几类捕食—食饵模型的定性分析

发布时间：2018-04-28 06:48

本文选题：捕食-食饵模型 + 非自治　；参考：《哈尔滨工业大学》2016年博士论文

【摘要】：捕食-食饵关系是种群生态学中最重要的关系之一,捕食-食饵生物动力模型是生物数学研究中的重要研究领域,考虑环境的季节性改变,捕食者和食饵的年龄结构,以及疾病在捕食者和食饵种间关系中的传播等性质具有重要的现实意义和理论意义。本学位论文主要对如下几类捕食-食饵动力模型进行定性分析:非自治生态流行病捕食-食饵模型,捕食者具有阶段结构的生态流行病模型,食饵具有年龄结构的捕食-食饵模型,捕食者和食饵均具有年龄结构的捕食-食饵模型。首先,研究了食饵患有SIS类型传染病的非自治生态流行病模型。运用比较原理和新的方法建立了患病食饵一致持久和灭绝的充分条件,以及患病食饵一致持久的必要条件。结果显示对患病食饵有倾向性的捕获可能会引发地方病,针对捕食者引入新的资源有益于消灭患病食饵。其次,考虑了捕食者具有阶段结构的生态流行病模型。通过构造Lyapunov泛函并运用LaSalle不变集原理,获得了模型的全局动力性态,同时由捕食者的净再生数确立了模型的阈值动力性态。基于最大存活率以及最大密度探讨了捕食者的捕获策略。作为应用,讨论了捕食者的净再生数达到最大时捕食者的捕获策略:结果表明当疾病是轻微的地方病时,捕食者仅捕获易感食饵,如果疾病是严重的地方病,则捕食者仅捕获患病食饵。此外,数值结果表明捕食者为达到自身密度最大,对易感食饵和患病食饵应分别保持适当的捕获概率。再次,提出了食饵具有年龄结构的捕食-食饵模型。通过定义捕食者的净再生数以及运用动力系统的一致持久生存理论证明了捕食者的一致持久性,同时结合比较原理得到了无捕食者平衡点的全局渐近稳定性。最后,分析了捕食者具有Beddington-DeAngelis型功能反应函数的年龄结构捕食-食饵模型。通过定义捕食者的净再生数,运用动力系统的一致持久生存理论得到了模型是一致持久的;运用比较原理及Krein-Rutman定理证明了无捕食者平衡点的全局渐近稳定性。数值模拟表明捕食者间合适的干扰可以促使共存状态趋于渐近稳定。
[Abstract]:Predator-prey relationship is one of the most important relationships in population ecology. Predator-prey biodynamic model is an important research field in biological mathematics, considering the seasonal change of environment, the age structure of predator and prey. The transmission of disease between predator and prey species has important practical and theoretical significance. In this dissertation, the following kinds of prey-prey dynamic models are analyzed qualitatively: nonautonomous ecological epidemic predator-prey model, predator ecological epidemic model with stage structure. Predator-prey model with age structure, predator and prey model with age structure. Firstly, the non-autonomous ecological epidemic model of prey with SIS type infectious diseases was studied. By using the principle of comparison and the new method, the sufficient conditions for the consistent persistence and extinction of sick prey and the necessary condition for consistent persistence of sick prey are established. The results show that the predation of diseased prey may lead to endemic disease, and the introduction of new resources to predators is beneficial to the elimination of diseased prey. Secondly, the ecological epidemic model of predator with stage structure is considered. By constructing the Lyapunov functional and using the LaSalle invariant set principle, the global dynamic state of the model is obtained, and the threshold dynamic state of the model is established from the net reproducing number of the predator. Based on the maximum survival rate and maximum density, the predator capture strategy is discussed. As an application, this paper discusses the predator capture strategy when the net regeneration number of the predator reaches the maximum. The results show that when the disease is a mild endemic disease, the predator catches only the susceptible bait, if the disease is a serious endemic disease, The predator catches only the sick prey. In addition, the numerical results show that in order to achieve the maximum density of the predator, the predatory prey and the diseased prey should be kept appropriate capture probability respectively. Thirdly, a prey-prey model with age structure is proposed. By defining the net regenerative number of predators and applying the uniformly persistent survival theory of dynamical systems, the uniformly persistent predator is proved, and the global asymptotic stability of the non-predator equilibrium is obtained by combining the principle of comparison. Finally, the age-structured predator-prey model with Beddington-DeAngelis type functional response function is analyzed. By defining the net reproducing number of predators and applying the uniformly persistent survival theory of dynamical systems, we obtain that the model is uniformly persistent, and prove the global asymptotic stability of the non-predator equilibrium by using the comparison principle and Krein-Rutman theorem. The numerical simulation shows that the proper interference between predators can promote the asymptotic stability of the coexistence state.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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