反應擴散方程組理論與相關領域之數學問題研討會
講 題 (1) Spreading speed revisited--A free boundary approach I and II
主講人
時
地
間
(2) Bifurcations in Reaction-Diffusion Systems
(1) Professor Yihong Du (University of New England, Australia)
(2) ProfessorJunping Shi (College of William and Mary, USA)
99 年 12 月 10 日(星期五)
(1)10:10-11:00; 14:10-15:00
點 靜安 327 演講廳
摘 要
(2)11:10-12:00; 15:10-16:00
(1) Much previous mathematical investigation on the spreading of population was based on the traveling
wave solutions, which were first introduced in the pioneering works of Fisher (1937) and
Kolmogorov et al (1937), and further developed later along several directions. In particular, making
use of the traveling wave solutions, Aronson and Weinberger (1978) established the existence of a
unique asymptotic spreading speed for the propagation front of a new or invasive species. This
classical work has seen much ground breaking further developments in recent years. I will report
some recent joint works with several collaborators (Z. Lin, Z. Guo, R. Peng, H. Matano, B. Lou) on
a new approach to this problem, where a free boundary is used to describe the spreading front,
instead of the level set of traveling wave solutions.
Part I of the talk is aimed at a general audience, in which I'll give an overview of the applications of
mathematics in ecology, and explain how the free boundary model arises, and give a description of
our results in general terms, finishing by showing some numerical simulations of the model.
In part II of the talk, I'll focus on the mathematics of the problem, with some key ideas used in our
model explained in detail.
(2) I will give two lectures on the applications of bifurcation theory in reaction-diffusion systems.
Lecture 1: Abstract local and global bifurcation theory of stationary problems. and Examples of
stationary bifurcations.
Lecture 2: Turing type bifurcation in systems. Local and global Hopf bifurcations.
For the stationary equations, I will describe basic local bifurcation theory based on implicit function
theorem and global bifurcation theorem based on topological degree. Examples include Turing
bifurcation induced by diffusion or cross-diffusion, loops of stationary solutions in predator-prey
systems, and also bounded branch connecting semi-trivial solutions in predator-prey systems. I will
also present Hopf bifurcation theorem for reaction-diffusion systems, including the case of delayed
reaction-diffusion equations. In particular, we show the existence of spatial non-homogeneous
periodic orbits in predator-prey systems. Global bifurcation of periodic orbits will also be discussed.
贊助單位
國科會數學中心
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