时间周期Lotka-Volterra竞争系统的空间扩张行为

It has always been the core topics for Mathematical ecology research to understand and describe the persistent existence and the spreading dynamics of alien invasive species in the unfamiliar environment by the qualitative and quantitative methods. Just for recent years, due to its strong explanatory and integration power, the reaction-diffusion models with the free boundary are increasingly focused by many researchers. Nevertheless, subject to the nontrivial feature produced by the time periodicity and their own coupling property, to investigate further on the time periodic systems, estimate the asymptotic spreading speeds precisely and character the asymptotic spreading profiles detailed are still the main purpose and difficulty home of current researches. By using principally the upper and lower solutions constructed precisely, together with other methods such as the time periodic principal eigenvalue theory, the zero number argument etc, the current project is dedicated to study the spatial spreading behaviours of two species Lotka-Volterra competition systems with the common free boundaries and the complete time periodic coefficients. On the basis of the established potential spreading-vanishing dynamics, we will investigate further on the existence and uniqueness of the relative semi-waves and the precise characterization of its speed, then, as result, the precise estimation of the asymptotic spreading speed and the detail characterization of the asymptotic spreading profile will be obtained as the key issues. The implement of the project will not only contribute to understand the deeper feature of the spatial spreading behaviours comprehensively, but also provide some helpful theoretical references to the practice in working.

定性定量地理解并刻画外来入侵物种在新环境中的持续生存和扩张动态一直是数学生态学研究的核心课题，而近年来，具有自由边界的反应扩散模型因其具有的强大的解释和整合能力，日益受到众多研究者的关注。然而，受制于由周期性所产生的非平凡特征和自身所具有的耦合性特点，对时间周期性系统予以深入考察，最终建立渐近扩张速度的精细估计和渐近扩张截面的细致刻画，仍是当前此类研究的主要宗旨和困难所在。本项目主要通过构造精巧的上下解，结合周期主特征理论、零数原理等方法，对具有共同自由边界的系数完全具有时间周期性的两种群Lotka-Volterra竞争系统的空间扩张行为进行集中考察，在建立其潜在的扩张-灭绝动态和判据的基础之上，通过深入考察对应半波问题的存在唯一性以及半波波速的精细刻画，重点给出此时渐近扩张速度的精细估计和渐近扩张截面的细致刻画，以求全面揭示此时系统空间扩张行为的深层次特征，为实际工作提供有益的理论启发。

定性定量地理解并刻画外来入侵物种在新环境中的持续生存和扩张动态是数学生态学研究的核心课题，而具有自由边界的反应扩散模型对此类问题具有的强大的解释和整合能力。本项目立足反应扩散方程全新理论，通过构造精巧的上下解，结合主特征理论、单调迭代等方法，先后克服了不同问题情景中由于超线性反应项、空间/时间异质性、非局部扩散和时间周期性等问题特征所产生的非平凡和部分合作性等困难，对(i)以自由边界问题为核心的超线性反应扩散方程的渐近行为；(ii)周期变化和具有自由边界的变化区域上时间周期鸟类-槲寄生系统的空时动态；(iii)具有非局部扩散模式的两种群 Lotka-Volterra 竞争系统的空间扩张等三个方面予以深入考察。项目的研究全面建立了相关问题的扩张-灭绝二分性或三分性结果，得到了利用初始条件表达的问题扩张-灭绝判据，对部分问题中的扩张情况给出了渐近扩张截面的精细刻画和渐近扩张速度较为细致的估计。研究的结果揭示了部分生物种群空间扩张行为的深层次特征，将为生物入侵预测、控制以及预防和濒危物种保护等实际工作提供有益的理论启发。

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