一类具有阶段结构的捕食模型性态分析
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摘要
本文主要考虑了捕食者具有一般性功能反应的阶段结构捕食模型以及阶段结构捕食模型中食饵采取鹰鸽对策的问题.全文共分三章,前两章均假设供予捕食者和食饵的资源是充裕的。第一章研究了具有一般性功能反应函数的阶段结构捕食模型,我们假设食饵种群中幼年个体和成年个体由一个确定年龄来划分开,并且捕食者仅捕食幼年食饵。我们以幼年食饵个体之间的密度制约率作为分支变量得到在一定条件下将出现超临界Hopf分支,并得到了系统一致持续生存和边界平衡点全局渐近稳定的充分条件。所得结论具有一般性,适用于具有Holling type-Ⅱ、Ⅲ,Rosenzweig和lvlev功能反应函数的捕食模型,并通过数值模拟验证了这些结论。
第二章研究了比率依赖的食饵带阶段结构捕食模型。在假设捕食者仅捕食幼年食饵基础上,讨论了平衡点的存在性。由于系统在原点处不能线性化,局部稳定性分析运用了线性化和系统变换的方法,结果发现原点总保持不稳定性,说明系统是持续生存的。此外,我们还证明了当正平衡点存在时系统是一致持续生存的,并利用极限系统理论和杜拉克判剧研究了边界平衡点全局稳定性,这表明此时捕食者种群将绝灭。所得结论适用于具有Holling type-Ⅱ、Ⅲ和Rosenzweig功能反应函数的捕食模型,最后通过数值模拟来验证所得结论。
第三章考虑了食饵具有阶段结构和鹰鸽对策的捕食模型。考虑到实际情况,假设成年食饵赖以生存和繁殖的资源有限,因此成年食饵为了生存和繁殖要发生争夺资源的竞争。在模型中我们考虑成年食饵采取鹰鸽对策,并且在竞争中采取鹰鸽对策获得的收益直接影响到成年食饵的生育率。对此模型,我们分收益大于和小于损伤代价(G＞C和G＜C)两种情况即在不同的存在区域内研究平衡点的存在性和稳定性态以及系统在两个区域内的一致持续生存性。最后对比两个模型的结论并进行生物解释,讨论了采取鹰鸽对策对传统捕食模型动力学行为的影响。
In this paper, we consider stage-structured predator-prey models with general functional response of predator and with prey using hawk and dove tactics. The paper consists of three parts, in the fist two of which the resources that are suppled for predator and prey are assumed to be aboundant. In chapter 1, a stage-structured predator-prey model with general functional response is studied. We consider immature and mature individuals of the prey population are divided by a fixed age and the predator individuals only prey on the immature prey individuals. We show that a supercritical Hopf bifurcation will occur under some conditions by using the density dependent rate in immature prey population as bifurcation parameter. Sufficient conditions which guarantee the uniform persistence and global stability of boundary equilibrium are also obtained. These conclusions are in general settings and are suitable for predator-prey models with Holling type-II, Holling type-III, Rosenzweig and lvlev functional response. Futhermore, numerical simulations are presented to illustrate these results.
In chapter 2, a ratio-dependent predator-prey model with stage structure for the prey is studied. We also assume that predator individuals only prey on the immature prey individuals. Existence of equilibria is discussed and their local asymptotical stability is analyzed by linearization and system transforming since the system can't be linearized at the origin. We find the trivial equilibrium is always unstable, which implies that the system is persistent. It is shown that the system is uniformly persistent when positive equilibrium exists. And global stability of boundary equilibrium, which implies the extinction of the predator population, is studied by the limit system theory and Dulac criterion. These conclusions are suitable for predator-prey models with Holling type-II, Holling type-III and Rosenzweig functional response. Lastly, numerical simulations are presented to illustrate these results.
In chapter 3, we consider a predator-prey model with stage structure and hawk-dove tactics for the prey. It is assumed that the resources for mature preys are limited and vital for their survival and reproduction. Then they usually compete fiercely for the resources in order to survive and reproduce. We choose the classical hawk and dove game for mature preys in the model and the gains that are obtained in the competition will influence the fecundity of mature prey individuals directly. Existence and stability of equilibria and uniform persistence of the system are studied in two different existence regions for G < C and G > C. Finally we compare and explain the results of two models from biological view and discuss the effects of different behaviors for mature prey individuals on traditional stage-structured predator-prey model dynamics.
第二章研究了比率依赖的食饵带阶段结构捕食模型。在假设捕食者仅捕食幼年食饵基础上,讨论了平衡点的存在性。由于系统在原点处不能线性化,局部稳定性分析运用了线性化和系统变换的方法,结果发现原点总保持不稳定性,说明系统是持续生存的。此外,我们还证明了当正平衡点存在时系统是一致持续生存的,并利用极限系统理论和杜拉克判剧研究了边界平衡点全局稳定性,这表明此时捕食者种群将绝灭。所得结论适用于具有Holling type-Ⅱ、Ⅲ和Rosenzweig功能反应函数的捕食模型,最后通过数值模拟来验证所得结论。
第三章考虑了食饵具有阶段结构和鹰鸽对策的捕食模型。考虑到实际情况,假设成年食饵赖以生存和繁殖的资源有限,因此成年食饵为了生存和繁殖要发生争夺资源的竞争。在模型中我们考虑成年食饵采取鹰鸽对策,并且在竞争中采取鹰鸽对策获得的收益直接影响到成年食饵的生育率。对此模型,我们分收益大于和小于损伤代价(G＞C和G＜C)两种情况即在不同的存在区域内研究平衡点的存在性和稳定性态以及系统在两个区域内的一致持续生存性。最后对比两个模型的结论并进行生物解释,讨论了采取鹰鸽对策对传统捕食模型动力学行为的影响。
In this paper, we consider stage-structured predator-prey models with general functional response of predator and with prey using hawk and dove tactics. The paper consists of three parts, in the fist two of which the resources that are suppled for predator and prey are assumed to be aboundant. In chapter 1, a stage-structured predator-prey model with general functional response is studied. We consider immature and mature individuals of the prey population are divided by a fixed age and the predator individuals only prey on the immature prey individuals. We show that a supercritical Hopf bifurcation will occur under some conditions by using the density dependent rate in immature prey population as bifurcation parameter. Sufficient conditions which guarantee the uniform persistence and global stability of boundary equilibrium are also obtained. These conclusions are in general settings and are suitable for predator-prey models with Holling type-II, Holling type-III, Rosenzweig and lvlev functional response. Futhermore, numerical simulations are presented to illustrate these results.
In chapter 2, a ratio-dependent predator-prey model with stage structure for the prey is studied. We also assume that predator individuals only prey on the immature prey individuals. Existence of equilibria is discussed and their local asymptotical stability is analyzed by linearization and system transforming since the system can't be linearized at the origin. We find the trivial equilibrium is always unstable, which implies that the system is persistent. It is shown that the system is uniformly persistent when positive equilibrium exists. And global stability of boundary equilibrium, which implies the extinction of the predator population, is studied by the limit system theory and Dulac criterion. These conclusions are suitable for predator-prey models with Holling type-II, Holling type-III and Rosenzweig functional response. Lastly, numerical simulations are presented to illustrate these results.
In chapter 3, we consider a predator-prey model with stage structure and hawk-dove tactics for the prey. It is assumed that the resources for mature preys are limited and vital for their survival and reproduction. Then they usually compete fiercely for the resources in order to survive and reproduce. We choose the classical hawk and dove game for mature preys in the model and the gains that are obtained in the competition will influence the fecundity of mature prey individuals directly. Existence and stability of equilibria and uniform persistence of the system are studied in two different existence regions for G < C and G > C. Finally we compare and explain the results of two models from biological view and discuss the effects of different behaviors for mature prey individuals on traditional stage-structured predator-prey model dynamics.
引文
[1]Britton Nicholas Ferris.Essential mathematical biology[M],Springer,2003.
[2]陈兰荪.数学生态学模型与研究方法[M].北京:科学出版社,1988.
[3]陈兰荪,孙键,非线性生物动力系统[M].北京:科学出版社,1993.
[4]马知恩.种群生态学的数学建模与研究.[M].合肥:安徽教育出版社,1996.
[5]Hofbauer J,Sigmund K.Evolutionary Games and Population Dynamics[M].Cambridge University Press,1998,Cambridge.(进化对策与种群动力学,四川科学技术出版社,陆征一,罗勇译,2002.)
[6]Arditi R,Ginzburg L R.Coupling in predator-prey dynamics:ratio-dependent[J].J Theor Biol,1989,139:311-326.
[7]Beretta E,Kuang Y.Global analysis in some delayed ratio-dependent predator-prey systems [J].Nonlinear Anal,1998,32:381-408.
[8]Berezovskaya F,Karev G,Arditi R.Parametric analysis of the ratio-dependent predatorprey model[J].J Math Biol,2001,43:221-246.
[9]Xiao Dongmei,Li Wenxia,Han Maoan.Dynamics in a ratio-dependent predator-prey model with predator harvesting[J].J Math Anal Appl,2006,324:14-29.
[10]Xu Rui,Ma Zhien.Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure[J].Chaos,Solitons and Fractals,2007.
[11]Saha Tapan,Bandyopadhyay Malay.Dynamical analysis of a delayed ratio-dependent preyprdator model with fluctuating environment[J].Appl Math and Comput,2008,196:458-478.
[12]陆征一,周义仓,数学生物学进展[M].北京:科学出版社,2006.
[13]Aiello W G,Freedman H I.A time delay model of single species growth with stage structure [J].Math Biosci,1990,101:139-156.
[14]Chen L,Song X,Lu Z.Mathematical models and methods in ecology[M].Sichuan Science and Technology Press,2003.
[15]Wang W,Chen L.A predator-prey system with stage- structure for predator[J].Comput Math Appl,1997,33:83-91.
[16]Magnusson K G.Destabilizing effect of cannibalism on a structured predator-prey system[J].Math Biosci,1999,155:61-75.
[17]Zhang X,Chen L,Neuman A U.The stage-structure predator-prey model and optimal harvesting policy[J].Math Biosci,2000,101:139-153.
[18]Wang W,Mulone G,Salemi F,Salone V.Permanence and stability of a stage-structured predator-prey model[J].J Math Anal Appl,2001,262:499-528.
[19]Song X Y,Cui J.The stage-structured predator-prey system with delay and harvesting[J].Appl Anal,2002,81:1127-1142.
[20]Xiao Y,Chen L.Global stability of a predator-prey system with stage structure for the predator[J].Acta Math Sinica,English Series,2003,19:1-11.
[21]Cui J A,Song X Y.Permanence of a predator-prey system with stage structure[J].Discrete Continuous Dynam Syst,Series B4(3),2004,547-554.
[22]Abrans Peter A,Quince Christopher.The impact of mortality on predator population size and stability in systems with stage structured prey[J].Theor Population Bio,2005,68:253-266.
[23]Cui Jing'an,Takeuchi Yasuhiro.A predator-prey system with a stage structure for the prey[J].Math Comput Modelling,2006,64:1126-1132.
[24]Cai Liming,Song Xinyu.Permanence and Stability of a predator-prey system with stage structure for predator[J].J Compu Appl Math,2007,201:356-366.
[25]Xu Rui,Ma Zhien.Stability and Hopf bifurcation in a predator-prey system with stage structure for the predator[J].Nonlinear Anal,2008.
[26]Auger P,Pointer D.Fast Game Dynamics Coupled to Slow Population Dynamics:A single Population with Hawk-Dove Strategies[J].Math Comput Modelling,1998,27:81-88.
[27]Auger Pierre,Pointer Dominique.Fast game theory coupled to slow population dynamics:the case of domestic cat populations[J].Math Biosci,1998,148:65-82.
[28]Pontier Dominique,Auger Pierre.The impact of behavioral plasticity at individual level on domestic cat population dynamics[J].Ecol Model,2000,133:117-124.
[29]Auger Pierre,Parra Rafael Bravo de la,Morand Serge,Sanchez Eva.A predator-prey model with predators using hawk and dove tactics[J].Math Biosci,2002,177-178:185-200.
[30]Auger Pierre,Kooi Bob W,Parra Rafael Bravo dela,Poggiale Jean-Christophe.Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics[J].J Theor Biol,2006,238:597-607.
[31]Lett Christophe,Auger Pierre,Gaillard Jean-Michel.Continuous cycling of grouped vs.solitary strategy frequencies in a predator-prey model[J].Theoretical Population Biology,2004,65:263-270.
[32]Mchich Rachid,Auger Pierre,Lett Christophe.Effects of aggregative and solitary individual behaviors on the dynamics of predator-prey game models[J].Ecologycal Modelling,2006,197:281-289.
[33]Zhang X,Chen L,Neuman A U.The stage-structure predator-prey model and optimal harvesting policy[J].Math Biosci,2000,101:139-153.
[34]Xu Rui,Chaplain M A J,Davidson F A.Persistence and global stability of a ratio-dependent predator-prey model with stage structure[J].Appl Math Comput,2004,158:729-744.
[35]Wang S.Research on the suitable living environment of the Rana temporaria chensinensis larva[J].Chinese J Zool,t997,32:38-41.
[36]王高雄,周之铭,朱思铭,王寿松,常微分方程(第三版)[M].北京:高等教育出版社,2006.
[37]Liu Wei-min.Criterion of Hopf bifurcations without using eigenvalues[J].J Math Anal Appl,1994,182:250-256.
[38]Hale J.K,Waltman P.Persistence in infinite-dimensional systems[J].SIAM J Nath Anal,1989,20:388-395.
[39]廖晓昕,稳定性的理论、方法和应用[M].武汉:华中理工大学出版社,1998.
[40]Castillo-Chavez C,Thieme H R.Asymptotically autonomous epidemic models,in:O.Arino,et al.(Eds.),Mathematical Population Dynamics:Analysis of Heterogeneity[J].Ⅰ Theory of Epidemics,Wuerz,Canada,1995,pp:33-50.
[41]Jost Christian.About Deterministic Extinction in Ratio-dependent Predator-Prey Models [J].Bull Math Biol,1999,61:19-32.
[42]Chambon-Dubreuil Estelle,Auger Pierre.Effect of aggressive behaviour on age-structured population dynamics[J].Ecol Model,2006,193:777-786.
[43]Iwasa Y,Andreasen V,Levin S A.Aggregation in model ecosystems Ⅰ.Perfect aggregation[J].Ecol Model,1987,37:287-302.
[44]Iwasa Y,Levin S A,Andreasen V.Aggregation in model ecosystems Ⅱ.Approximate aggregation [J].IMA J Math Appl Med Biol,1989,6:1-23.
[45]Auger P,Roussarie R,Complex ecological models with simple dynamics:From individuals to populations[J].Acta Biotheoretica,1994,42:111-136.
[46]Bravo de la Parra Rafael,Auger P,Sanchez E,Aggregation methods in time discrete models [J].J Biological Systems,1995,3:603-612.
[47]Auger P,Poggiale J C,Emerging properties in population dynamics with different time scales[J].J Biological Systems,1995,3:591-602.
[48]Auger Pierre,Bravo de la Parra Rafael,R.Methods of aggregation of variables in population dynamics[J].Comptes rendus de l'Academie des science series Ⅲ-Sciences de la vie-life,2000,323:665-674.
[49]张锦炎,冯贝叶,常微分方程几何理论与分支问题(第三版)[M].北京:北京大学出版社,2000.
[2]陈兰荪.数学生态学模型与研究方法[M].北京:科学出版社,1988.
[3]陈兰荪,孙键,非线性生物动力系统[M].北京:科学出版社,1993.
[4]马知恩.种群生态学的数学建模与研究.[M].合肥:安徽教育出版社,1996.
[5]Hofbauer J,Sigmund K.Evolutionary Games and Population Dynamics[M].Cambridge University Press,1998,Cambridge.(进化对策与种群动力学,四川科学技术出版社,陆征一,罗勇译,2002.)
[6]Arditi R,Ginzburg L R.Coupling in predator-prey dynamics:ratio-dependent[J].J Theor Biol,1989,139:311-326.
[7]Beretta E,Kuang Y.Global analysis in some delayed ratio-dependent predator-prey systems [J].Nonlinear Anal,1998,32:381-408.
[8]Berezovskaya F,Karev G,Arditi R.Parametric analysis of the ratio-dependent predatorprey model[J].J Math Biol,2001,43:221-246.
[9]Xiao Dongmei,Li Wenxia,Han Maoan.Dynamics in a ratio-dependent predator-prey model with predator harvesting[J].J Math Anal Appl,2006,324:14-29.
[10]Xu Rui,Ma Zhien.Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure[J].Chaos,Solitons and Fractals,2007.
[11]Saha Tapan,Bandyopadhyay Malay.Dynamical analysis of a delayed ratio-dependent preyprdator model with fluctuating environment[J].Appl Math and Comput,2008,196:458-478.
[12]陆征一,周义仓,数学生物学进展[M].北京:科学出版社,2006.
[13]Aiello W G,Freedman H I.A time delay model of single species growth with stage structure [J].Math Biosci,1990,101:139-156.
[14]Chen L,Song X,Lu Z.Mathematical models and methods in ecology[M].Sichuan Science and Technology Press,2003.
[15]Wang W,Chen L.A predator-prey system with stage- structure for predator[J].Comput Math Appl,1997,33:83-91.
[16]Magnusson K G.Destabilizing effect of cannibalism on a structured predator-prey system[J].Math Biosci,1999,155:61-75.
[17]Zhang X,Chen L,Neuman A U.The stage-structure predator-prey model and optimal harvesting policy[J].Math Biosci,2000,101:139-153.
[18]Wang W,Mulone G,Salemi F,Salone V.Permanence and stability of a stage-structured predator-prey model[J].J Math Anal Appl,2001,262:499-528.
[19]Song X Y,Cui J.The stage-structured predator-prey system with delay and harvesting[J].Appl Anal,2002,81:1127-1142.
[20]Xiao Y,Chen L.Global stability of a predator-prey system with stage structure for the predator[J].Acta Math Sinica,English Series,2003,19:1-11.
[21]Cui J A,Song X Y.Permanence of a predator-prey system with stage structure[J].Discrete Continuous Dynam Syst,Series B4(3),2004,547-554.
[22]Abrans Peter A,Quince Christopher.The impact of mortality on predator population size and stability in systems with stage structured prey[J].Theor Population Bio,2005,68:253-266.
[23]Cui Jing'an,Takeuchi Yasuhiro.A predator-prey system with a stage structure for the prey[J].Math Comput Modelling,2006,64:1126-1132.
[24]Cai Liming,Song Xinyu.Permanence and Stability of a predator-prey system with stage structure for predator[J].J Compu Appl Math,2007,201:356-366.
[25]Xu Rui,Ma Zhien.Stability and Hopf bifurcation in a predator-prey system with stage structure for the predator[J].Nonlinear Anal,2008.
[26]Auger P,Pointer D.Fast Game Dynamics Coupled to Slow Population Dynamics:A single Population with Hawk-Dove Strategies[J].Math Comput Modelling,1998,27:81-88.
[27]Auger Pierre,Pointer Dominique.Fast game theory coupled to slow population dynamics:the case of domestic cat populations[J].Math Biosci,1998,148:65-82.
[28]Pontier Dominique,Auger Pierre.The impact of behavioral plasticity at individual level on domestic cat population dynamics[J].Ecol Model,2000,133:117-124.
[29]Auger Pierre,Parra Rafael Bravo de la,Morand Serge,Sanchez Eva.A predator-prey model with predators using hawk and dove tactics[J].Math Biosci,2002,177-178:185-200.
[30]Auger Pierre,Kooi Bob W,Parra Rafael Bravo dela,Poggiale Jean-Christophe.Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics[J].J Theor Biol,2006,238:597-607.
[31]Lett Christophe,Auger Pierre,Gaillard Jean-Michel.Continuous cycling of grouped vs.solitary strategy frequencies in a predator-prey model[J].Theoretical Population Biology,2004,65:263-270.
[32]Mchich Rachid,Auger Pierre,Lett Christophe.Effects of aggregative and solitary individual behaviors on the dynamics of predator-prey game models[J].Ecologycal Modelling,2006,197:281-289.
[33]Zhang X,Chen L,Neuman A U.The stage-structure predator-prey model and optimal harvesting policy[J].Math Biosci,2000,101:139-153.
[34]Xu Rui,Chaplain M A J,Davidson F A.Persistence and global stability of a ratio-dependent predator-prey model with stage structure[J].Appl Math Comput,2004,158:729-744.
[35]Wang S.Research on the suitable living environment of the Rana temporaria chensinensis larva[J].Chinese J Zool,t997,32:38-41.
[36]王高雄,周之铭,朱思铭,王寿松,常微分方程(第三版)[M].北京:高等教育出版社,2006.
[37]Liu Wei-min.Criterion of Hopf bifurcations without using eigenvalues[J].J Math Anal Appl,1994,182:250-256.
[38]Hale J.K,Waltman P.Persistence in infinite-dimensional systems[J].SIAM J Nath Anal,1989,20:388-395.
[39]廖晓昕,稳定性的理论、方法和应用[M].武汉:华中理工大学出版社,1998.
[40]Castillo-Chavez C,Thieme H R.Asymptotically autonomous epidemic models,in:O.Arino,et al.(Eds.),Mathematical Population Dynamics:Analysis of Heterogeneity[J].Ⅰ Theory of Epidemics,Wuerz,Canada,1995,pp:33-50.
[41]Jost Christian.About Deterministic Extinction in Ratio-dependent Predator-Prey Models [J].Bull Math Biol,1999,61:19-32.
[42]Chambon-Dubreuil Estelle,Auger Pierre.Effect of aggressive behaviour on age-structured population dynamics[J].Ecol Model,2006,193:777-786.
[43]Iwasa Y,Andreasen V,Levin S A.Aggregation in model ecosystems Ⅰ.Perfect aggregation[J].Ecol Model,1987,37:287-302.
[44]Iwasa Y,Levin S A,Andreasen V.Aggregation in model ecosystems Ⅱ.Approximate aggregation [J].IMA J Math Appl Med Biol,1989,6:1-23.
[45]Auger P,Roussarie R,Complex ecological models with simple dynamics:From individuals to populations[J].Acta Biotheoretica,1994,42:111-136.
[46]Bravo de la Parra Rafael,Auger P,Sanchez E,Aggregation methods in time discrete models [J].J Biological Systems,1995,3:603-612.
[47]Auger P,Poggiale J C,Emerging properties in population dynamics with different time scales[J].J Biological Systems,1995,3:591-602.
[48]Auger Pierre,Bravo de la Parra Rafael,R.Methods of aggregation of variables in population dynamics[J].Comptes rendus de l'Academie des science series Ⅲ-Sciences de la vie-life,2000,323:665-674.
[49]张锦炎,冯贝叶,常微分方程几何理论与分支问题(第三版)[M].北京:北京大学出版社,2000.