一、課程背景與目的：

Mathematical modeling can play an important role in the understanding of mechanisms in biological systems. Motivated by this, we intend to introduce mathematical models of HIV infection, spatial segregation, and ecological systems with internal storage. Recently, Ergodic Theory has been successfully used to investigate the weakly repelling property of a compact and invariant set on the boundary, which is crucial when we utilize persistence theory to establish the possibility of coexistence for an ecosystem. Thus, we will also provide a brief introduction of Ergodic Theory in this course.

 

二、課程之大綱與講者：

Lecture 1

Lecturer: Professor Chih-Hung Chang (張志鴻) , National University of Kaohsiung

Title: A Short Course for Ergodic Theory

Abstract:

Ergodic theory studies the long time behaviour of dynamical systems. This line of investigation has its origin in Poincare's investigations in statistical physics more than one hundred years ago. However, in the meantime ergodic theory has found many remarkable applications in various branches of mathematics. In this short course, I will give an introduction to some classical results, such as Poincare's recurrence theorem, Birkhoff ergodic theorem, and Furstenberg's multiple recurrence theorem. Meanwhile, some applications like the ergodic viewpoint of Borel's normal number theorem will also be covered.

 

Lecture 2

Lecturer: Professor Chang-Yuan Cheng (鄭昌源), National Pingtung University

Title: Mathematical Studies for HIV infection

Abstract:

Human immunodeficiency virus type 1 (HIV-1) is one of the most and intensely studied viral pathogens in the history of science. Mathematical modeling has helped to improve our understanding of the infection as well as the pharmacodynamics within the host under a drug therapy. However, from recent reports, the spacial heterogeneity related to either the virus environments or the pharmacodynamics is a significant issue that deserves more attentions. We begin the sequel from introducing the routes of infection and virus environments, then formulating the drug efficacy, and studying its critical effect on viral dynamics via a perturbation technique. The lecture will include four parts:

(1) Basic viral dynamics in ODE/DDE/aged models

(2) Multi-compartmental virus models

(3) Pharmacodynamics within the host under a drug therapy

(4) A perturbation technique to determine viral extinction/persistence

 

Lecture 3

Lecturer: Professor Chang-Hong Wu (吳昌鴻), National Chiao Tung University

Title: Spatial Segregation in Competition Models

Abstract:

The understanding of the spatial behavior of the interacting species is one fundamental issue in mathematical ecology. In particular, spatial segregation of species is a commonly observed phenomenon.This phenomenon has been studied widely in the literature. Generally speaking, it may occur when the competition between species is very strong. We will introduce some reaction-diffusion systems to approximate such a phenomenon. Furthermore, the limiting problems may help us understand the formation of spreading front from the modeling viewpoint and bring some mathematical models that may describe the invasion of species. In this lecture, we plan to divide the discussion into three parts:

(1) A brief introduction regarding the spatial segregation phenomenon

(2) The Fisher-Stefan model and some competition models 

(3) Some applications.

 

Lecture 4

Lecturer: Professor Feng-Bin Wang (王埄彬), Chang Gung University

Title: Applications of Theory of Monotone Dynamical System and Uniform Persistence to Ecological Models with Internal Storage

Abstract:

Competition for resources is a fundamental interaction between species and there has been a lot of experimental and theoretical analyses of nutrient-limited phytoplankton growth and competition studies. The simplest competition models use one ordinary differential equation to govern the dynamics of each species. These population dynamics are coupled to dynamics of one or more resources by assuming a constant quota of nutrient per individual, or equivalently, a constant yield of individuals from consumption of a unit of resource. In fact, quotas may vary, leading to variable-internal-stores models. In this talk, I shall introduce several systems modeling nutrient consumption, storage, and population growth in temporally homogeneous/varying environments (ODEs system involved in constant/time-periodic coefficients). In order to obtain the analytical analysis of the models, I shall further introduce the theory of monotone dynamical system and uniform persistence.

 

三、課程詳細時間地點以及方式：

課程主題與時間：

Lecture 1　A Short Course for Ergodic Theory

Time : 10:00-12:00,14:00-16:00, July 27-28, 2020.

 

Lecture 2　Mathematical studies for HIV infection

Time : 10:00-12:00,14:00-16:00, July 29-30, 2020.

 

Lecture 3　Spatial segregation in competition models

Time : 10:00-12:00,14:00-16:00, July 31 and August 3, 2020.

 

Lecture 4　Applications of theory of monotone dynamical system and uniform persistence to ecological models with internal storage.

Time : 10:00-12:00,14:00-16:00, August 4-5, 2020.

 

地點：Room B, 4F, Third General Building, National Tsing Hua University

 

方式：The course will include lectures