| 具有功能性反应的捕食系统的周期解及动力学行为 |
| Alternative Title | Periodic Solutions and Dynamics of Predator-Prey Systems with Functional Responses
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| 樊永红 |
| Thesis Advisor | 李万同
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| 2005-05-25
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 博士
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| Keyword | 持久性
周期解
全局渐进稳定性
分歧
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| Abstract | 关于时滞周期捕食系统的持久性问题, 主要研究了具单调功能性反应的捕食系统在连续与离散不同情形下的持久性. 在连续情形下(这里的功能性反应函数只需满足一般的单调性条件即可)，通过建立与周期系统对应的比较系统，其中比较系统与以往总选择自治系统不同，这里选择另一简单周期系统. 获得了在平均条件下，周期系统自身的持久性. 对于离散周期系统(这里的功能性反应函数是Holling III型)的持久性的研究，建立了与之对应的比较系统，利用半环理论克服了离散系统复杂甚至出现混沌的情形，得到在几乎平均意义下系统是持久的. 这种研究方法也为研究有界时变系数的生态系统持久性提供了一条有意义的途径.
关于时滞周期捕食系统的周期解问题, 当周期捕食系统(不含时滞)是持久时，则可通过构造
Poincare映射，利用Bronwer不动点定理获得系统周期解的存在性.但若考察时滞周期捕食系统，此时由于系统是一个无穷维动力系统，经典的Leray-Schauder不动点定理则需要所构造的Poincare映射的紧性条件，而这一点一般很难证明.本文通过利用由Mawhin等发展的重合度理论，克服了用不动点定理需要验证映射紧性条件这一困难，获得了对于一般的具单调功能性反应的捕食系统，保证该系统持久的平均条件同样也保证了它至少有一个正周期解.另一方面，对于非单调功能性反应的捕食-食饵系统，注意到以往文献对此鲜有提及，这是因为与具有单调功能性反应的捕食-食饵系统只有一个正平衡点不同,具有非单调功能性反应的捕食-食饵系统（无论是食饵依赖型, 还是比率依赖型）可以存在两个正平衡点, 这样在计算算子的度的时候就会出现一定的困难, 本文通过选择不同的象空间，坐标平移，积分估计等手段，克服了这一困难，
分别建立了易于验证的一系列充分条件， 并通过数值模拟实现了上述结论. |
| Other Abstract | For the investigations on permanent properties for periodic delay predator-prey systems, we mainly study the permanent property for predator-prey systems with monotonic functional responses in their continuous forms or discrete forms respectively. For continuous systems (here the functional response functions only need to satisfy the general monotonic conditions), by constructing the corresponding comparison systems according to the periodic systems, we obtain the permanence for the periodic systems
themselves. The comparison systems we select are different from the autonomous systems usually used before, they are simple periodic systems. In the study of the permanence for discrete periodic systems (here the functional response functions are of Holling type III), by using the theory for semicycles and the comparison theorems being constructed, we conquer the difficulties that difference equations have more complicated dynamics or even “chaotic”, and obtain the permanence for the periodic systems in the sense of almost average. Also our method offers a road to study the permanence for ecological systems with bounded time variable coefficients.
On the problem for periodic solutions of delay periodic predator-prey systems, one can get the existence of periodic solutions by constructing Poincare map and using Bronwer fixed point theorem when the periodic predator-prey systems (without delay) are permanent. But when delays exist, the systems become infinite dimensional dynamical systems, the classical Leray-Schauder fixed point theorem need compact conditions of the Poincare map constructed, while it is difficult to prove this. Through the coincidence degree theory developed by Mawhin etc, we conquer this difficulty and obtain that: for the general predator-prey systems with monotonic functional responses, the
conditions that ensure the permanence can also guarantee the existence of at least one positive periodic solution. On the other hand, most investigations concentrate on predator-prey systems with monotonic functional responses, predator-prey systems with
nonmonotonic functional responses are seldom mentioned. The reason is: different from the fact that predator-prey systems with monotonic functional responses have a unique positive equilibrium, predator-prey systems with nonmonotonic functional responses
(either for systems with prey-dependent or ratio-dependent functional responses) may have two positive equilibria, thus certain difficulties occur... |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/224972
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
樊永红. 具有功能性反应的捕食系统的周期解及动力学行为[D]. 兰州. 兰州大学,2005.
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