三类生态模型解的渐近性
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摘要
本文通过构造一致持久生存域，利用Liapunov泛函、代数理论、特征方程等方法研究了三类生态模型解的渐近性，其中包括一致持久生存性、全局吸引性、稳定性和Hopf-分支等解的性态。
在生态学研究中，对于生命周期短、世代不重叠或者虽然生命周期长、世代重叠，但在数量上较少的种群，研究离散模型比连续模型更具有实际意义。本文首先研究了一类具有时滞的离散非自治捕食链模型。通过构造持久生存域得到了模型一致持久生存的充分条件和必要条件，当系统是自治系统时，给出了模型一致持久生存的充要条件。得出的定理结论说明了时滞对该模型的一致持久生存性没有影响。
在现实的野生动物保护过程中人们通常采用扩散的方法让濒危种群迁移改变其生存环境来保护野生动物，近年来许多学者研究了生物种群在不同斑块中扩散的生态模型，但是对于扩散具有时滞效应的模型很少研究。本文其次研究了一类扩散项具有时滞的两斑块环境的捕食与被捕食模型。首先利用微分不等式讨论了模型的一致持久生存性，得到了模型一致持久生存的充分条件，当系统的系数是正ω周期函数、时滞是ω的非负整数倍时，利用Brouwer不动点原理得到了系统存在一个正ω周期解，利用构造V函数的方法以及Barbǎlat引理得到了系统的正周期解全局吸引的充分条件。
在自然界中，许多种群个体在一生中都要经历两个阶段，即幼年和成年两个阶段。而且种群在不同的生理阶段其生理特征有着明显的差异，如幼年种群没有生育能力，捕食能力较弱，而成年种群不仅有生育能力，而且生存能力较强。随着对传染病研究的深入，数学模型日益成为分析和控制传染病的重要工具。传染病模型日益受到科学家们的重视，近年来许多学者研究了传染病模型，但这些模型总是假定各个年龄阶段的种群个体对某种传染病具有相同的传染率。然而对于某些传染病事实并非如此，如麻疹、天花等传染病多发于幼年阶段，而伤寒、副伤寒等传染病多在成年人之间流行。因此，考虑具有不同传染率的阶段结构的传染病模型更具有实际意义。本文最后研究了一类成年种群具有疾病而幼年种群不具有疾病的SI传染病模型。在模型中假定种群具有两个阶段，即幼年阶段和成年阶段，且幼年种群转化为成年种群的数量与幼年种群的数量成正比其比率为常数，幼年种群的死亡率为常数且成年种群具有密度制约，幼年种群不感染疾病仅
成年种群感染疾病，疾病具有一个潜伏期丁。本文利用代数理论及特征方程讨论
了模型非负平衡态的局部稳定性，得到了系统正平衡态绝对稳定的充分条件。当
把时滞作为分支参数时，给出了系统出现分支的条件及分支值，利用Liapunov函
数及比较原理得出了疾病消除的阂值。
In this paper, we consider the asymptotic behavior of three ecological systems by establishing region of permanence, using Liapunov function, algebraic theory and characteristic equation.
In the study of ecology, discrete model is more significant in practice than differential model as for these species which are short in life and non-overlapping in generations or long in life and overlapping in generations but fewer in quantity. In this paper, first, we consider a discrete nonautonmous food-chain system with delays. The sufficent condition and necessary condition are obtained by means of constructing the region of permanence and necessary and sufficient condition for permanence is obtained when the system is autonomous. As a result the mathematic methods in this paper can be used to research food-chain systems with multi-species.
One of the most important questions in population ecology is to find the permanence conditions for the species. Disperse is used making the extinct species migrate so that they are saved for protecting species. In recent years, many ecological systems have been investigated by ecological scientists. But we find few studies on the systems with delayed effect in disperse. In this paper, second, we consider a nonautonomous predator-prey system with dispersal delays in two habitats. The persistence of system and the sufficient condition of persistence is obtained by means of using the differential inequality and when the coefficents of system are positive periodic functions with common periodic u> and delays are integro-multiple of w, the w-periodic positive solution of solutions are exist by means of using of Brouwer's fixed point theorem and sufficient condition is obtained for the global attracticity of w-periodic positive solution of system by means of constructing suitable Lyapunov function.
Individual's growth of many species have two stage that are juvenile stage and adult stage. In each stage of its development, it always shows different characteristic. For instance, the immature species cannot have reproductive ability and predative ability while the mature species not only have reproductive ability but also have more powerful survival capacity. Therefore studing of stage-structured systems have much practical significance. Mathematic models have been used to analysis and control epidemic models with the
further investigating epidemic. Epidemic models have been studied by many scientist. But they always propose that the individuals have same infection conversing in different stage. However this is not the thing for some disease transmission. For instance, some epidemic (measles, smallpox) always transmite in juvenile stage while some epidemic (typhoid fever, paratyphoid fever)are transmite in adult stage. Therefore, it has much practical significance towards investgating epidemic models with stage-structured. In this paper, final, we investigate a SI epidemic model with stage-structured. We propose that individual of species has two stage that are immature and mature stage and the mature rate of immature species is proportional to existing immature specie density in proportion to the constant d and the death rate of immature species is proportional to existing immature specie density, the number of mature specie depents on the density of exiting mature specie and we assume that immature species dosn't infect
disease but only mature species infect disease and disease has the latent period r. In this paper we investigated the local stability of non-negative equilibria and the sufficient condition of locally asymptotically stable is obtaied by means of using algebra theory and characteristic equations. We show that positive equilibrium will loss of stabillity with the delay increased and a Hopf biffur-cation will occur and the global attractivity of disease-free equilibrum is surveyed and the threshold of disease disppearing is obtained by means of using Liapunov function.
在生态学研究中，对于生命周期短、世代不重叠或者虽然生命周期长、世代重叠，但在数量上较少的种群，研究离散模型比连续模型更具有实际意义。本文首先研究了一类具有时滞的离散非自治捕食链模型。通过构造持久生存域得到了模型一致持久生存的充分条件和必要条件，当系统是自治系统时，给出了模型一致持久生存的充要条件。得出的定理结论说明了时滞对该模型的一致持久生存性没有影响。
在现实的野生动物保护过程中人们通常采用扩散的方法让濒危种群迁移改变其生存环境来保护野生动物，近年来许多学者研究了生物种群在不同斑块中扩散的生态模型，但是对于扩散具有时滞效应的模型很少研究。本文其次研究了一类扩散项具有时滞的两斑块环境的捕食与被捕食模型。首先利用微分不等式讨论了模型的一致持久生存性，得到了模型一致持久生存的充分条件，当系统的系数是正ω周期函数、时滞是ω的非负整数倍时，利用Brouwer不动点原理得到了系统存在一个正ω周期解，利用构造V函数的方法以及Barbǎlat引理得到了系统的正周期解全局吸引的充分条件。
在自然界中，许多种群个体在一生中都要经历两个阶段，即幼年和成年两个阶段。而且种群在不同的生理阶段其生理特征有着明显的差异，如幼年种群没有生育能力，捕食能力较弱，而成年种群不仅有生育能力，而且生存能力较强。随着对传染病研究的深入，数学模型日益成为分析和控制传染病的重要工具。传染病模型日益受到科学家们的重视，近年来许多学者研究了传染病模型，但这些模型总是假定各个年龄阶段的种群个体对某种传染病具有相同的传染率。然而对于某些传染病事实并非如此，如麻疹、天花等传染病多发于幼年阶段，而伤寒、副伤寒等传染病多在成年人之间流行。因此，考虑具有不同传染率的阶段结构的传染病模型更具有实际意义。本文最后研究了一类成年种群具有疾病而幼年种群不具有疾病的SI传染病模型。在模型中假定种群具有两个阶段，即幼年阶段和成年阶段，且幼年种群转化为成年种群的数量与幼年种群的数量成正比其比率为常数，幼年种群的死亡率为常数且成年种群具有密度制约，幼年种群不感染疾病仅
成年种群感染疾病，疾病具有一个潜伏期丁。本文利用代数理论及特征方程讨论
了模型非负平衡态的局部稳定性，得到了系统正平衡态绝对稳定的充分条件。当
把时滞作为分支参数时，给出了系统出现分支的条件及分支值，利用Liapunov函
数及比较原理得出了疾病消除的阂值。
In this paper, we consider the asymptotic behavior of three ecological systems by establishing region of permanence, using Liapunov function, algebraic theory and characteristic equation.
In the study of ecology, discrete model is more significant in practice than differential model as for these species which are short in life and non-overlapping in generations or long in life and overlapping in generations but fewer in quantity. In this paper, first, we consider a discrete nonautonmous food-chain system with delays. The sufficent condition and necessary condition are obtained by means of constructing the region of permanence and necessary and sufficient condition for permanence is obtained when the system is autonomous. As a result the mathematic methods in this paper can be used to research food-chain systems with multi-species.
One of the most important questions in population ecology is to find the permanence conditions for the species. Disperse is used making the extinct species migrate so that they are saved for protecting species. In recent years, many ecological systems have been investigated by ecological scientists. But we find few studies on the systems with delayed effect in disperse. In this paper, second, we consider a nonautonomous predator-prey system with dispersal delays in two habitats. The persistence of system and the sufficient condition of persistence is obtained by means of using the differential inequality and when the coefficents of system are positive periodic functions with common periodic u> and delays are integro-multiple of w, the w-periodic positive solution of solutions are exist by means of using of Brouwer's fixed point theorem and sufficient condition is obtained for the global attracticity of w-periodic positive solution of system by means of constructing suitable Lyapunov function.
Individual's growth of many species have two stage that are juvenile stage and adult stage. In each stage of its development, it always shows different characteristic. For instance, the immature species cannot have reproductive ability and predative ability while the mature species not only have reproductive ability but also have more powerful survival capacity. Therefore studing of stage-structured systems have much practical significance. Mathematic models have been used to analysis and control epidemic models with the
further investigating epidemic. Epidemic models have been studied by many scientist. But they always propose that the individuals have same infection conversing in different stage. However this is not the thing for some disease transmission. For instance, some epidemic (measles, smallpox) always transmite in juvenile stage while some epidemic (typhoid fever, paratyphoid fever)are transmite in adult stage. Therefore, it has much practical significance towards investgating epidemic models with stage-structured. In this paper, final, we investigate a SI epidemic model with stage-structured. We propose that individual of species has two stage that are immature and mature stage and the mature rate of immature species is proportional to existing immature specie density in proportion to the constant d and the death rate of immature species is proportional to existing immature specie density, the number of mature specie depents on the density of exiting mature specie and we assume that immature species dosn't infect
disease but only mature species infect disease and disease has the latent period r. In this paper we investigated the local stability of non-negative equilibria and the sufficient condition of locally asymptotically stable is obtaied by means of using algebra theory and characteristic equations. We show that positive equilibrium will loss of stabillity with the delay increased and a Hopf biffur-cation will occur and the global attractivity of disease-free equilibrum is surveyed and the threshold of disease disppearing is obtained by means of using Liapunov function.
引文
[1] H.M.Anderson, V.Hutson, R.Law, On the conditions for permanence of species in ecological communities, Am.Nat,1992,139,663-668.
[2] R.Kon, Y.Takeuchi, Permanence ofhost-parasitoid systems,Nonl.Anal,2001,47,1383-1393.
[3] V.Hutson, R.Law, Permanence coexistence in obeying diffence equations, J. Math.Biol,1982,15,203-213.
[4] J.Hofbauer, K.Sigmund, Evolutionary games and population dynamics,Cambridge University Press,Cambridge, 1998.
[5] Z.Lu, W.Wang, Permanence and global attractivity for Lotka-Volterra difference systems, J.Math.Biol,1999,39,269-282.
[6] Y.Saito, W.Ma, T.Hara, A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays, J.Math.Anal.Appl, 2001, 256,162-174.
[7] L.S.Chen, Mathematical models and methods in ecology, Science Press, Beijing, 1988(Chinese).
[8] Y.Kang, " Delay differential equations with appliations in population dynamics, " Academic York, 1990.
[9] K.Gopalsamy, Stability and oscillations in linear delay differential equations of population dynamics, Kluwer Academic, Dordrecht, 1992.
[10] J.K.Hale, S.M.Verduyn Lunel, Introduction to functional differential equations, Springer-verlag, New york, 1993.
[11] Y.Cao, H.I.Freedman, Global attractivity in time-delayed predator-prey systems, J.Austrol.Math.Soc.Ser.B 1996,38,149-162.
[12] 陈兰荪，陈健．非线性生物动力系统[M]．北京：科学出版社，1993．
[13] J.K.Hale, In theory of function differential equations,Springer,New York,1997.
[14] Y.Takeuchi, Global stability in generalized Lotke-volterra diffusion systems, J.Math.Anal.Appl,1986,116,209-221.
[15] R.Mahbuba, L.Chen, On the nonautomous Lottka-Volterra conpetition system with diffucion, J.Diff.Equa.Dynsyst,1994,2,243-253.
[16] H.I.Freedman, Y.Takeuchi, Predator survial versus extinction as a function of dispersal in a predator-prey model with patchy enviroment, J.Appl.Anal,1989, 31,247-266.
[17] M.G.Neubert, P.Klepac, P.V.Driessche, Stabilizing dispersal delays in predator-prey metapopulation models, Theor.Popul.Biol,2002,61,339-347.
[18] H.Zhu, K.Duan, Global stability and periodic orbits for a two-patch diffusion predator-prey model with time delays, Nonlinear.Anal,2000,41, 1083-1096.
[19] Y.Kuang, Y.Jakeuchi, Predator-prey dynamics in models of prey dispersal in two-patch enviroments, Math Biosci,1994,120,77-98.
[20] V.Hutson, K.Schmitt, Permanence and the dynamics of biological systems, Math. Biosci, 1990,111,1-17.
[21] 蒋威，退化时滞微分系统．安徽大学出版社，1998．3．
[22] 罗定军等，动力系统的定性与分支理论．科学出版社，2001．
[23] X.Song, L.Chen, Conditions for global attractivity of n-patches predator-prey dispersion-delay models,J.Math.Anal.Appl,2001,253,1-15.
[24] J.M.Cushing, A simple model of cannibalism,Math.Biosci,1991,107-147.
[25] K.J.Magnussion,Destabilizing effect of cannibalism on a structured predatorprey system,Math Biosci,1999,155,61-75.
[26] J.M.Cushing, A size-structured model for cannibalism,Theor.Popul.Biol,1992, 42,347-361.
[27] Y.Xiao, L.Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math Biosci,2001,171,59-82
[28] W.G.Aiello, H.I.Freedman, A time delay model of single-species growth with stage structure, Math Biosci,1990,101,139-153
[29] F.Van den Bosch, W.Gabriel, Cannibalism in an age structured predator-prey system, Bull.Math.Biol,1997,59,551-567.
[30] H.I.Freedman, J.Wu, Persistence and global asympotical stability of single species dispersal models with stage structure, Quart.Appl.Math,1991,49,351-371.
[31] 秦元勋，常差分方程，科学出版社，1991．
[32] Lecture notes for summer course on epidemiological models and population dynamics in xi'an 2000，种群动力学与流行病动力学，西安交通大学，2000．7．
[33] E.Beretta, Y.Takuchi, Global stability of an SIR epidemic model with time delays, J.Math.Biol,1995,33,250-260.
[34] E.Beretta, Y.Takuchi, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2001,2,35-74.
[35] H.I.Freedman, V.Sree Hari Rao, The trade-off betwee mutual interference and time lags in predator-prey systems, Bull.Math.Biol,1983,45,991.
[36] 郑祖庥，泛函微分方程．安徽教育出版社．1992．
[37] W.Wang, Global behavior of an SEIRS epidemic model with time delays, Appl. Math.Lett,2002,15,423-428.
[38] K.L.Cooke, P.Van den Driessche, Analysis of an SEIRS epidemic model with two delays, J.Math.Biol,1996,35,240-260.
[39] 张芷芬等，微分方程的定性理论．科学出版社．1992．
[40] 廖晓昕，稳定性理论、方法和应用，华中科学技术大学出版社，1991．
[41] S.Busenberg, P.Van Driessche, Analysis of a disease transmission model in a population with varing size, J.Math.Biol,1990,28,275-270
[2] R.Kon, Y.Takeuchi, Permanence ofhost-parasitoid systems,Nonl.Anal,2001,47,1383-1393.
[3] V.Hutson, R.Law, Permanence coexistence in obeying diffence equations, J. Math.Biol,1982,15,203-213.
[4] J.Hofbauer, K.Sigmund, Evolutionary games and population dynamics,Cambridge University Press,Cambridge, 1998.
[5] Z.Lu, W.Wang, Permanence and global attractivity for Lotka-Volterra difference systems, J.Math.Biol,1999,39,269-282.
[6] Y.Saito, W.Ma, T.Hara, A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays, J.Math.Anal.Appl, 2001, 256,162-174.
[7] L.S.Chen, Mathematical models and methods in ecology, Science Press, Beijing, 1988(Chinese).
[8] Y.Kang, " Delay differential equations with appliations in population dynamics, " Academic York, 1990.
[9] K.Gopalsamy, Stability and oscillations in linear delay differential equations of population dynamics, Kluwer Academic, Dordrecht, 1992.
[10] J.K.Hale, S.M.Verduyn Lunel, Introduction to functional differential equations, Springer-verlag, New york, 1993.
[11] Y.Cao, H.I.Freedman, Global attractivity in time-delayed predator-prey systems, J.Austrol.Math.Soc.Ser.B 1996,38,149-162.
[12] 陈兰荪，陈健．非线性生物动力系统[M]．北京：科学出版社，1993．
[13] J.K.Hale, In theory of function differential equations,Springer,New York,1997.
[14] Y.Takeuchi, Global stability in generalized Lotke-volterra diffusion systems, J.Math.Anal.Appl,1986,116,209-221.
[15] R.Mahbuba, L.Chen, On the nonautomous Lottka-Volterra conpetition system with diffucion, J.Diff.Equa.Dynsyst,1994,2,243-253.
[16] H.I.Freedman, Y.Takeuchi, Predator survial versus extinction as a function of dispersal in a predator-prey model with patchy enviroment, J.Appl.Anal,1989, 31,247-266.
[17] M.G.Neubert, P.Klepac, P.V.Driessche, Stabilizing dispersal delays in predator-prey metapopulation models, Theor.Popul.Biol,2002,61,339-347.
[18] H.Zhu, K.Duan, Global stability and periodic orbits for a two-patch diffusion predator-prey model with time delays, Nonlinear.Anal,2000,41, 1083-1096.
[19] Y.Kuang, Y.Jakeuchi, Predator-prey dynamics in models of prey dispersal in two-patch enviroments, Math Biosci,1994,120,77-98.
[20] V.Hutson, K.Schmitt, Permanence and the dynamics of biological systems, Math. Biosci, 1990,111,1-17.
[21] 蒋威，退化时滞微分系统．安徽大学出版社，1998．3．
[22] 罗定军等，动力系统的定性与分支理论．科学出版社，2001．
[23] X.Song, L.Chen, Conditions for global attractivity of n-patches predator-prey dispersion-delay models,J.Math.Anal.Appl,2001,253,1-15.
[24] J.M.Cushing, A simple model of cannibalism,Math.Biosci,1991,107-147.
[25] K.J.Magnussion,Destabilizing effect of cannibalism on a structured predatorprey system,Math Biosci,1999,155,61-75.
[26] J.M.Cushing, A size-structured model for cannibalism,Theor.Popul.Biol,1992, 42,347-361.
[27] Y.Xiao, L.Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math Biosci,2001,171,59-82
[28] W.G.Aiello, H.I.Freedman, A time delay model of single-species growth with stage structure, Math Biosci,1990,101,139-153
[29] F.Van den Bosch, W.Gabriel, Cannibalism in an age structured predator-prey system, Bull.Math.Biol,1997,59,551-567.
[30] H.I.Freedman, J.Wu, Persistence and global asympotical stability of single species dispersal models with stage structure, Quart.Appl.Math,1991,49,351-371.
[31] 秦元勋，常差分方程，科学出版社，1991．
[32] Lecture notes for summer course on epidemiological models and population dynamics in xi'an 2000，种群动力学与流行病动力学，西安交通大学，2000．7．
[33] E.Beretta, Y.Takuchi, Global stability of an SIR epidemic model with time delays, J.Math.Biol,1995,33,250-260.
[34] E.Beretta, Y.Takuchi, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2001,2,35-74.
[35] H.I.Freedman, V.Sree Hari Rao, The trade-off betwee mutual interference and time lags in predator-prey systems, Bull.Math.Biol,1983,45,991.
[36] 郑祖庥，泛函微分方程．安徽教育出版社．1992．
[37] W.Wang, Global behavior of an SEIRS epidemic model with time delays, Appl. Math.Lett,2002,15,423-428.
[38] K.L.Cooke, P.Van den Driessche, Analysis of an SEIRS epidemic model with two delays, J.Math.Biol,1996,35,240-260.
[39] 张芷芬等，微分方程的定性理论．科学出版社．1992．
[40] 廖晓昕，稳定性理论、方法和应用，华中科学技术大学出版社，1991．
[41] S.Busenberg, P.Van Driessche, Analysis of a disease transmission model in a population with varing size, J.Math.Biol,1990,28,275-270