具有脉冲干扰的生态数学模型周期解的存在性与全局吸引性
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摘要
本篇博士学位论文由五章组成。
第一章,简述种群生态学的研究发展概况；概述时滞微分方程、脉冲微分方程的研究发展状况；简述本文问题产生的背景和本文的主要工作以及一些预备知识。
第二章,我们讨论了具有食饵扩散和脉冲干扰的捕食者-食饵模型。第一节我们利用迭合度理论中的Mawhin连续定理得到了一类具有脉冲干扰和扩散的非自治半基于比率捕食者-食饵模型至少有一个正周期解存在的充分条件,我们的结果改进和推广了已有的结果。第二节利用迭合度理论中的Mawhin连续定理、Lyapunov泛函方法以及一些分析技巧,得到了一类具有脉冲干扰和扩散的捕食者-食饵模型正周期解的存在性、全局吸引性和持续生存的充分条件,已有文献中的结果得到了改进和推广。在应用方面,我们举了几个实例来说明我们的结果的可行性。
第三章,我们考虑了一类具有单调(或者非单调)功能反应函数和脉冲干扰的捕食者-食饵模型。利用迭合度理论中的Mawhin连续定理,我们得到了该脉冲模型存在多个正周期解的充分条件。特别地,所得的结果改进和推广了已有的结果。
第四章,我们研究了两类具有时滞和脉冲干扰的n维种群竞争模型。第一节利用已有文献给出的一个新的不动点定理,我们得到了一类具有脉冲干扰和时滞n维Lotka-Volterra竞争模型正周期解的存在性的充分条件,并将获得的主要结果应用到几个生物模型当中,由此得到了一些有应用价值的结果,所得结果改进了已有的结果。第二节我们利用巴拿赫空间中锥上一个不动点定理,获得了一类n维Gilpin-Ayala型竞争脉冲模型的正周期解存在性和唯一性的判别准则,改进和推广了已有的结果。
第五章,我们研究了具有多时滞和脉冲干扰的对数人口模型。第一节我们利用k-集压缩算子的抽象延拓定理得到了一类具有多时滞和脉冲干扰中立型单种群对数人口模型正周期解存在性,通过分析的技巧得到正周期解全局吸引的充分条件,所得结果改进了文已有的结果。在应用方面,我们举了一个实例来证明我们的结果的可行性。第二节我们利用压缩映射原理和不等式技巧得到了一类n维具有多时滞和脉冲干扰对数人口模型正周期解存在性和全局吸引性的充分条件。我们所需的条件比已有结果所需的条件弱些,能更好地运用到一些特殊情形。最后,我们也用一个实例来证明我们的结果的可行性。第三节我们利用压缩映射原理和不等式技巧得到了一类n维具有多时滞、脉冲干扰和反馈控制中立型对数人口模型正周期解存在性和全局吸引性的充分条件,所得结果改进了已有的结果。最后,我们也用一个实例来说明我们的结果的可行性。
This Ph.D.thesis is divided into five chapters and main contents are as follows:
In Chapter 1, we give a survey to the developments of population ecology, delay differential equations and impulsive differential equations. Then we intro-duce the background of problems, the main results of this dissertation and some preliminaries are also summarized.
In Chapter 2, we discuss the predator-prey model with diffusion and im-pulse. In the section 2.1, we obtain some sufficient conditions which guarantee the existence of at least one positive periodic solution for an impulsive semi-ratio-dependent predator-prey model with dispersion and time delays by using Mawhin's continuation theorem of coincidence degree theory, the result not only improves but also generalizes the known one. In the section 2.2, by applying Mawhin's continuation theorem of coincidence degree theory and a suitable Lyapunov func-tional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness and global stability of positive periodic solution for a delayed predator-prey model with dispersion and impulses. Some known results are im-proved and generalized. As an application, we also give some examples to illustrate the feasibility of our main results.
In Chapter 3, we investigate a delayed predator-prey model with monotonic or non-monotonic functional response and impulse. By using Mawhin's continuation theorem of coincidence degree theory, we obtain some sufficient conditions which guarantee the existence of multiple positive periodic solutions for the model. In particular, the presented criteria improve and extend some previous results.
In Chapter 4, we consider two classes of non-autonomous delay n-species com-petitive models with impulses. In the section 4.1, we obtain some sufficient and realistic conditions for the existence of positive periodic solutions of a general neutral delay n-species competitive model with impulses by using some analysis techniques and a new existence theorem, which is different from Mawhin's contin-uation theorem of coincidence degree theory and abstract continuation theory for k-set contraction. As an application, we also examine some special cases which have been studied extensively in the literature, some known results are improved and generalized. In the section 4.2, we apply the fixed point theorem in a cone of Banach space to obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solution of generalized n-species Gilpin-Ayala type competition system with multiple delays and impulses, which improves and generalizes the known ones.
In Chapter 5, we study some classes of Logarithmic population models with multi delays and impulses. In the section 5.1, we use the theory of abstract contin-uous theorem of k-set contractive operator and some inequality techniques to get sufficient and realistic conditions which are established for the existence, global attractivity of positive periodic solution to a neutral single-species Logarithmic population model with multi delays and impulse. The results improve and gen-eralize the known ones. As an application, we give an example to illustrate the feasibility of our main results. In the section 5.2, by using the contraction map-ping principle and some analysis techniques, we establish a set of easily applicable criteria for the existence, uniqueness and global attractivity of positive periodic solution for a multi-species Logarithmic population system with impulses. The conditions we obtained are weaker than the previously known ones, which can be easily reduced to several special cases. At last, we also give an example to illus-trate the feasibility of our main results. In the section 5.3, we investigate a neutral multi-species Logarithmic population model with feedback control and impulse. By applying the contraction mapping principle and some inequality techniques, a set of easily applicable criteria for the existence,uniqueness and global attractivity of positive periodic solution is established. The results improve and generalize the known ones. At last, we also give an example to illustrate the applicability of our results.
第一章,简述种群生态学的研究发展概况；概述时滞微分方程、脉冲微分方程的研究发展状况；简述本文问题产生的背景和本文的主要工作以及一些预备知识。
第二章,我们讨论了具有食饵扩散和脉冲干扰的捕食者-食饵模型。第一节我们利用迭合度理论中的Mawhin连续定理得到了一类具有脉冲干扰和扩散的非自治半基于比率捕食者-食饵模型至少有一个正周期解存在的充分条件,我们的结果改进和推广了已有的结果。第二节利用迭合度理论中的Mawhin连续定理、Lyapunov泛函方法以及一些分析技巧,得到了一类具有脉冲干扰和扩散的捕食者-食饵模型正周期解的存在性、全局吸引性和持续生存的充分条件,已有文献中的结果得到了改进和推广。在应用方面,我们举了几个实例来说明我们的结果的可行性。
第三章,我们考虑了一类具有单调(或者非单调)功能反应函数和脉冲干扰的捕食者-食饵模型。利用迭合度理论中的Mawhin连续定理,我们得到了该脉冲模型存在多个正周期解的充分条件。特别地,所得的结果改进和推广了已有的结果。
第四章,我们研究了两类具有时滞和脉冲干扰的n维种群竞争模型。第一节利用已有文献给出的一个新的不动点定理,我们得到了一类具有脉冲干扰和时滞n维Lotka-Volterra竞争模型正周期解的存在性的充分条件,并将获得的主要结果应用到几个生物模型当中,由此得到了一些有应用价值的结果,所得结果改进了已有的结果。第二节我们利用巴拿赫空间中锥上一个不动点定理,获得了一类n维Gilpin-Ayala型竞争脉冲模型的正周期解存在性和唯一性的判别准则,改进和推广了已有的结果。
第五章,我们研究了具有多时滞和脉冲干扰的对数人口模型。第一节我们利用k-集压缩算子的抽象延拓定理得到了一类具有多时滞和脉冲干扰中立型单种群对数人口模型正周期解存在性,通过分析的技巧得到正周期解全局吸引的充分条件,所得结果改进了文已有的结果。在应用方面,我们举了一个实例来证明我们的结果的可行性。第二节我们利用压缩映射原理和不等式技巧得到了一类n维具有多时滞和脉冲干扰对数人口模型正周期解存在性和全局吸引性的充分条件。我们所需的条件比已有结果所需的条件弱些,能更好地运用到一些特殊情形。最后,我们也用一个实例来证明我们的结果的可行性。第三节我们利用压缩映射原理和不等式技巧得到了一类n维具有多时滞、脉冲干扰和反馈控制中立型对数人口模型正周期解存在性和全局吸引性的充分条件,所得结果改进了已有的结果。最后,我们也用一个实例来说明我们的结果的可行性。
This Ph.D.thesis is divided into five chapters and main contents are as follows:
In Chapter 1, we give a survey to the developments of population ecology, delay differential equations and impulsive differential equations. Then we intro-duce the background of problems, the main results of this dissertation and some preliminaries are also summarized.
In Chapter 2, we discuss the predator-prey model with diffusion and im-pulse. In the section 2.1, we obtain some sufficient conditions which guarantee the existence of at least one positive periodic solution for an impulsive semi-ratio-dependent predator-prey model with dispersion and time delays by using Mawhin's continuation theorem of coincidence degree theory, the result not only improves but also generalizes the known one. In the section 2.2, by applying Mawhin's continuation theorem of coincidence degree theory and a suitable Lyapunov func-tional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness and global stability of positive periodic solution for a delayed predator-prey model with dispersion and impulses. Some known results are im-proved and generalized. As an application, we also give some examples to illustrate the feasibility of our main results.
In Chapter 3, we investigate a delayed predator-prey model with monotonic or non-monotonic functional response and impulse. By using Mawhin's continuation theorem of coincidence degree theory, we obtain some sufficient conditions which guarantee the existence of multiple positive periodic solutions for the model. In particular, the presented criteria improve and extend some previous results.
In Chapter 4, we consider two classes of non-autonomous delay n-species com-petitive models with impulses. In the section 4.1, we obtain some sufficient and realistic conditions for the existence of positive periodic solutions of a general neutral delay n-species competitive model with impulses by using some analysis techniques and a new existence theorem, which is different from Mawhin's contin-uation theorem of coincidence degree theory and abstract continuation theory for k-set contraction. As an application, we also examine some special cases which have been studied extensively in the literature, some known results are improved and generalized. In the section 4.2, we apply the fixed point theorem in a cone of Banach space to obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solution of generalized n-species Gilpin-Ayala type competition system with multiple delays and impulses, which improves and generalizes the known ones.
In Chapter 5, we study some classes of Logarithmic population models with multi delays and impulses. In the section 5.1, we use the theory of abstract contin-uous theorem of k-set contractive operator and some inequality techniques to get sufficient and realistic conditions which are established for the existence, global attractivity of positive periodic solution to a neutral single-species Logarithmic population model with multi delays and impulse. The results improve and gen-eralize the known ones. As an application, we give an example to illustrate the feasibility of our main results. In the section 5.2, by using the contraction map-ping principle and some analysis techniques, we establish a set of easily applicable criteria for the existence, uniqueness and global attractivity of positive periodic solution for a multi-species Logarithmic population system with impulses. The conditions we obtained are weaker than the previously known ones, which can be easily reduced to several special cases. At last, we also give an example to illus-trate the feasibility of our main results. In the section 5.3, we investigate a neutral multi-species Logarithmic population model with feedback control and impulse. By applying the contraction mapping principle and some inequality techniques, a set of easily applicable criteria for the existence,uniqueness and global attractivity of positive periodic solution is established. The results improve and generalize the known ones. At last, we also give an example to illustrate the applicability of our results.
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