污染环境中具有捕获的种群生存分析
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摘要
随着环境污染的日益严重,研究污染环境中种群生存状况的变化规律,成为数学生态学领域中的一个热点问题。污染与捕获是影响种群生存和发展的两个重要因素。本文通过对污染环境中具有变收获量的生态模型定性分析,以此预测种群的发展变化及人们的捕获行为对种群生存产生的影响。由此判断出应把污染控制到什么程度,才能够维持生态系统的平衡。研究的结论对于控制生态系统的稳定性具有一定的参考价值。本文研究内容如下:
第一部分,考虑了捕获以及死亡对消费者种群和环境中毒素浓度的影响,改进了原有Gallopin资源—消费者模型。利用积分均值法和比较定理,给出消费者种群和资源量的有界性,以及环境容量较小情形下种群弱平均持续生存的充分条件。在假设消费者种群持续生存时,引入控制函数,借助其极值范围的讨论,得到了种群绝灭的几组充分条件。
第二部分,考虑了捕获、死亡以及毒素在种群间流动对种群生存和环境中毒素浓度的影响,改进了原有二维Volterra模型。针对竞争模型,在两种群均不受毒素影响和至少有一种种群受毒素影响情况下,以积分均值和比较定理为手段,得到种群弱平均持续生存性的充分条件。在假定两种群均受毒素影响时,用同样方法得到了种群绝灭的充分条件。针对互惠模型,得到种群β-绝灭和β-持续生存时体内毒素最值及控制函数最值。利用分析法和比较定理,得到两种群分别一致持续生存以及整个系统一致持续生存的充分条件。
第三部分,引入基于比率思想,并且假定每种群体内毒素浓度是不同的,建立了有两种竞争食饵和一种捕食者的模型。通过比较定理得到了种群的有界性,及种群体内毒素和环境中毒素的有界性。利用分析法,从环境内毒素的变化率出发,来计算单位生物体内毒素的变化率。推导过程中主要应用了比较定理,并且通过严密的计算,得到了三种群持续生存性和绝灭性的充分条件。
As the pollution of our environment is more serious day by day, research on changes of population's survival condition in a polluted environment becomes a hot topic in mathematical ecology. Pollution and capture are two important factors that affect the survival and development of population. In this paper, by means of qualitative analysis of the ecological model with variable capture in a polluted environment, population's development and the influence of people's capture behavior on the population survival can be forecasted. Besides, it can be estimated how we should control contamination in order to maintain the balance of ecosystem. The results in this paper can be referenced in controlling the stability of ecosystem. The present paper is organized as follows:
First, influence of capture and the death on consumer population and the toxin density of environment is discussed, and the original Gallopin resources-consumer model is improved. By integral in mean and comparison theorem, boundednesses of consumer population and resources, as well as a sufficient condition for population’s weak persistent in mean are given when the environment is with a small capacity. Under the assumption of persistent of consumer population, control function is introduced and some sufficient conditions for population’s extinction are obtained based on the discussion of the range of extremum.
Sencond, influence of capture and the death as well as toxin flowing between populations on the survival of population and toxin density of environment is discussed, and the original two-dimensional Volterra model is improved. Furhermore, by integral in mean and comparison principle, a sufficient condition for population’s weak persistent in mean in a competing model is presented in the case that none of two populatons is affected by toxin and that at least one population is under the influence of toxin. Under the assumption that two populations both receive the influence of toxin, a sufficient condition for population extinct is obtained by similar methods. Moreover, Extremums of toxin and control function in a cooperative model are shown forβ-extinction andβ-persistence of population. By analytical method and comparison theorem, some sufficient conditions for uniformly persistent of two populations and the whole system are obtained.
Third, ratio-dependent idea is introduced and a model, which has two kinds of competed predators and one prey, is established based on the fact that toxin density of each population is different. Boundednesses of population and toxin in population and in the environment are shown by comparison theorem. By means of analytic method, the rate of change of toxin in unit organism is calculated by the rate of change of the toxin in environment. In the process of calculation, by comparison theorem and strict calculation, some sufficient conditions for persistent and extinct of three populations are obtained.
第一部分,考虑了捕获以及死亡对消费者种群和环境中毒素浓度的影响,改进了原有Gallopin资源—消费者模型。利用积分均值法和比较定理,给出消费者种群和资源量的有界性,以及环境容量较小情形下种群弱平均持续生存的充分条件。在假设消费者种群持续生存时,引入控制函数,借助其极值范围的讨论,得到了种群绝灭的几组充分条件。
第二部分,考虑了捕获、死亡以及毒素在种群间流动对种群生存和环境中毒素浓度的影响,改进了原有二维Volterra模型。针对竞争模型,在两种群均不受毒素影响和至少有一种种群受毒素影响情况下,以积分均值和比较定理为手段,得到种群弱平均持续生存性的充分条件。在假定两种群均受毒素影响时,用同样方法得到了种群绝灭的充分条件。针对互惠模型,得到种群β-绝灭和β-持续生存时体内毒素最值及控制函数最值。利用分析法和比较定理,得到两种群分别一致持续生存以及整个系统一致持续生存的充分条件。
第三部分,引入基于比率思想,并且假定每种群体内毒素浓度是不同的,建立了有两种竞争食饵和一种捕食者的模型。通过比较定理得到了种群的有界性,及种群体内毒素和环境中毒素的有界性。利用分析法,从环境内毒素的变化率出发,来计算单位生物体内毒素的变化率。推导过程中主要应用了比较定理,并且通过严密的计算,得到了三种群持续生存性和绝灭性的充分条件。
As the pollution of our environment is more serious day by day, research on changes of population's survival condition in a polluted environment becomes a hot topic in mathematical ecology. Pollution and capture are two important factors that affect the survival and development of population. In this paper, by means of qualitative analysis of the ecological model with variable capture in a polluted environment, population's development and the influence of people's capture behavior on the population survival can be forecasted. Besides, it can be estimated how we should control contamination in order to maintain the balance of ecosystem. The results in this paper can be referenced in controlling the stability of ecosystem. The present paper is organized as follows:
First, influence of capture and the death on consumer population and the toxin density of environment is discussed, and the original Gallopin resources-consumer model is improved. By integral in mean and comparison theorem, boundednesses of consumer population and resources, as well as a sufficient condition for population’s weak persistent in mean are given when the environment is with a small capacity. Under the assumption of persistent of consumer population, control function is introduced and some sufficient conditions for population’s extinction are obtained based on the discussion of the range of extremum.
Sencond, influence of capture and the death as well as toxin flowing between populations on the survival of population and toxin density of environment is discussed, and the original two-dimensional Volterra model is improved. Furhermore, by integral in mean and comparison principle, a sufficient condition for population’s weak persistent in mean in a competing model is presented in the case that none of two populatons is affected by toxin and that at least one population is under the influence of toxin. Under the assumption that two populations both receive the influence of toxin, a sufficient condition for population extinct is obtained by similar methods. Moreover, Extremums of toxin and control function in a cooperative model are shown forβ-extinction andβ-persistence of population. By analytical method and comparison theorem, some sufficient conditions for uniformly persistent of two populations and the whole system are obtained.
Third, ratio-dependent idea is introduced and a model, which has two kinds of competed predators and one prey, is established based on the fact that toxin density of each population is different. Boundednesses of population and toxin in population and in the environment are shown by comparison theorem. By means of analytic method, the rate of change of toxin in unit organism is calculated by the rate of change of the toxin in environment. In the process of calculation, by comparison theorem and strict calculation, some sufficient conditions for persistent and extinct of three populations are obtained.
引文
1 徐克学.生物数学.科学出版社,1991:1~2
2 马知恩.种群生态学的数学建模与研究.安徽教育出版社,1996:2,179,268,56
3 陈兰荪.数学生态学模型与研究方法.科学出版社,1991:1,19~20
4 T.G.Hallam,C.E.Clark and R.R.Lassider. Effects of Toxicants on Popu-lations:A Qualitative Approach I. Equilibrium Environmental Exposure. Ecol.Model.,1983,8(1):291~304
5 T.G.Hallam,C.E.Clark and G.S.Jordan. Effects of Toxicants on Popu-lations:A Qualitative Approach II. First Order Kinetics. J.Math.Biol.,1983,18(2):25~37
6 T.G.Hallam,J.T.Deluna.Effects of Toxicants on Populations:A Qualita- tive Approach III. Environmental and Food Chain Pathways. J.Theor. Biol.,1984,109(3):411~429
7 P.D.N.Srinivasu. Control of Environmental Pollution to Conserve a Popu-lation. Nonlinear Anal:Real World Appl.,2002,(3):397~411
8 G.C.Gallopin. A Generalized Model of Resource-population System. Oeco- logia,1971,(7):382~432
9 何泽荣,马知恩.污染环境中 Leslie 模型的生存分析. 生物数学学报,1999,14(3):281~287
10 何泽荣,马知恩.污染环境中 Gallopin 模型的生存分析.应用数学和力学,2000,21(8):830~835
11 Debasis MukherJee. Persistence and Global Stability of a Population in a Pollu- ted Environment with Delay. J.Biol.Sys.,2002,10(3):225~232
12 何泽荣,马知恩.污染与捕获对 Logistic 种群的影响.生物数学学报,1997,12(3):230~237
13 于永泉,游雄,俞军.一类带收获率的消费者种群的 β -生存与灭绝条件.南京理工大学学报,2002,26(6):662~665
14 B.Buonomo,A.Di Liddo and I.Sgura.A Diffusive-convective Model for The Dynamics of Population-toxicant Interactions:Some Analytical and Numerical Results.Math.Biosci.,l999,(157):37~64
15 冯由玲. 污染环境中单种群的生存分析与经济策略. 东北师范大学硕士论文,2005:35~47
16 L.Huaping,Ma Zhien. The Threshold of Survival System of Two Species in a Polluted Environment.J.Math.Biol.,1991,(30):49~61
17 阎慧臻.具有毒素影响的非简化二维 Lotka-Volterra 模型.大连轻工业学院学报,1995,14(1):54~59
18 阎慧臻,马知恩.非简化模型的 β -持续生存及 β -绝灭. 大连轻工业学院学报,1996,15(2):1~8
19 Wang Wendi,Ma Zhien. Permanence of a Nonautonomous Population Model. ACTA Math.Appl.Sinica,1998,(14):86~95
20 宋保军.污染环境中两种群的持续性.生物数学学报,1993,(2):46~55
21 Ma Zhien,Song BaoJun and T.G.Hallam. The Threshold of Survival for Systems in a Fluctuating Environment.Bull.Math.Biol.,1989,51(3):31 1~323
22 赖菲,马知恩,周佃民,王庆.污染环境中两种群的永久生存与灭绝.西安交通大学学报,2000,34(12):88~90
23 Ma Zhien,Cui Guirong and Wang Wendi.Persistence and Extinction of a Population in a Polluted Environment.Math.Biosci.,1990,(101):75~97
24 T.G.Hallam,Ma Zhien. Persistence in Population Models with Demographic Fluctuations. J.Math.Biol.,1986,24(3):327~339
25 Ma Zhien,Luo Zhixue. The Threshold of Survival for Volterra Systems of Two Preys and One Predator in Polluted Environment. J.Math.Biol.,2001,(22):27~39
26 Ma Zhien,Luo Zhixue,Jin Zhen and Zhao Hengbao. The Threshold of Survival for Predator-prey Volterra System of Three Species in a Polluted Environment. Sys.Sci.Math.Sci.,1995,8(4):373~382
27 靳祯,马知恩. 污染环境中三维时变 Volterra 捕食-被捕食模型的持续生存.生物数学学报,2000,15(4):413~419
28 钟云娇,王克.污染环境中二维 Volterra 捕食者-食饵模型生存分析.哈尔滨师范大学学报,2005,21(6):4~8
29 靳祯,马知恩. 污染环境中三维 Volterra 系统持续生存与绝灭的阈值. 系统科学与数学,1998,18(3):366~370
30 潘晋孝,王峰.污染环境中三维 Volterra 互惠系统持续生存与绝灭的阈值.工程数学学报,2000,17(1):93~98
31 雒志学,王绵森,何泽荣.体内毒素不同的三维时变 Volterra 系统的持续生存.生物数学学报,2003,18(1):43~49
32 M.Gamez,R.Carreno. Persistence in the Mean of Some Competitive Sys- tems. Nonlinear Studies,1999,6(1):65~72
33 李展.两类生态系统的研究.东北师范大学硕士学位论文,2005:37~46
34 杨晓峰.脉冲污染环境中种群生态模型的持续性研究.中北大学硕士学位论文,2006:35~42
35 R.Arditi,L.R.Ginzburg. Coupling in Predator-Prey Dynamics:Ratio-Dep- endence. J.Theor.Biol.,1989,(139):311~326
36 张惠英,陈凤德.基于比率的三种群混合模型的研究.福州大学学报,2003,31(4):391~395
37 Zhan Li,Zhisheng Shuai and Ke Wang.Persistence and Extinction of Single Population in a Polluted Enviroment . Electronic Journal of Differential Equations,2004,108:1~5
38 R.Arditi,L.R.Ginzburg and H.R.Akcakaya. Variation in Plankton Densities among Lakes:A Case for Ratio-Dependent Models. American Naturalist, 1991,(138):1287~1296
39 R.Arditi,N.Perrin and H.Saiah. Functional Response and Heterogeneities:An Experiment Test with Cladocerans. Oikos,1991,(60):69~75
40 I.Hanski. The Functional Response of Predator:Worries about Scale. TREE,1991,(6):141~142
41 马知恩,周义仓.常微分方程定性与稳定性方法.科学出版社,2003:13~14
42 何继伟,王克.污染环境中的 Lesile 模型.模型科学与数学,2006,26(1):31~41
43 Jiang Liqiang,Ma Zhien.Persistence of Population for a Predator-Prey Systerm in a Polluted Environment. Ann.of Diff .Eqs.,1999,15(2):127~142
44 He Jiwei,Wang Ke.The Survival for a Nonautomous Population Model in a Polluted Environment.College Mathmatics,2005,21(1):30~36
45 程亚焕,李冬梅. 小容量污染环境中二维 Volterra 互惠模型的生存分析. 黑龙江大学自然科学学报,2006,(23):269~272
46 G.Kirlinger. Permanence of Some Ecological Systerms with Several Predator and one Prey Species. J.Math.Biol.,1998,(26):217~232
2 马知恩.种群生态学的数学建模与研究.安徽教育出版社,1996:2,179,268,56
3 陈兰荪.数学生态学模型与研究方法.科学出版社,1991:1,19~20
4 T.G.Hallam,C.E.Clark and R.R.Lassider. Effects of Toxicants on Popu-lations:A Qualitative Approach I. Equilibrium Environmental Exposure. Ecol.Model.,1983,8(1):291~304
5 T.G.Hallam,C.E.Clark and G.S.Jordan. Effects of Toxicants on Popu-lations:A Qualitative Approach II. First Order Kinetics. J.Math.Biol.,1983,18(2):25~37
6 T.G.Hallam,J.T.Deluna.Effects of Toxicants on Populations:A Qualita- tive Approach III. Environmental and Food Chain Pathways. J.Theor. Biol.,1984,109(3):411~429
7 P.D.N.Srinivasu. Control of Environmental Pollution to Conserve a Popu-lation. Nonlinear Anal:Real World Appl.,2002,(3):397~411
8 G.C.Gallopin. A Generalized Model of Resource-population System. Oeco- logia,1971,(7):382~432
9 何泽荣,马知恩.污染环境中 Leslie 模型的生存分析. 生物数学学报,1999,14(3):281~287
10 何泽荣,马知恩.污染环境中 Gallopin 模型的生存分析.应用数学和力学,2000,21(8):830~835
11 Debasis MukherJee. Persistence and Global Stability of a Population in a Pollu- ted Environment with Delay. J.Biol.Sys.,2002,10(3):225~232
12 何泽荣,马知恩.污染与捕获对 Logistic 种群的影响.生物数学学报,1997,12(3):230~237
13 于永泉,游雄,俞军.一类带收获率的消费者种群的 β -生存与灭绝条件.南京理工大学学报,2002,26(6):662~665
14 B.Buonomo,A.Di Liddo and I.Sgura.A Diffusive-convective Model for The Dynamics of Population-toxicant Interactions:Some Analytical and Numerical Results.Math.Biosci.,l999,(157):37~64
15 冯由玲. 污染环境中单种群的生存分析与经济策略. 东北师范大学硕士论文,2005:35~47
16 L.Huaping,Ma Zhien. The Threshold of Survival System of Two Species in a Polluted Environment.J.Math.Biol.,1991,(30):49~61
17 阎慧臻.具有毒素影响的非简化二维 Lotka-Volterra 模型.大连轻工业学院学报,1995,14(1):54~59
18 阎慧臻,马知恩.非简化模型的 β -持续生存及 β -绝灭. 大连轻工业学院学报,1996,15(2):1~8
19 Wang Wendi,Ma Zhien. Permanence of a Nonautonomous Population Model. ACTA Math.Appl.Sinica,1998,(14):86~95
20 宋保军.污染环境中两种群的持续性.生物数学学报,1993,(2):46~55
21 Ma Zhien,Song BaoJun and T.G.Hallam. The Threshold of Survival for Systems in a Fluctuating Environment.Bull.Math.Biol.,1989,51(3):31 1~323
22 赖菲,马知恩,周佃民,王庆.污染环境中两种群的永久生存与灭绝.西安交通大学学报,2000,34(12):88~90
23 Ma Zhien,Cui Guirong and Wang Wendi.Persistence and Extinction of a Population in a Polluted Environment.Math.Biosci.,1990,(101):75~97
24 T.G.Hallam,Ma Zhien. Persistence in Population Models with Demographic Fluctuations. J.Math.Biol.,1986,24(3):327~339
25 Ma Zhien,Luo Zhixue. The Threshold of Survival for Volterra Systems of Two Preys and One Predator in Polluted Environment. J.Math.Biol.,2001,(22):27~39
26 Ma Zhien,Luo Zhixue,Jin Zhen and Zhao Hengbao. The Threshold of Survival for Predator-prey Volterra System of Three Species in a Polluted Environment. Sys.Sci.Math.Sci.,1995,8(4):373~382
27 靳祯,马知恩. 污染环境中三维时变 Volterra 捕食-被捕食模型的持续生存.生物数学学报,2000,15(4):413~419
28 钟云娇,王克.污染环境中二维 Volterra 捕食者-食饵模型生存分析.哈尔滨师范大学学报,2005,21(6):4~8
29 靳祯,马知恩. 污染环境中三维 Volterra 系统持续生存与绝灭的阈值. 系统科学与数学,1998,18(3):366~370
30 潘晋孝,王峰.污染环境中三维 Volterra 互惠系统持续生存与绝灭的阈值.工程数学学报,2000,17(1):93~98
31 雒志学,王绵森,何泽荣.体内毒素不同的三维时变 Volterra 系统的持续生存.生物数学学报,2003,18(1):43~49
32 M.Gamez,R.Carreno. Persistence in the Mean of Some Competitive Sys- tems. Nonlinear Studies,1999,6(1):65~72
33 李展.两类生态系统的研究.东北师范大学硕士学位论文,2005:37~46
34 杨晓峰.脉冲污染环境中种群生态模型的持续性研究.中北大学硕士学位论文,2006:35~42
35 R.Arditi,L.R.Ginzburg. Coupling in Predator-Prey Dynamics:Ratio-Dep- endence. J.Theor.Biol.,1989,(139):311~326
36 张惠英,陈凤德.基于比率的三种群混合模型的研究.福州大学学报,2003,31(4):391~395
37 Zhan Li,Zhisheng Shuai and Ke Wang.Persistence and Extinction of Single Population in a Polluted Enviroment . Electronic Journal of Differential Equations,2004,108:1~5
38 R.Arditi,L.R.Ginzburg and H.R.Akcakaya. Variation in Plankton Densities among Lakes:A Case for Ratio-Dependent Models. American Naturalist, 1991,(138):1287~1296
39 R.Arditi,N.Perrin and H.Saiah. Functional Response and Heterogeneities:An Experiment Test with Cladocerans. Oikos,1991,(60):69~75
40 I.Hanski. The Functional Response of Predator:Worries about Scale. TREE,1991,(6):141~142
41 马知恩,周义仓.常微分方程定性与稳定性方法.科学出版社,2003:13~14
42 何继伟,王克.污染环境中的 Lesile 模型.模型科学与数学,2006,26(1):31~41
43 Jiang Liqiang,Ma Zhien.Persistence of Population for a Predator-Prey Systerm in a Polluted Environment. Ann.of Diff .Eqs.,1999,15(2):127~142
44 He Jiwei,Wang Ke.The Survival for a Nonautomous Population Model in a Polluted Environment.College Mathmatics,2005,21(1):30~36
45 程亚焕,李冬梅. 小容量污染环境中二维 Volterra 互惠模型的生存分析. 黑龙江大学自然科学学报,2006,(23):269~272
46 G.Kirlinger. Permanence of Some Ecological Systerms with Several Predator and one Prey Species. J.Math.Biol.,1998,(26):217~232