几类捕食系统的分支问题与周期解
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摘要
本文考虑了几类捕食系统.运用微分方程的定性理论,分支理论和泛函微分方程理论研究其动力学行为,并探讨了系统中参数(如时滞,扩散及非单调功能反应函数等)对其动力学行为的影响,为解释,预测和控制生态学中的一些现象提供相应的理论依据.具体而言,本文做了以下工作.
首先,研究了具有多个离散时滞的二维Lotka-Volterra捕食系统,给出了系统出现Hopf分支的条件;运用规范形理论和中心流形定理,讨论了分支周期解的性质.结果表明,在适当的假设下,即使捕食系统的时滞选取不同,但系统产生Hopf分支的临界时滞参数是相同的.进一步,研究了系统分支周期解的大范围存在性.对于一般的Holling型捕食系统,从理论上证明了只有具有非单调功能反应函数的Holling型捕食系统才会出现退化的Bogdanov-Takens奇点.基于此,我们考虑了同时含有时滞和非单调功能反应函数的捕食系统,研究了系统的Hopf分支和Bogdanov-Takens分支,并给出Hopf分支的方向和分支周期解的稳定性,同时计算了Bogdanov-Takens分支的普适开折.结果表明,系统在选取不同的参数值时,分别会出现极限环和同宿轨.
其次,考虑具有扩散影响的Leslie型时滞捕食系统.通过分析正常数平衡态处的线性化系统和相应的特征方程,研究了正常数平衡态的渐近稳定性和系统存在Hopf分支的条件.运用偏泛函微分方程的规范形理论和已知的相应结果,讨论空间齐次Hopf分支的性质.特别地,我们研究了扩散对Hopf分支的影响,发现大扩散不影响系统对应的泛函微分方程的Hopf分支,而小扩散可使系统在正平衡点附近分支出空间非齐次的周期解,同时获得了决定空间非奇次Hopf分支的方向以及分支周期解的稳定性的公式.
最后,考虑具有非单调功能反应函数的Leslie-Gower型捕食系统.虽然这类系统中不含时滞,但是由于系统的正平衡点无法显式表出,所以研究系统的动力学行为是比较困难的.我们运用微分方程定性理论讨论了系统的Bogdanov-Takens分支.数值模拟表明,非单调功能反应函数导致系统出现复杂的动力学行为,如随着参数的变化,系统会出现两个极限环共存,或者极限环和同宿轨共存的现象.
This thesis is concerned with several predator-prey systems.By using the qualitative theory,bifurcation theory for differential equations and theory for functional differential equations,we study the effects of parameters on these systems(for example, time delay,diffusion,non-monotonic functional response and so on),which provide corresponding theory basis for explaining,predicting and controlling some phenomena arising in ecology.Concretely speaking,this paper have done the following works.
Firstly,we study two dimension Lotka-Volterra predator-prey system with multiple discrete time delays and some conditions for the appearance of Hopf bifurcation are given.By applying the normal form theory and center manifold theorem,we discuss the property of bifurcation periodic solutions.Under suitable assumption,even if the time delays in predator-prey are different,the system has the same critical delay parameter at which the Hopf bifurcation appears.Furthermore,we give the global existence of bifurcation periodic solutions for the system.As to the general Holling type predator-prey system,we prove theoretically that Bogdanov-Takens singularity appears only in the predator-prey system with non-monotonic functional response.In view of this,we consider the Hopf bifurcation and Bogdanov-Takens bifurcation for the predator-prey system with delay and non-monotonic functional response,and investigate the direction of Hopf bifurcation and stability of bifurcation periodic solutions.What's more,we calculate the universal fold of Bogdanov-Takens bifurcation.Our results shows that there are different parameter values for which system has a limit cycle and homoclinic loop.
Secondly,we consider Leslie type predator-prey system with delay and diffusion. By analyzing the linearized system at the positive constant steady state and the corresponding characteristic equation,we study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. By applying the normal form theory of partial functional differential equations and some known results,the property of spatially homogeneous Hopf bifurcation is discussed.Particularly,we investigate the effect of diffusion on Hopf bifurcation. Our results show that large diffusivity has no effect on the Hopf bifurcation of the corresponding functional differential equations,while small diffusivity can lead to the fact that the system bifurcates a spatially inhomogeneous periodic solutions at the positive equilibrium.Meantime,we obtain a formula which can determine the direction and stability of spatially inhomogeneous Hopf bifurcation.
Finally,we consider the Leslie-Gower predator-prey system with non-monotonic functional response.Even if this system has no delays,it is difficult to discuss the dynamical behaviors of this system since the positive equilibrium can not be expressed explicitly.By applying the qualitative theory for differential equations,we discuss the Bogdanov-Takens bifurcation of this system.Numerical simulations show that non-monotonic functional response can lead to complex dynamical behavior for this system.For example,with the change of parameters,there are some new phenomena in this system,such as the coexistence of two limit cycles,or coexistence of limit cycle and homoclinic loop.
首先,研究了具有多个离散时滞的二维Lotka-Volterra捕食系统,给出了系统出现Hopf分支的条件;运用规范形理论和中心流形定理,讨论了分支周期解的性质.结果表明,在适当的假设下,即使捕食系统的时滞选取不同,但系统产生Hopf分支的临界时滞参数是相同的.进一步,研究了系统分支周期解的大范围存在性.对于一般的Holling型捕食系统,从理论上证明了只有具有非单调功能反应函数的Holling型捕食系统才会出现退化的Bogdanov-Takens奇点.基于此,我们考虑了同时含有时滞和非单调功能反应函数的捕食系统,研究了系统的Hopf分支和Bogdanov-Takens分支,并给出Hopf分支的方向和分支周期解的稳定性,同时计算了Bogdanov-Takens分支的普适开折.结果表明,系统在选取不同的参数值时,分别会出现极限环和同宿轨.
其次,考虑具有扩散影响的Leslie型时滞捕食系统.通过分析正常数平衡态处的线性化系统和相应的特征方程,研究了正常数平衡态的渐近稳定性和系统存在Hopf分支的条件.运用偏泛函微分方程的规范形理论和已知的相应结果,讨论空间齐次Hopf分支的性质.特别地,我们研究了扩散对Hopf分支的影响,发现大扩散不影响系统对应的泛函微分方程的Hopf分支,而小扩散可使系统在正平衡点附近分支出空间非齐次的周期解,同时获得了决定空间非奇次Hopf分支的方向以及分支周期解的稳定性的公式.
最后,考虑具有非单调功能反应函数的Leslie-Gower型捕食系统.虽然这类系统中不含时滞,但是由于系统的正平衡点无法显式表出,所以研究系统的动力学行为是比较困难的.我们运用微分方程定性理论讨论了系统的Bogdanov-Takens分支.数值模拟表明,非单调功能反应函数导致系统出现复杂的动力学行为,如随着参数的变化,系统会出现两个极限环共存,或者极限环和同宿轨共存的现象.
This thesis is concerned with several predator-prey systems.By using the qualitative theory,bifurcation theory for differential equations and theory for functional differential equations,we study the effects of parameters on these systems(for example, time delay,diffusion,non-monotonic functional response and so on),which provide corresponding theory basis for explaining,predicting and controlling some phenomena arising in ecology.Concretely speaking,this paper have done the following works.
Firstly,we study two dimension Lotka-Volterra predator-prey system with multiple discrete time delays and some conditions for the appearance of Hopf bifurcation are given.By applying the normal form theory and center manifold theorem,we discuss the property of bifurcation periodic solutions.Under suitable assumption,even if the time delays in predator-prey are different,the system has the same critical delay parameter at which the Hopf bifurcation appears.Furthermore,we give the global existence of bifurcation periodic solutions for the system.As to the general Holling type predator-prey system,we prove theoretically that Bogdanov-Takens singularity appears only in the predator-prey system with non-monotonic functional response.In view of this,we consider the Hopf bifurcation and Bogdanov-Takens bifurcation for the predator-prey system with delay and non-monotonic functional response,and investigate the direction of Hopf bifurcation and stability of bifurcation periodic solutions.What's more,we calculate the universal fold of Bogdanov-Takens bifurcation.Our results shows that there are different parameter values for which system has a limit cycle and homoclinic loop.
Secondly,we consider Leslie type predator-prey system with delay and diffusion. By analyzing the linearized system at the positive constant steady state and the corresponding characteristic equation,we study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. By applying the normal form theory of partial functional differential equations and some known results,the property of spatially homogeneous Hopf bifurcation is discussed.Particularly,we investigate the effect of diffusion on Hopf bifurcation. Our results show that large diffusivity has no effect on the Hopf bifurcation of the corresponding functional differential equations,while small diffusivity can lead to the fact that the system bifurcates a spatially inhomogeneous periodic solutions at the positive equilibrium.Meantime,we obtain a formula which can determine the direction and stability of spatially inhomogeneous Hopf bifurcation.
Finally,we consider the Leslie-Gower predator-prey system with non-monotonic functional response.Even if this system has no delays,it is difficult to discuss the dynamical behaviors of this system since the positive equilibrium can not be expressed explicitly.By applying the qualitative theory for differential equations,we discuss the Bogdanov-Takens bifurcation of this system.Numerical simulations show that non-monotonic functional response can lead to complex dynamical behavior for this system.For example,with the change of parameters,there are some new phenomena in this system,such as the coexistence of two limit cycles,or coexistence of limit cycle and homoclinic loop.
引文
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