兰州大学机构库 >数学与统计学院
一类时间周期Lotka-Volterra竞争扩散系统的入侵整体解
Alternative TitleInvasion Entire Solutions in a Time Periodic Lotka-Volterra Competition-Diffusion System
杜丽君
Thesis Advisor李万同
2016-05-18
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name硕士
Keyword时间周期行波解 指数衰减行为 比较原理 入侵整体解
Abstract自治情形下,有关Lotka-Volterra竞争扩散系统的行波解和整体解的研究成果已经较完善,但对于时间周期情形, 相关研究结果则较少。不同于自治情形,时间周期非线性项的出现带给我们显著的困难,这使得相关研究方法有很大的不同。本文将从一类时间周期的Lotka-Volterra竞争扩散系统出发,主要研究单稳情形下一类入侵整体解的存在性。首先,利用Harnack不等式估计和合作型抛物系统的比较原理并结合上下解方法,得到时间周期行波解在无穷远处的指数型衰减估计。其次,我们利用动力系统的方法来研究时间周期行波解在无穷远处的精确指数衰减率。通过视复合变量z=x-ct为发展变量,结合双边Laplace变换和谱分析的方法,得到时间周期行波解精确的指数衰减率。最后,我们研究系统的整体解。先利用时间周期行波解的指数渐近行为建立一些先验估计,据此构造出一对合适的上下解,再结合比较原理,得到一类新型整体解的存在性及其定性性质,其表现为沿实轴两端以不同波速相向而行的时间周期波前解。在生物意义上表示优势物种从生存区域的两端入侵,导致劣势物种灭绝的过程,即为物种提供了一种新的入侵方式。
Other AbstractTraveling wave solutions and entire solutions for the Lotka-Volterra competition-diffusion system in the autonomous case have been studied intensively, while there are still very few studies devoted to time periodic traveling waves for systems with periodic reaction terms. Unlike the autonomous case, the presence of time periodic nonlinearities poses significant difficulties and requires new approaches. We deal with the time periodic Lotka-Volterra competition-diffusion system and study the so-called entire solutions in the monostable case in this paper. Firstly, by the Harnack's inequality and the comparison principle coupled with the sub- and supersolution, we study the exponential decay estimate of the time periodic traveling waves as it tends towards its limiting state.Secondly, we establish the exact exponential decay rate of the time periodic traveling waves by employing a dynamical system approach. Using the moving coordinate frame z=x-ct as an evolution variable, we utilize the Laplace transform method and spectral theory to obtain the exponential decay rate.Finally, we study the entire solution. We firstly obtain some estimates by using the exponential asymptotic behaviors, secondly we construct a pair of appropriate (irregular) supersolution and subsolution, we then establish the existence and various qualitative properties of an entire solution by using the comparison argument,which behaves as two pairs of time periodic wave fronts with different speeds coming from the both sides of x-axis. This indicates that the entire solution exhibits the extinction of the inferior species v by the superior one u invading from both sides of x-axis, which provides a new invasion way of u to v other than traveling waves in a time periodic environment.
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224454
Collection数学与统计学院
Recommended Citation
GB/T 7714
杜丽君. 一类时间周期Lotka-Volterra竞争扩散系统的入侵整体解[D]. 兰州. 兰州大学,2016.
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