扩散捕食系统正平衡态的定性分析
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摘要
具有空间结构的生态模型已成为近几十年来最为活跃的研究领域之一,引起了众多数学家和生物学家的广泛兴趣.特别的,由于自然界中能量传递形式的差异,带有不同功能反应的扩散捕食系统的长时间行为和非常数正平衡态的存在性与不存在性是种群动态模型研究的重要内容.本文的主要工作是对几类带有不同功能反应项的扩散捕食系统进行定性分析.
首先,研究了两类带有Robin边界条件的捕食模型,一类是捕食者种群带有修正的Leslie-Gower型功能反应,而食饵种群中则含有一般的功能反应函数p(u)；另一类是带有Beddinton-DeAngelis功能反应的捕食系统.运用锥映射的不动点指数理论,给出了这两类系统共存解的存在性与不存在性的一些充分必要条件.这些条件依赖于系统中的某些参数和Robin边界条件下特征值问题的主特征值.并且还讨论了共存解的稳定性和抛物系统的渐近行为.特别的,对于第一类捕食系统给出了一个具体应用,即取p(u)为Holling-II型功能反应,获得了相应的结果.
其次,研究了一类修正的Holling-Tanner浦食系统,带有齐次Neumann边界条件.讨论了抛物系统的全局吸引性、持久性；分别运用线性化方法和Lyapunov函数法,当系统中参数满足一定条件时,研究了正常数平衡态的局部与全局渐近稳定性；通过最大值原理和Harnack不等式给出了椭圆问题正解的先验上、下界估计；进而分别运用能量方法和Leray-Schauder度理论证明了系统在大扩散时非常数正平衡态的不存在性和存在性.再次,研究了一类带有扩散项的三种群捕食系统,其中两捕食者猎获同一种食饵.当系统中参数满足一定条件时,讨论了系统唯一正常数平衡态的局部与全局渐近稳定性；通过对椭圆问题正解的先验估计,运用能量方法证明了当扩散系数d1充分大时系统非常数正平衡态的不存在性.
最后,鉴于脉冲微分方程在描述某些生态系统的重要意义,研究了含有脉冲和扩散项的比例依赖Holling-III型捕食系统,给出系统的正向不变集、解的最终有界性与持久性等结果,以及捕食者种群灭绝的充分条件.并且还将这些结果推广到了多种群情形.
In the past decades, ecological models with spacial structure have been one of the most active fields and received extensive concerns from many mathematicians and biologists. In particular, due to the differences in energy transformation, the long-time behaviors and the existence/nonexistence of nonconstant positive steady states for diffusive predator-prey systems with different functional responses are im-portant themes in studying population dynamical models. This thesis is concerned with the qualitative property for some diffusive predator-prey systems with different functional responses.
Firstly, we study two classes of predator-prey models with Robin boundary conditions. One incorporates the modified Leslie-Gower functional response for the predator and the general functional response p(u) for the prey. The other one involves the Beddington-DeAngelis functional response. By applying the theory of fixed point in cones, we give some sufficient and necessary conditions for the existence and nonexistence of coexistence solutions to these systems, which are dependent on some parameters and the principle eigenvalues under Robin boundary conditions. Furthermore, we will discuss the stability of coexistence solutions and the asymptotic behavior for the parabolic systems. In particular, we give an application of the first class of predator-prey system, that is to say, we replace the functional response p(u) with Holling-II type functional response and obtain the corresponding results.
Secondly, we study a kind of modified Holling-Tanner type predator-prey sys-tem under homogeneous Neumann boundary conditions and discuss the global at-tractivity and persistence property of the parabolic system. By using the linearized method and Lyapunov function method, we investigate the locally and globally asymptotic stability of the positive constant steady state when the parameters sat-isfy suitable conditions. After this, we apply the maximal principle and Harnack inequality to give a priori upper and lower bounds estimate for the positive solu-tions to elliptic system. Then, we establish the nonexistence and existence of the nonconstant positive steady states by energy method and Leray-Schauder degree theory, respectively, in the case of large diffusion. Furthermore, we study a class of three-species predator-prey system with diffusion, in which the two predators con-sume the common prey. The locally and globally asymptotic stability of the unique positive constant steady state is discussed. Through giving a priori estimate on the positive solution to the corresponding elliptic problem, the nonexistence of noncon- stant positive steady state is established by applying the energy method when the diffusion coefficient d1 is sufficiently large.
Finally, in view of the important role of impulsive differential equations in mod-eling some ecological systems, we discuss a class of ratio-dependent Holling-III type predator-prey system with diffusion and impulses. Some sufficient conditions for the existence of positively invariant set, ultimate boundedness of solutions, persis-tence property and the extinction of predator population are given. Furthermore, we generalize these results to multi-species predator-prey systems.
首先,研究了两类带有Robin边界条件的捕食模型,一类是捕食者种群带有修正的Leslie-Gower型功能反应,而食饵种群中则含有一般的功能反应函数p(u)；另一类是带有Beddinton-DeAngelis功能反应的捕食系统.运用锥映射的不动点指数理论,给出了这两类系统共存解的存在性与不存在性的一些充分必要条件.这些条件依赖于系统中的某些参数和Robin边界条件下特征值问题的主特征值.并且还讨论了共存解的稳定性和抛物系统的渐近行为.特别的,对于第一类捕食系统给出了一个具体应用,即取p(u)为Holling-II型功能反应,获得了相应的结果.
其次,研究了一类修正的Holling-Tanner浦食系统,带有齐次Neumann边界条件.讨论了抛物系统的全局吸引性、持久性；分别运用线性化方法和Lyapunov函数法,当系统中参数满足一定条件时,研究了正常数平衡态的局部与全局渐近稳定性；通过最大值原理和Harnack不等式给出了椭圆问题正解的先验上、下界估计；进而分别运用能量方法和Leray-Schauder度理论证明了系统在大扩散时非常数正平衡态的不存在性和存在性.再次,研究了一类带有扩散项的三种群捕食系统,其中两捕食者猎获同一种食饵.当系统中参数满足一定条件时,讨论了系统唯一正常数平衡态的局部与全局渐近稳定性；通过对椭圆问题正解的先验估计,运用能量方法证明了当扩散系数d1充分大时系统非常数正平衡态的不存在性.
最后,鉴于脉冲微分方程在描述某些生态系统的重要意义,研究了含有脉冲和扩散项的比例依赖Holling-III型捕食系统,给出系统的正向不变集、解的最终有界性与持久性等结果,以及捕食者种群灭绝的充分条件.并且还将这些结果推广到了多种群情形.
In the past decades, ecological models with spacial structure have been one of the most active fields and received extensive concerns from many mathematicians and biologists. In particular, due to the differences in energy transformation, the long-time behaviors and the existence/nonexistence of nonconstant positive steady states for diffusive predator-prey systems with different functional responses are im-portant themes in studying population dynamical models. This thesis is concerned with the qualitative property for some diffusive predator-prey systems with different functional responses.
Firstly, we study two classes of predator-prey models with Robin boundary conditions. One incorporates the modified Leslie-Gower functional response for the predator and the general functional response p(u) for the prey. The other one involves the Beddington-DeAngelis functional response. By applying the theory of fixed point in cones, we give some sufficient and necessary conditions for the existence and nonexistence of coexistence solutions to these systems, which are dependent on some parameters and the principle eigenvalues under Robin boundary conditions. Furthermore, we will discuss the stability of coexistence solutions and the asymptotic behavior for the parabolic systems. In particular, we give an application of the first class of predator-prey system, that is to say, we replace the functional response p(u) with Holling-II type functional response and obtain the corresponding results.
Secondly, we study a kind of modified Holling-Tanner type predator-prey sys-tem under homogeneous Neumann boundary conditions and discuss the global at-tractivity and persistence property of the parabolic system. By using the linearized method and Lyapunov function method, we investigate the locally and globally asymptotic stability of the positive constant steady state when the parameters sat-isfy suitable conditions. After this, we apply the maximal principle and Harnack inequality to give a priori upper and lower bounds estimate for the positive solu-tions to elliptic system. Then, we establish the nonexistence and existence of the nonconstant positive steady states by energy method and Leray-Schauder degree theory, respectively, in the case of large diffusion. Furthermore, we study a class of three-species predator-prey system with diffusion, in which the two predators con-sume the common prey. The locally and globally asymptotic stability of the unique positive constant steady state is discussed. Through giving a priori estimate on the positive solution to the corresponding elliptic problem, the nonexistence of noncon- stant positive steady state is established by applying the energy method when the diffusion coefficient d1 is sufficiently large.
Finally, in view of the important role of impulsive differential equations in mod-eling some ecological systems, we discuss a class of ratio-dependent Holling-III type predator-prey system with diffusion and impulses. Some sufficient conditions for the existence of positively invariant set, ultimate boundedness of solutions, persis-tence property and the extinction of predator population are given. Furthermore, we generalize these results to multi-species predator-prey systems.
引文
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