具有时滞和空间离散的Lotka-Volterra竞争系统的渐近传播速度的线性确定性

The Lotka-Volterra competition diffusive model has been extensively used in various fields such as biology,chemistry and physics.Traveling waves and the spreading speed play a crucial role in the study of Lotka-Volterra competition diffusive model. Especially, the spreading speed is an important threshold to predict whether the introduced species can invade successfully and how fast they invade. The most famous linear conjecture believe that the spreading speed can compute from the appropriate linearization. However, studies have shown that the linear conjecture is not always true. Hence, it is necessary to provide some sufficient conditions to guarantee the validity of linear conjecture, which is called linear determinacy. To the best of our knowledge, most of the existing work focused on the linear determinacy of the Lotka-Volterra competition model without delay, while the result for the delayed model was quite limited due to the difficulty arising from the time delay. This project is mainly concerned with the linear determinacy of spreading speed for a delayed and spatial discrete Lotka-Volterra competition system. More precisely, we shall first establish the spreading speed of traveling waves. Next, we will give sufficient conditions under which the spreading speed of traveling waves is precisely equal to the minimal speed of the corresponding linearized model. Finally, we try to provide examples for which the linear conjecture is not true. That is, the spreading speed cannot be linearly determined in general.

Lotka-Volterra竞争扩散系统广泛应用于物理、化学、生物等众多学科中.行波解及渐近传播速度是Lotka-Volterra竞争扩散系统的重要研究课题.其中渐近传播速度是决定外来种群是否能成功入侵的重要阈值.著名的“线性化猜想”认为,系统的渐近传播速度与其线性化模型的最小波速相等.然而已有研究结果表明此结论并不总是成立.线性确定性问题致力于研究系统的渐近传播速度与其线性化模型的最小波速相等的充分条件.目前,具有时滞和空间离散的Lotka-Volterra竞争系统渐近传播速度的线性确定性方面的研究结果十分有限,许多还是公开问题.时滞的加入,使模型更加符合实际,但同时也导致数学上的研究困难.本项目致力于研究具有时滞和空间离散的Lotka-Volterra竞争系统的渐近传播速度的线性确定性,建立其渐近传播速度满足线性确定性的充分条件,并寻找渐近传播速度不满足线性确定性的系统参数范围.

本项目围绕多种群交互（竞争或合作）扩散模型的渐近传播动力学展开了研究。具体主要结果包括：（1）一类具有时滞和空间离散的三种群竞争合作系统的双稳行波解的存在性和单调性。（2）一类空间离散和反应扩散三种群竞争合作系统的行波解的波速符号和渐近传播性质。（3）一类退化单稳情形下的时滞（时间）周期方程的渐近传播性质。从生态学角度来看，本项目具有重要的研究意义和科学价值。本项目的主要研究结果可用于解释生态系统中多种群交互（竞争或合作）过程中的一些生物入侵现象。生物入侵（外来物种增多而使原有物种灭绝的过程）现象在生态系统的进化过程中扮演着极其重要的角色，其将影响生物种群的共存性,进而影响生态系统的多样性和持久性。因此，本项目的研究结果对维持生态多样化和持久性具有重要的理论指导意义。