食系统。在第一章里介绍了两种群的食饵捕食系统的研究现状及研究意义。第二章研究了一类食饵—捕食者系统当食饵种群的增长率和捕食率都为非线性情形的定性行为,分析了该系统的平衡点的性态,给出了该系统极限环不存在、存在的充分条件,补充完善了程荣福、蔡淑云(2002)及颜向平、张存华(2004)的结论。第三章研究捕食者具有Beddington-Deangelis功能反应函数和周期脉冲扰动的捕食者-食饵系统,利用脉冲微分方程的Floquet理论和比较定理,得到了系统灭绝和持续生存的充分条件。在第四章里我们讨论了一个具有性别偏食的食饵捕食系统,得出了系统灭绝和一致持久的充分条件。并且利用重合度理论中的延拓定理,给出了周期解存在的充分条件,特别的,得到了具有性别偏食的自治的食饵捕食系统周期解存在的充分条件,补充了刘秀湘,冯佑和的结论。
A kind of food-predator system are considered in this paper. Some background knowledge and the state of study are introduced in the first chapter. Based on the fact, the predator-prey system with functional response are considered, the quality of the equilibrium is discussed, the conditions of the existence and the nonexistence of the limit cycle are provided, and the conclusions of Cheng Rongfu and Yan Tonghua are bettered in the second chapter. The predator-prey system with Beddington-Deangelis functional response and periodic impulsive perturbations on the predator are considered by using the floquet theory of impulsive equation and comparison theorem, sufficient conditions for the system to be extinct and permanence are given in the third chapter. A prey-predator system with sexual favoritism are considered by means of a continuous theorem of coincidence degree theory, a sufficient condition of existence for periodic solution is presented in the fourth chapter. Especially, a sufficient condition of existence for periodic solution for automatic system is given out. The conclusions of Xiuixiang Liu and Youhe Feng are developed.
A kind of food-predator system are considered in this paper. Some background knowledge and the state of study are introduced in the first chapter. Based on the fact, the predator-prey system with functional response are considered, the quality of the equilibrium is discussed, the conditions of the existence and the nonexistence of the limit cycle are provided, and the conclusions of Cheng Rongfu and Yan Tonghua are bettered in the second chapter. The predator-prey system with Beddington-Deangelis functional response and periodic impulsive perturbations on the predator are considered by using the floquet theory of impulsive equation and comparison theorem, sufficient conditions for the system to be extinct and permanence are given in the third chapter. A prey-predator system with sexual favoritism are considered by means of a continuous theorem of coincidence degree theory, a sufficient condition of existence for periodic solution is presented in the fourth chapter. Especially, a sufficient condition of existence for periodic solution for automatic system is given out. The conclusions of Xiuixiang Liu and Youhe Feng are developed.
引文
[1] Holling CS. The functional response of predator to prey density and its role in mimicry and population regulation [J]. Men.Ent Soc.Can, 45:1-60.
[2] Li Yong Kun. Periodic solutions of n-species competion system with time delays [J]. Biomath. 1997,12(1):1-18.
[3] Zhengqiu Zhang, Zhicheng Wang. Periodic solutions for nonautonomous predator-prey system witn diffusion and time delay[J]. Hiroshima Math. 2001, 31:371-381.
[4] Zhang Xiao-ying, Bai ling, Fan Meng, Wang Ke. Existence of positive periodic solution for predator-prey difference system with Holling Ⅲ functional response [J]. Mathematica Applicata. 2002,15(3):25-31
[5] J.F.Andrews. A mathematical model for the continuous culture of microorganisms utilising inhibitory substrates[J]. Biotechnol.Bioeng. 10(1968),707-723.
[6] J.B.Collings. The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model[J]. J.Math.Biol. 36(1997),149-168.
[7] W. Sokol, J. A. Howell. Kinetics of phenol oxidation by washed cells[J], Biotechnol Bioeng. 1980.23(1). 2039-2049.
[8] J.R.Beddington. Mutual interference between parasites or predators and its effect on searching efficiency [J]. J.Animal Ecol. 44(1975) 331-340.
[9] D.L.DeAngelis, R.A.Goldstein, R.V.O Neill. A model for trophic interaction [J]. Ecology. 56(1975) 881—892.
[10] Dobromir T. Dimitror, Hristo V.Kojouharov. Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-Deangelis functional response [J]. Applied Mathmatics and Computation. 2005,162(2):523-538.
[11] Tzy-wei Huang. Global analysis of the predator-prey system with Beddington-DeAngelis functional response [J]. Math. Anal. Appl. 2003,281(1): 113-122.
[12] Robert Stephen Cantrell,Chris Cosner. On the Dnamics of predator-prey models with the Beddington-DeAngelis functional response [J]. Math. Anal. Appl. 2001, 257(l):206-222.
[13]Zhihua Liu,Rong Yuan. Stability and bifurcation in a delayed predator-prey system with the Beddington-DeAngelis functional response [J]. Math. Anal. Appl. 2004, 296(2):521-537
[14] Qiu Zhipeng,Yu Jun,Zou Yun. The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response [J]. Mathematical Biosciences. 2004,187(2): 175-187.
[15] Tzy-wei Huang. Uniqueness of limit cycles of the predator-prey system with Beddington- DeAngelis functional response [J]. Math. Anal. Appl. 290(1): 113-122.
[16] Meng Fan, Yang Kuang. Dynamics of a nonautonomous predator-prey system with Beddington- DeAngelis functional response [J]. Math. Anal. Appl. 2004, 295(1): 15-39.
[17J Kaill Norrdahl, Erkki Korpimaki. Des mobility or sex of voles affect risk of predation by mammalian predators [J].Ecology. 1998, 79(l):226-232.
[18] B.Shulgin,L.stone,Z.P.Agur. Pulse vaccination strategy in the SIR epidemic model [J]. Bull math Biol.1998, 44:203-224.
[19] A.Lakmeche,O.Arino. Birfurcation of non-trivial periodic solutions of impusive differential equations arising chemotherapeutic treament [J]. Dynamics of Continoous, Discrete and Impulsive System . 2000, 7:265-287.
[20] M.GRoberts, R.R.Kao .The dynamics of an infectious disease in a population with birth pulses [J]. Math Biosci. 1998, 149:23-26.
[21] S.Tang, L,Chen. Density-dependent birth rate, birth pluses and their population dynamic consequences[J]. Math Bio 2002, 44:2185-199.
[22] 程荣福,蔡淑云.一类具功能反应的食饵——捕食者两种群模型的定性分析[J].生物数学学报,2002,17(4):406—410.
[23] 颜向平,张存华.一类具功能反应的食饵——捕食者两种群模型的定性分析[J].生物数学学报,2004,19(3):323—327.
[24] 刘秀湘,冯佑和.具有性别偏食的二种群捕食者—食饵系统的模型[J].生物数学学报,2002,17(1):5-11.
[25] 张平光,赵申琪.一类三次系统极限环的惟一性[J].高校应用数学学报A辑 2003,18(1):27—32.
[26] 张芷芬,丁同仁,黄文灶等.微分方程定性理论[M].北京:科学出版社.1985,119.
[27] V. Lakshmikantham, D. Bainov, P. S. Simeonov. Theory of impulsive differential equations[M]. Singapore: World Scientific, 1989.
[28] O. Bemar, S. Souissi. qualitative behavior of stage-structure populations: application to structural validation[J]. J Math Biol, 1998, 37: 291-308.
[29] Kuang Y. Delay Differential Equations with Applications in Population Dynamics[M]. Academic Press, New York. 1993.
[2] Li Yong Kun. Periodic solutions of n-species competion system with time delays [J]. Biomath. 1997,12(1):1-18.
[3] Zhengqiu Zhang, Zhicheng Wang. Periodic solutions for nonautonomous predator-prey system witn diffusion and time delay[J]. Hiroshima Math. 2001, 31:371-381.
[4] Zhang Xiao-ying, Bai ling, Fan Meng, Wang Ke. Existence of positive periodic solution for predator-prey difference system with Holling Ⅲ functional response [J]. Mathematica Applicata. 2002,15(3):25-31
[5] J.F.Andrews. A mathematical model for the continuous culture of microorganisms utilising inhibitory substrates[J]. Biotechnol.Bioeng. 10(1968),707-723.
[6] J.B.Collings. The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model[J]. J.Math.Biol. 36(1997),149-168.
[7] W. Sokol, J. A. Howell. Kinetics of phenol oxidation by washed cells[J], Biotechnol Bioeng. 1980.23(1). 2039-2049.
[8] J.R.Beddington. Mutual interference between parasites or predators and its effect on searching efficiency [J]. J.Animal Ecol. 44(1975) 331-340.
[9] D.L.DeAngelis, R.A.Goldstein, R.V.O Neill. A model for trophic interaction [J]. Ecology. 56(1975) 881—892.
[10] Dobromir T. Dimitror, Hristo V.Kojouharov. Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-Deangelis functional response [J]. Applied Mathmatics and Computation. 2005,162(2):523-538.
[11] Tzy-wei Huang. Global analysis of the predator-prey system with Beddington-DeAngelis functional response [J]. Math. Anal. Appl. 2003,281(1): 113-122.
[12] Robert Stephen Cantrell,Chris Cosner. On the Dnamics of predator-prey models with the Beddington-DeAngelis functional response [J]. Math. Anal. Appl. 2001, 257(l):206-222.
[13]Zhihua Liu,Rong Yuan. Stability and bifurcation in a delayed predator-prey system with the Beddington-DeAngelis functional response [J]. Math. Anal. Appl. 2004, 296(2):521-537
[14] Qiu Zhipeng,Yu Jun,Zou Yun. The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response [J]. Mathematical Biosciences. 2004,187(2): 175-187.
[15] Tzy-wei Huang. Uniqueness of limit cycles of the predator-prey system with Beddington- DeAngelis functional response [J]. Math. Anal. Appl. 290(1): 113-122.
[16] Meng Fan, Yang Kuang. Dynamics of a nonautonomous predator-prey system with Beddington- DeAngelis functional response [J]. Math. Anal. Appl. 2004, 295(1): 15-39.
[17J Kaill Norrdahl, Erkki Korpimaki. Des mobility or sex of voles affect risk of predation by mammalian predators [J].Ecology. 1998, 79(l):226-232.
[18] B.Shulgin,L.stone,Z.P.Agur. Pulse vaccination strategy in the SIR epidemic model [J]. Bull math Biol.1998, 44:203-224.
[19] A.Lakmeche,O.Arino. Birfurcation of non-trivial periodic solutions of impusive differential equations arising chemotherapeutic treament [J]. Dynamics of Continoous, Discrete and Impulsive System . 2000, 7:265-287.
[20] M.GRoberts, R.R.Kao .The dynamics of an infectious disease in a population with birth pulses [J]. Math Biosci. 1998, 149:23-26.
[21] S.Tang, L,Chen. Density-dependent birth rate, birth pluses and their population dynamic consequences[J]. Math Bio 2002, 44:2185-199.
[22] 程荣福,蔡淑云.一类具功能反应的食饵——捕食者两种群模型的定性分析[J].生物数学学报,2002,17(4):406—410.
[23] 颜向平,张存华.一类具功能反应的食饵——捕食者两种群模型的定性分析[J].生物数学学报,2004,19(3):323—327.
[24] 刘秀湘,冯佑和.具有性别偏食的二种群捕食者—食饵系统的模型[J].生物数学学报,2002,17(1):5-11.
[25] 张平光,赵申琪.一类三次系统极限环的惟一性[J].高校应用数学学报A辑 2003,18(1):27—32.
[26] 张芷芬,丁同仁,黄文灶等.微分方程定性理论[M].北京:科学出版社.1985,119.
[27] V. Lakshmikantham, D. Bainov, P. S. Simeonov. Theory of impulsive differential equations[M]. Singapore: World Scientific, 1989.
[28] O. Bemar, S. Souissi. qualitative behavior of stage-structure populations: application to structural validation[J]. J Math Biol, 1998, 37: 291-308.
[29] Kuang Y. Delay Differential Equations with Applications in Population Dynamics[M]. Academic Press, New York. 1993.