污染环境中两类生物模型的定性分析
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摘要
当今世界污染日益严重,研究污染环境中生物种群的生存问题成为许多学者关注的热点问题。捕获也是影响生物种群生存的重要因素,尤其是常数捕获对生物种群造成的影响更难控制。本文利用积分均值、比较定理等方法分别对污染环境中Smith模型和Gallopin模型进行分析研究,以此预测种群的发展以及人为捕获对种群生存的影响。具体工作如下:
考虑死亡种群对体内毒素及环境毒素的影响,建立了一类污染环境中的单种群Smith模型。对外界毒素不变时,利用比较定理、Ruth-Hurwitz准则和复合矩阵,分别给出平衡点的存在性和稳定性的充分条件。当外界毒素变化时,在无捕获情况下,利用比较定理进行放缩寻找到体内毒素和外界毒素合适的界,得到种群在有限时间内β-生存和β-绝灭的充分条件。利用积分均值,引入控制函数,借助其极值范围的讨论,得到种群在有限时间内β-生存和β-绝灭的几组充分条件。在具有常数捕获下,利用比较定理和微分不等式的方法,得到种群在有限时间内β-生存和β-绝灭充分条件,并得到保证种群生存的最大捕获量。
考虑死亡种群对体内毒素及环境毒素的影响,对一类污染环境中的单种群Gallopin资源-消费者模型进行改进。针对无捕获时的Gallopin模型,利用微分不等式和比较定理,得到消费者种群和资源的有界性,并利用体内毒素和外界毒素合适的界得到种群在有限时间内β-生存和β-绝灭的充分条件。针对具有常数捕获时的Gallopin模型利用类似方法,得到种群在有限时间内β-生存、β-绝灭以及零绝灭的几组充分条件,并得到保证消费者种群生存的最大捕获量。
通过对污染环境中种群稳定性,生存状况的研究可以更好地指导人们保护环境,控制污染,这对于保持生态平衡和开发利用生物资源有着广泛的理论和现实意义。
In today,s world, our environment is more serious day and day. So research on changes of population,s survival condition in a polluted environment becomes a hot topic to many scholars. In addition to pollution, capture is also a important factor that affect the survival of population. Especially, the impact of constant capture is more difficult to control. In this paper, Smith model and Gallopin mod-el which live in polluted environment are studied by methods of integral in mean, comparison theorem and so on. So we can predict the development of population as well as the impact of man-made capture on population survival. Main results are as follows:
Firstly, under the effect of the variance of the dead population on the internal and environment toxicant density, a kind of single species Smith model is established. This paper gives sufficient conditions of the existence and stability of equilibrium under the constant external toxin by comparison theorem, Ruth-Hurwitz criterion and complex matrix. Under changed external toxin it gives some sufficient conditions forβ- persistence andβ- extinction of the Smith model with no capture by tow methodds. One is looking for the right boundary of internal and environment toxicant density by comparison theorem. The other is introudcting control function and discussing its extreme range by integral in mean. For the model with constant capture, it gives the sufficient conditions forβ-persistence andβ-extinction and obtains the maximal harvesting rate when the population is persistence.
Secondly, considering the influence of death on consumer population and the toxin density of environment, the original Gallopin model is improved. For the model with no capture, this paper gives the boundary of consumer population and resoures population. And the sufficient conditions forβ- persistence andβ-extinction are gives by differential inequality and comparison theorem. For the model with constant capture, the sufficient conditions forβ- persistence,β- extinction and zero-extinction are given by the same methods. And the maximal harvesting rate is obtained when the consumer population is persistence.
The research about stability and persistence of the population can be used to guide people to protect the environment and control pollution. Those will have very extensive theoretical and practical significance to maintain the diversity of ecosystems and make full use of the nature.
考虑死亡种群对体内毒素及环境毒素的影响,建立了一类污染环境中的单种群Smith模型。对外界毒素不变时,利用比较定理、Ruth-Hurwitz准则和复合矩阵,分别给出平衡点的存在性和稳定性的充分条件。当外界毒素变化时,在无捕获情况下,利用比较定理进行放缩寻找到体内毒素和外界毒素合适的界,得到种群在有限时间内β-生存和β-绝灭的充分条件。利用积分均值,引入控制函数,借助其极值范围的讨论,得到种群在有限时间内β-生存和β-绝灭的几组充分条件。在具有常数捕获下,利用比较定理和微分不等式的方法,得到种群在有限时间内β-生存和β-绝灭充分条件,并得到保证种群生存的最大捕获量。
考虑死亡种群对体内毒素及环境毒素的影响,对一类污染环境中的单种群Gallopin资源-消费者模型进行改进。针对无捕获时的Gallopin模型,利用微分不等式和比较定理,得到消费者种群和资源的有界性,并利用体内毒素和外界毒素合适的界得到种群在有限时间内β-生存和β-绝灭的充分条件。针对具有常数捕获时的Gallopin模型利用类似方法,得到种群在有限时间内β-生存、β-绝灭以及零绝灭的几组充分条件,并得到保证消费者种群生存的最大捕获量。
通过对污染环境中种群稳定性,生存状况的研究可以更好地指导人们保护环境,控制污染,这对于保持生态平衡和开发利用生物资源有着广泛的理论和现实意义。
In today,s world, our environment is more serious day and day. So research on changes of population,s survival condition in a polluted environment becomes a hot topic to many scholars. In addition to pollution, capture is also a important factor that affect the survival of population. Especially, the impact of constant capture is more difficult to control. In this paper, Smith model and Gallopin mod-el which live in polluted environment are studied by methods of integral in mean, comparison theorem and so on. So we can predict the development of population as well as the impact of man-made capture on population survival. Main results are as follows:
Firstly, under the effect of the variance of the dead population on the internal and environment toxicant density, a kind of single species Smith model is established. This paper gives sufficient conditions of the existence and stability of equilibrium under the constant external toxin by comparison theorem, Ruth-Hurwitz criterion and complex matrix. Under changed external toxin it gives some sufficient conditions forβ- persistence andβ- extinction of the Smith model with no capture by tow methodds. One is looking for the right boundary of internal and environment toxicant density by comparison theorem. The other is introudcting control function and discussing its extreme range by integral in mean. For the model with constant capture, it gives the sufficient conditions forβ-persistence andβ-extinction and obtains the maximal harvesting rate when the population is persistence.
Secondly, considering the influence of death on consumer population and the toxin density of environment, the original Gallopin model is improved. For the model with no capture, this paper gives the boundary of consumer population and resoures population. And the sufficient conditions forβ- persistence andβ-extinction are gives by differential inequality and comparison theorem. For the model with constant capture, the sufficient conditions forβ- persistence,β- extinction and zero-extinction are given by the same methods. And the maximal harvesting rate is obtained when the consumer population is persistence.
The research about stability and persistence of the population can be used to guide people to protect the environment and control pollution. Those will have very extensive theoretical and practical significance to maintain the diversity of ecosystems and make full use of the nature.
引文
[1] T G HALLAM, C E CLARK, R R LASSIDER. Effects of Toxicants on Populat-ions: A Qualitative ApproachⅠ. Equilibrium Environmental Exposure[J]. Ecol. Model., 1983, 8(1): 291-304.
[2] T G HALLAM, C E CLARK, G S JORDAN. Effects of Toxicants on Pppulatio- ns: A Qualitative ApproachⅠ. Equilibrium Environmental Exposure[J]. Ecol. Model., 1983, 8(1): 291-304.
[3] T G HALLAM, J T DELUNA. Effects of Toxicants on Populations: A Qualitati- ve ApproachⅢ. Environmental and Food Chain Pathways[J]. J. Theor. Biol., 1984, 109(3): 411-429.
[4] T G HALLAM, Z MA. Persistence in Population Models With Demographic[J]. Math. Biol., 1986, 24(2): 327-339.
[5] Z MA, B SONG, T G HALLAM. The Threshold Between Persistence and Extinction of a Population in a Polluted Environment[J]. Bull of Math. Biol., 1989, 51(3): 311-323.
[6] LIU HUAPING, MA ZHIEN. Threshold of Survival for System of Two Species in a Polluted Environment[J]. Math. Biol., 1991, (30): 49-61.
[7]闫惠臻.具有毒素影响的非简化二维Lotka-Volterra模型[J].大连轻工业学院学报, 1995, 14(1): 54-59.
[8] WANG WENDI, MA ZHIEN. Permanence of a Nonautomous Population Model [J]. ACTA Math., Appl. Sinica., 1998, (14): 86-95.
[9]何泽荣,马知恩.污染环境中Leslie模型的生存分析[J].生物数学学报, 1999, 14(3): 281-287.
[10] HE ZERONG, MA ZHIEN. The Survival Analysis for Gallopin,s System in a Polluted Environment [J]. Applied Mathematics and Mechanics., 2000, 21(8): 830-835.
[11] P D N Srinivasu. Control of Environmental Pollution to Conserve a Population [J]. Nonlinear Anal: Real World Appl., 2002, (3): 397-411.
[12]颜刚,王寿松,李英华.在小容量的污染环境中种群的持续生存与绝灭[J].生物数学学报, 2002, 17(3): 299-304.
[13]钟云娇,王克.污染环境中二维Volterra捕食-食饵模型生存分析[J].哈尔滨师范大学学报, 2005, 21(6): 4-8.
[14] HE JIWEI, WANG KE. The Surival Analysis for Smith,s System in a Polluted Environment[J]. Math. Biol., 2006, 21(1): 9-17.
[15]冯由玲,王克,孙静懿.具污染与捕获的Logistic单种群的持续生存及绝灭[J].生物数学学报, 2006, 21(3): 365-369.
[16]何继伟,王克.污染环境中Leslie系统[J].系统科学与数学, 2006, 26(1): 31-41.
[17]刘宇红,刘志美.具有毒素影响的二维Kolmogorov模型的定性分析[J].数学的实践与认识, 2007, 37(16): 3-6.
[18] YAN HUIZHEN, MA ZHIEN. Persistence and Extinction of an Individual in a Polluted Environment[J]. Engineering Math., 2007, 24(1):145-156.
[19]杨晓峰.脉冲污染环境中种群生态模型的持续性研究[D].山西:中北大学, 2006: 35-42.
[20]刘潇.污染环境中具有生育脉冲的单种群模型的最优捕获问题[J].生物数学学报, 2007, 22(2): 265-271.
[21]张弘,孟新柱,陈兰荪.脉冲作用对环境污染中单种群动力学行为影响[J].大连理工大学学报, 2008, 48(1): 147-153.
[22] JIAO JIANJUN, CHEN LANSUN. The Extinction Threshold on a Single Population Model with Pulse Input of Environmental Toxin in a Polluted Enviroment[J]. Mathematica Applicata., 2009, 22(1): 11-19.
[23]张晓颖,唐宗明,翁世有.污染环境中具有阶段结构的种群模型的生存分析[J].长春大学学报, 2007, 17(4): 1-6.
[24] MENG XINZHU, ZHAO QIULAN, CHEN LANSUN. Glabal Qualitative Analysis of new Monod Type Chemostat Model with Delayed Growth Resonse and Pulsed Input in Polluted Environment[J]. Appl. Math. Mech., 2008, 29(1): 75-87.
[25]李晓艳,姚频.污染斑块上具有扩散的Lotka-Volterra系统的持续生存与绝灭[J].重庆工学院学报, 2008, 22(4): 45-48.
[26]杨卓琴,陆启超,杨在中.扩散对污染斑块上Smith种群生存的影响[J].生物数学学报, 2003, 18(4):427-432.
[27]陈兰荪.数学生态学模型与研究方法[M].北京:科学出版社, 1988, 391-393.
[28]马知恩.种群生态学的建模与研究[M].合肥:安徽教育出版社, 1996, 169-301.
[29] T G HALLAM, Z MA. On Density and Oxtinction in Contionuous Population Moels[J]. Math. Biol., 1987, 25: 191-201.
[30]马知恩,宋保军,王稳地.环境污染对生物种群持续生存的影响[J].西安交通大学学报, 1991, 25(2): 37-47.
[31]闫慧臻,马知恩.非简化模型的β?持续生存及β?绝灭[J].大连轻工业学院学报, 1996, 15(2): 1-8.
[32]何泽荣,马知恩.污染与捕获对Logistic种群的影响[J].生物数学学报, 1997, 12(3): 230-237.
[33]于永泉,游雄,俞军.一类带收获的消费者种群的β?生存与绝灭条件[J].南京理工大学学报, 2002, 26(6): 662-665.
[34]程亚焕,李冬梅.污染与捕获条件下一类广义Logistic种群的生存分析[J].哈尔滨理工大学学报, 2004, 9(6): 119-121.
[35]刚毅,雒志学.污染环境中Leslie系统的收获模型[J].重庆工学院学报, 2008, 22(8): 62-65.
[36]马知恩,周义仓.常微分方程定性与稳定性方法[M].北京:科学出版社, 2001, 41-96.
[37] MICHAEL Y. LI, HAL L. SMITH, LIANCHENG WANG. Global Dynamics of an SEIR Epidemic Model with Vertical Transmission[J]. SIAM J. MATH., 2001(62): 58-69.
[38]尤秉礼.常微分方程补充教程[M].北京:人民教育出版社, 1981, 34-50.
[39]廖晓昕.稳定性理论、方法和应用[M].武汉华中理工大学出版社, 1999.
[40] M Y, Li, J S Muldowney. A Geometric Approach to Global-stability Problems [J]. SIAM M.Y, J. Math. Anal., 1996(1): 1070-1083.
[41] M Fan, M Y, Li, K Wang. Global Stability of an SEIR Epidemic Model with Recruitment and a Varying Total Population Size[J]. Math.Biosci., 2001.
[42] R H Martin, Jr. Logarithmic Norms and Projections Applied to Linear Differential Systems[J]. J. Math. Anal. Appl, 1974(45):432-454.
[2] T G HALLAM, C E CLARK, G S JORDAN. Effects of Toxicants on Pppulatio- ns: A Qualitative ApproachⅠ. Equilibrium Environmental Exposure[J]. Ecol. Model., 1983, 8(1): 291-304.
[3] T G HALLAM, J T DELUNA. Effects of Toxicants on Populations: A Qualitati- ve ApproachⅢ. Environmental and Food Chain Pathways[J]. J. Theor. Biol., 1984, 109(3): 411-429.
[4] T G HALLAM, Z MA. Persistence in Population Models With Demographic[J]. Math. Biol., 1986, 24(2): 327-339.
[5] Z MA, B SONG, T G HALLAM. The Threshold Between Persistence and Extinction of a Population in a Polluted Environment[J]. Bull of Math. Biol., 1989, 51(3): 311-323.
[6] LIU HUAPING, MA ZHIEN. Threshold of Survival for System of Two Species in a Polluted Environment[J]. Math. Biol., 1991, (30): 49-61.
[7]闫惠臻.具有毒素影响的非简化二维Lotka-Volterra模型[J].大连轻工业学院学报, 1995, 14(1): 54-59.
[8] WANG WENDI, MA ZHIEN. Permanence of a Nonautomous Population Model [J]. ACTA Math., Appl. Sinica., 1998, (14): 86-95.
[9]何泽荣,马知恩.污染环境中Leslie模型的生存分析[J].生物数学学报, 1999, 14(3): 281-287.
[10] HE ZERONG, MA ZHIEN. The Survival Analysis for Gallopin,s System in a Polluted Environment [J]. Applied Mathematics and Mechanics., 2000, 21(8): 830-835.
[11] P D N Srinivasu. Control of Environmental Pollution to Conserve a Population [J]. Nonlinear Anal: Real World Appl., 2002, (3): 397-411.
[12]颜刚,王寿松,李英华.在小容量的污染环境中种群的持续生存与绝灭[J].生物数学学报, 2002, 17(3): 299-304.
[13]钟云娇,王克.污染环境中二维Volterra捕食-食饵模型生存分析[J].哈尔滨师范大学学报, 2005, 21(6): 4-8.
[14] HE JIWEI, WANG KE. The Surival Analysis for Smith,s System in a Polluted Environment[J]. Math. Biol., 2006, 21(1): 9-17.
[15]冯由玲,王克,孙静懿.具污染与捕获的Logistic单种群的持续生存及绝灭[J].生物数学学报, 2006, 21(3): 365-369.
[16]何继伟,王克.污染环境中Leslie系统[J].系统科学与数学, 2006, 26(1): 31-41.
[17]刘宇红,刘志美.具有毒素影响的二维Kolmogorov模型的定性分析[J].数学的实践与认识, 2007, 37(16): 3-6.
[18] YAN HUIZHEN, MA ZHIEN. Persistence and Extinction of an Individual in a Polluted Environment[J]. Engineering Math., 2007, 24(1):145-156.
[19]杨晓峰.脉冲污染环境中种群生态模型的持续性研究[D].山西:中北大学, 2006: 35-42.
[20]刘潇.污染环境中具有生育脉冲的单种群模型的最优捕获问题[J].生物数学学报, 2007, 22(2): 265-271.
[21]张弘,孟新柱,陈兰荪.脉冲作用对环境污染中单种群动力学行为影响[J].大连理工大学学报, 2008, 48(1): 147-153.
[22] JIAO JIANJUN, CHEN LANSUN. The Extinction Threshold on a Single Population Model with Pulse Input of Environmental Toxin in a Polluted Enviroment[J]. Mathematica Applicata., 2009, 22(1): 11-19.
[23]张晓颖,唐宗明,翁世有.污染环境中具有阶段结构的种群模型的生存分析[J].长春大学学报, 2007, 17(4): 1-6.
[24] MENG XINZHU, ZHAO QIULAN, CHEN LANSUN. Glabal Qualitative Analysis of new Monod Type Chemostat Model with Delayed Growth Resonse and Pulsed Input in Polluted Environment[J]. Appl. Math. Mech., 2008, 29(1): 75-87.
[25]李晓艳,姚频.污染斑块上具有扩散的Lotka-Volterra系统的持续生存与绝灭[J].重庆工学院学报, 2008, 22(4): 45-48.
[26]杨卓琴,陆启超,杨在中.扩散对污染斑块上Smith种群生存的影响[J].生物数学学报, 2003, 18(4):427-432.
[27]陈兰荪.数学生态学模型与研究方法[M].北京:科学出版社, 1988, 391-393.
[28]马知恩.种群生态学的建模与研究[M].合肥:安徽教育出版社, 1996, 169-301.
[29] T G HALLAM, Z MA. On Density and Oxtinction in Contionuous Population Moels[J]. Math. Biol., 1987, 25: 191-201.
[30]马知恩,宋保军,王稳地.环境污染对生物种群持续生存的影响[J].西安交通大学学报, 1991, 25(2): 37-47.
[31]闫慧臻,马知恩.非简化模型的β?持续生存及β?绝灭[J].大连轻工业学院学报, 1996, 15(2): 1-8.
[32]何泽荣,马知恩.污染与捕获对Logistic种群的影响[J].生物数学学报, 1997, 12(3): 230-237.
[33]于永泉,游雄,俞军.一类带收获的消费者种群的β?生存与绝灭条件[J].南京理工大学学报, 2002, 26(6): 662-665.
[34]程亚焕,李冬梅.污染与捕获条件下一类广义Logistic种群的生存分析[J].哈尔滨理工大学学报, 2004, 9(6): 119-121.
[35]刚毅,雒志学.污染环境中Leslie系统的收获模型[J].重庆工学院学报, 2008, 22(8): 62-65.
[36]马知恩,周义仓.常微分方程定性与稳定性方法[M].北京:科学出版社, 2001, 41-96.
[37] MICHAEL Y. LI, HAL L. SMITH, LIANCHENG WANG. Global Dynamics of an SEIR Epidemic Model with Vertical Transmission[J]. SIAM J. MATH., 2001(62): 58-69.
[38]尤秉礼.常微分方程补充教程[M].北京:人民教育出版社, 1981, 34-50.
[39]廖晓昕.稳定性理论、方法和应用[M].武汉华中理工大学出版社, 1999.
[40] M Y, Li, J S Muldowney. A Geometric Approach to Global-stability Problems [J]. SIAM M.Y, J. Math. Anal., 1996(1): 1070-1083.
[41] M Fan, M Y, Li, K Wang. Global Stability of an SEIR Epidemic Model with Recruitment and a Varying Total Population Size[J]. Math.Biosci., 2001.
[42] R H Martin, Jr. Logarithmic Norms and Projections Applied to Linear Differential Systems[J]. J. Math. Anal. Appl, 1974(45):432-454.