阶段结构和脉冲效应在种群模型中的应用
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摘要
微分方程数学模型在描述种群动力学行为中起到了非常重要的作用,它从数学的角度解释各种种群动力学行为,使人们科学地认识种群动力学,从而对某些种群相互作用进行有目的地控制。特别是用脉冲微分方程来描述种群动力学模型能够更合理、更精确地反映各种变化规律,因为现实世界中的许多生命现象和人类的开发行为几乎都是脉冲的。本文针对害虫控制问题提出了单种群和两种群控制的几个问题,并且利用脉冲微分方程的相关理论和方法研究了相应的动力学模型,讨论了所提模型的各种动力学行为,包括平衡点的存在性和稳定性、周期解的存在性和吸引性、系统的持久性与灭绝等。本文的主要结果概括如下:
1.第三章讨论了两个具有阶段结构的单种群模型。第一节研究幼年染病的具有阶段结构和时滞的单种群模型。所考虑的种群具有两个年龄阶段,幼年期和成年期,并且从幼年到成年的平均成熟期为一个常数,在模型中用时滞来表示。在幼年种群中存在流行病,并且这个流行病只在幼年种群中传播。讨论了几个平衡点的存在性和稳定性,并且研究了时滞对模型动力学行为的影响。第二节研究具有阶段结构和脉冲的单种群SI流行病模型。为了控制害虫的数量,在固定时刻脉冲式投放染病的害虫,从而让流行病在害虫种群中传播以达到控制害虫数量的目的,因为染病害虫是不危害农作物的。得到了易感害虫绝灭周期解的存在性和全局渐进稳定性条件,也得到了系统持久的充分条件。为利用流行病控制害虫提供了理论依据。
2.第四章讨论了两个具有阶段结构的两种群捕食模型。第一节研究了一个食饵具有阶段结构的Lotka-Volterra捕食模型。假设食饵分为两个年龄阶段,幼年卵和成虫;捕食者种群不分年龄,并且只捕食成年食饵,因为幼年食饵被它们的卵壳所保护。定期的投放天敌达到控制害虫的目的,捕食功能函数是最基本的Lotka-Volterra型。得到了害虫绝灭周期解局部稳定和全局稳定的充分条件;也得到了系统持久的充分条件,也就是害虫没有被有效地控制的条件。得到的释放天敌的阈值为合理的利用天敌控制害虫提供了理论依据。第二节研究了食饵具有阶段结构的比率依赖捕食模型。在此模型中,仍然假设食饵分为两个年龄段,幼年卵和成虫;捕食者种群不分年龄,并且只捕食成年食饵,捕食功能函数是比率依赖的。研究结果表明:在特殊的情况下,无论投放多少天敌,都不能有效的控制害虫。这也揭示了比率依赖对模型动力学行为的影响,从而也解释了为什么有些害虫很难被控制住。
3.第五章讨论了具有两次脉冲的食饵染病的捕食模型。假设食饵(害虫)种群分为两类:易感食饵(易感害虫)和染病食饵(染病害虫)。染病害虫是不危害农作物的,所以为了控制食饵种群的数量,采取释放染病害虫和释放天敌(捕食者)相结合的方法。在某些固定时刻脉冲式的释放天敌,在另外一些固定时刻脉冲式的释放染病害虫。基于这些假设建立了具有两次脉冲的捕食模型。通过使用脉冲微分方程的Floquet定理,小扰动方法以及比较技巧,研究了易感害虫绝灭周期解的全局渐近稳定性以及系统持久生存的条件。
Mathematical models of differential equations play an important role in describing population dynamic behavior.Mathematically,these models explain all kinds of population dynamic behaviors,which allows people to understand population dynamics scientifically so that some interactions of population can be intend to control.Especially, impulsive differential equations describe population dynamic models,which is more reasonable and precise on reflecting all kinds of change orderliness,since many life phenomena and human exploitation are almost impulsive in the natural world.In this dissertation, population dynamic models for pest management are established to consider several problems in population controls by means of the theory and method of impulsive differential equations.Dynamic behaviors,including the existence and stability of equilibriums,the existence of periodic solution and its global attractivity,the permanence and extinction of system,are investigated.The main results of this dissertation may be summarized as follows:
In Chapter 3,two simple species models with stage-structure are formulated and investigated.In section 3.1,a stage-structured simple species model with time-delay and disease in the infant is studied.It is assumed that the simple species has two stages, immature and mature.And the time from immature to mature is a constant which is expressed by a time delay.There is a disease among the immature population,and the disease will only infect the immature population,while the mature population will never be infected.The existence and stability of some potential equilibriums and the effect of time-delay on the dynamics of the model are studied.In section 3.2,a stage-structured SI epidemic model is constructed and studied.In order to control the number of pest, some infected pests are impulsively released at fixed time,so as to the pest number can be suppressed below economical injury level.The sufficient condition for the existence and stability of the susceptible pest eradication periodic solution is obtained,and the sufficient condition for the permanence of the system is also obtained.The results provide some theoretical bases for pest management.
In Chapter 4,two stage-structured predator-prey models are studied.In section 4.1, a Lotka-Volterra predator-prey model with stage structure in the prey is investigated.It is assumed that the prey,population has two stages,immature egg and mature pest.The predator population only capture mature pest,since immature prey is protected by the eggshell.In the model,the traditional Lotka-Volterra predation response is considered. The natural enemy(predator) is impulsively released to control the pest.The sufficient condition for the existence and stability of the pest eradication periodic solution is gotten; and the sufficient condition for the permanence of the system is also obtained.A threshold is obtained,which provide theoretical base for pest management.In section 4.2,a ratiodependent predator-prey model with stage-structure in the prey is studied.In this model, it is also assumed that the prey population has two stages,immature egg and mature pest. The predator only capture mature pest,since the immature is protected by the eggshell. And,the ratio-dependent predation response is considered.The results show that under some special situation,the pest population will never be controlled,no matter how many natural enemy(predator) is released.This result shows the difference between ratiodependent predation and other predation response,and it also explains why some pests are hard to be eradicated.
In Chapter 5,a predator-prey model with disease in the prey and two impulses for integrated pest management is investigated.The prey(pest) population is divided into two classes:susceptible prey(susceptible pest)and infected prey(infected prey).The infected pest will not do harm to the crops.Thus,infected pest and predator are impulsively released at different fixed time to control the number of the pest.Based on the above assumptions,a predator-prey model with two impulses is constructed.By use of the Floquet theorem,small interruption and comparative skills,the sufficient condition for the global stability of the pest-eradication periodic solution is obtained,and the permanence of the system is also obtained.
1.第三章讨论了两个具有阶段结构的单种群模型。第一节研究幼年染病的具有阶段结构和时滞的单种群模型。所考虑的种群具有两个年龄阶段,幼年期和成年期,并且从幼年到成年的平均成熟期为一个常数,在模型中用时滞来表示。在幼年种群中存在流行病,并且这个流行病只在幼年种群中传播。讨论了几个平衡点的存在性和稳定性,并且研究了时滞对模型动力学行为的影响。第二节研究具有阶段结构和脉冲的单种群SI流行病模型。为了控制害虫的数量,在固定时刻脉冲式投放染病的害虫,从而让流行病在害虫种群中传播以达到控制害虫数量的目的,因为染病害虫是不危害农作物的。得到了易感害虫绝灭周期解的存在性和全局渐进稳定性条件,也得到了系统持久的充分条件。为利用流行病控制害虫提供了理论依据。
2.第四章讨论了两个具有阶段结构的两种群捕食模型。第一节研究了一个食饵具有阶段结构的Lotka-Volterra捕食模型。假设食饵分为两个年龄阶段,幼年卵和成虫;捕食者种群不分年龄,并且只捕食成年食饵,因为幼年食饵被它们的卵壳所保护。定期的投放天敌达到控制害虫的目的,捕食功能函数是最基本的Lotka-Volterra型。得到了害虫绝灭周期解局部稳定和全局稳定的充分条件;也得到了系统持久的充分条件,也就是害虫没有被有效地控制的条件。得到的释放天敌的阈值为合理的利用天敌控制害虫提供了理论依据。第二节研究了食饵具有阶段结构的比率依赖捕食模型。在此模型中,仍然假设食饵分为两个年龄段,幼年卵和成虫;捕食者种群不分年龄,并且只捕食成年食饵,捕食功能函数是比率依赖的。研究结果表明:在特殊的情况下,无论投放多少天敌,都不能有效的控制害虫。这也揭示了比率依赖对模型动力学行为的影响,从而也解释了为什么有些害虫很难被控制住。
3.第五章讨论了具有两次脉冲的食饵染病的捕食模型。假设食饵(害虫)种群分为两类:易感食饵(易感害虫)和染病食饵(染病害虫)。染病害虫是不危害农作物的,所以为了控制食饵种群的数量,采取释放染病害虫和释放天敌(捕食者)相结合的方法。在某些固定时刻脉冲式的释放天敌,在另外一些固定时刻脉冲式的释放染病害虫。基于这些假设建立了具有两次脉冲的捕食模型。通过使用脉冲微分方程的Floquet定理,小扰动方法以及比较技巧,研究了易感害虫绝灭周期解的全局渐近稳定性以及系统持久生存的条件。
Mathematical models of differential equations play an important role in describing population dynamic behavior.Mathematically,these models explain all kinds of population dynamic behaviors,which allows people to understand population dynamics scientifically so that some interactions of population can be intend to control.Especially, impulsive differential equations describe population dynamic models,which is more reasonable and precise on reflecting all kinds of change orderliness,since many life phenomena and human exploitation are almost impulsive in the natural world.In this dissertation, population dynamic models for pest management are established to consider several problems in population controls by means of the theory and method of impulsive differential equations.Dynamic behaviors,including the existence and stability of equilibriums,the existence of periodic solution and its global attractivity,the permanence and extinction of system,are investigated.The main results of this dissertation may be summarized as follows:
In Chapter 3,two simple species models with stage-structure are formulated and investigated.In section 3.1,a stage-structured simple species model with time-delay and disease in the infant is studied.It is assumed that the simple species has two stages, immature and mature.And the time from immature to mature is a constant which is expressed by a time delay.There is a disease among the immature population,and the disease will only infect the immature population,while the mature population will never be infected.The existence and stability of some potential equilibriums and the effect of time-delay on the dynamics of the model are studied.In section 3.2,a stage-structured SI epidemic model is constructed and studied.In order to control the number of pest, some infected pests are impulsively released at fixed time,so as to the pest number can be suppressed below economical injury level.The sufficient condition for the existence and stability of the susceptible pest eradication periodic solution is obtained,and the sufficient condition for the permanence of the system is also obtained.The results provide some theoretical bases for pest management.
In Chapter 4,two stage-structured predator-prey models are studied.In section 4.1, a Lotka-Volterra predator-prey model with stage structure in the prey is investigated.It is assumed that the prey,population has two stages,immature egg and mature pest.The predator population only capture mature pest,since immature prey is protected by the eggshell.In the model,the traditional Lotka-Volterra predation response is considered. The natural enemy(predator) is impulsively released to control the pest.The sufficient condition for the existence and stability of the pest eradication periodic solution is gotten; and the sufficient condition for the permanence of the system is also obtained.A threshold is obtained,which provide theoretical base for pest management.In section 4.2,a ratiodependent predator-prey model with stage-structure in the prey is studied.In this model, it is also assumed that the prey population has two stages,immature egg and mature pest. The predator only capture mature pest,since the immature is protected by the eggshell. And,the ratio-dependent predation response is considered.The results show that under some special situation,the pest population will never be controlled,no matter how many natural enemy(predator) is released.This result shows the difference between ratiodependent predation and other predation response,and it also explains why some pests are hard to be eradicated.
In Chapter 5,a predator-prey model with disease in the prey and two impulses for integrated pest management is investigated.The prey(pest) population is divided into two classes:susceptible prey(susceptible pest)and infected prey(infected prey).The infected pest will not do harm to the crops.Thus,infected pest and predator are impulsively released at different fixed time to control the number of the pest.Based on the above assumptions,a predator-prey model with two impulses is constructed.By use of the Floquet theorem,small interruption and comparative skills,the sufficient condition for the global stability of the pest-eradication periodic solution is obtained,and the permanence of the system is also obtained.
引文
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[76] Song X Y and Chen L S. Optimal harvesting policy and stability for single-species growth model with stage structure. J. Sys. Sci. Complexity. 2002. 15: 194-201.
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[81]Wang W D and Chert L S.A predator-prey system with stage-structure for predator.Computers Math.Applic.33(1997) 83-91.
[82]Liu S Q,Chert L S and Liu Z J.Extinction and permanence in nonautonomous competitive system with stage structure.Journal of Mathematical Analysis and Applications 274(2002) 667-684.
[83]Liu S Q,Chen L S and Agarwal R.Recent progess on stage-structured population dynamics.Mathematical and Computer Modelling 36(2002) 1319-1360.
[84]Arditi R and Saiah H.Empirical evidence of the role of heterogeneity in ratio-dependent consumption.Ecology,73(1992),1544-1551.
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[91]Wang Y M.Numerical solutions of a Michaelis-Menten type ratio-dependent predator-prey system with diffusion,Applied Numerical Mathematics(2008),doi:10.1016/j.apnum.2008.05.003.
[92]Krisztina Kiss and Sandor Kovacs.Qualitative behavior of n-dimensional ratio-dependent predator-prey systems.Applied Mathematics and Computation 199(2008) 535-546.
[93] Xiao Y N and Chen L S. A ratio-dependent predator-prey model with disease in the prey. Applied Mathematics and Computation 131 (2002), 397-414.
[94] Song X Y, Cai L M and Neumann Avidan U. Ratio-dependent predator-prey system with stage structure for prey. Discrete and Continuous Dynamical Systems - Series B (DCDS-B) 4(3) (2004) 747-758.
[95] R. Xu and Z. Ma. Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure. Chaos, Solitons & Fractals 38 (2008) 669-684.
[96] Song X Y and Chen L S. Optimal harvesting and stability for a predator-prey system with stage struture. Acta Math. Appl. Sinica. 18 (2002) 423-430.
[97] Xiao Y N and Chen L S. Stabilizing effect of cannibalism on a structured competitive system. Acta Mathematics (in Chinese), 22A(2) (2002) 210-216.
[98] McEwen F L and Stephenson R G. The use and significance of pesticides in the environment. New York: Wiley, 1979.
[99] Phelps W. Pesticide environmental fate: bridging the gap between laboratory and field studies. USA: American Chemical Society, 2002.
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[101] Davis P E, Myers K, Hoy J B. Biological control among vertebrates. Theory and practice of biological control. (Huffaker C B and Messenger P S, Editors). Plenum Press, New York, (1976).
[102] Falcon L A. Problem associated with the use of arthropod viruses in pest control. Annu. Rev. Entomol. 21, 305-324, (1976).
[103] Tanada Y. Epizootiology of insect disease. Bilogical control of insect pests and weeds (DeBach P, editor). Chapman & Hall, London, (1964).
[104] Van Lenteren J C. Measures of success in biological control of anthropoids by augmentation of natural enemies. Measures of Success in Biological Control (Wratten S, Gurr G, editors). Kluwer Academic Publishers, Dordrdcht. p.77-89, (2000).
[105] Van Lenteren J C, Woets J, Biological and integrated pest control in greenhouses. Ann. Rev. Ent. 239-250, (1988).
[106] Grasman J, Van Herwarrden O A. Hemerik L, et al. A two-component model of host- parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control. Math. Biosci. 196. 207-216, (2001).
[107] Debach P, Rosen D. Biological control by natural enemies (second edition). Cambridge University Press. Cambridge. (1991).
[108] Luff M L. The potential of predators for pest control. Agri. Ecos. Environ. 10, 159-181,( 1983).
[109] Cherry A J, Lomer. C J, Djegui D, Schulthess F. Pathogen incidence and their potential as microbial control agents in IPM of maize stemborers in West Africa. Biocontrol 44(a), 301-327,(1999).
[110] Gao S J, Chen L S, Sun L H. Optimal pulse fishing policy in stage-structured models with birth pulses. Chaos, Solitons & Fractals, 25, 1209-1219, (2005).
[111] Zhang Wei. Multiply methods for pest control. (in Chinese) http://biological.aweb.com.cn/news/2007/9/12/10495060.shtml, 2007-09-12.
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