### 動態系統與生物數學

### Dynamical Systems and Biomathematics

### 地 點 ： M 4 1 7 數 學 館

TMS Annual Meeting

### 數 學 年 會

## 2018 ^{數 學 年 會}

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### 演講摘要

Speech Abstracts

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### Travelling curved waves in two dimensional excitable media

### Chang-Hong Wu

### Department of Applied Mathematics National University of Tainan E-mail: changhong@mail.nutn.edu.tw

Wave propagation can occur in various area such as physics, biology, chemi-cal kinetics, and so on. In particular, excitable media, which are often modeled by nonlinear PDEs, can support abundant spatiotemporal dynamics. In this talk, we focus on a free boundary problem in two-dimensional excitable media arising from a singular limiting problem of a FitzHugh-Nagumo-type reaction-diffusion system. The existence, uniqueness and stability of traveling curved waves will be discussed. This is a joint work with Hirokazu Ninomiya (Meiji University).

### 33

### Dynamics on membranes mediated by bulk diffusion

### Eliot Fried

### Mathematics, Mechanics, and Materials Unit

### Okinawa Institute of Science and Technology Graduate University eliot.fried@oist.jp

Coupling between an active membrane and bulk diffusion arises in many bio-logical and chemical processes, cell polarization, protein activation waves on cell membranes, cell-to-cell signaling, membrane bound Turing patterns, Belousov-Zhabotinsky reactions on beads, and so on. Although experimental observa-tions demonstrate the presence of collective synchronous and asynchronous be-haviors along with other more complicated spatiotemporal pattern dynamics, very little theoretical research has been directed at exploring the processes that govern these phenomena. Assuming Van der Pol oscillator dynamics on the membrane, we identify two essential parameters governing the system. In this two-dimensional parameter space we observe various stable asymptotic dynam-ics, including synchronous, asynchronous, and decaying behavior. In addition to parameter regions where only one type of asymptotic dynamics occurs, there are also bistable domains. The parameter space contains a degenerate critical point which is considered to be the organizing center of the various dynamical regimes. This is joint work with Johannes Schönke and Toshiyuki Ogawa.

**Keywords: pattern formation, collective behavior, bistability**

### 34

### Threshold dynamics of a periodic parabolic system modeling the influence of salinity and

### nutrient recycling on the growth of algae

### Feng-Bin Wang

### Department of Natural Science Center for General Education Chang Gung University

### E-mail: fbwang@mail.cgu.edu.tw, fbwang0229@gmail.com

In this talk, we shall study a periodic advection-dispersion-reaction system
incorporating the effects o f s alinity, n utrient r ecycling, t emperature, a nd
spa-tial variations for the growth of harmful algae in riverine ecosystems. We can
introduce* the basic reproduction number R*0 for* algae and show that R*0 serves
as a threshold value for persistence and extinction of the algae. More precisely,
we* prove that the washout state is globally attractive if R*0 *<* 1, while there
exists* a positive periodic state and the algae is uniformly persistent if R*0*>* 1.

This talk is based on ongoing projects joint with Drs. James P. Grover and Xiao-Qiang Zhao.

**Keywords:** harmful algae, salinity, nutrient recycling, the basic
reproduc-tion number

### 35

### Existence and Instability of Traveling Pulses of Generalized Keller-Segel Equations

### Chueh-Hsin Chang

### Department of Applied Mathematics Tunghai University

### E-mail: changjuexin@thu.edu.tw

Keller-Segel Equations exhibit the phenomenon of chemotaxis. It is difficult to find the traveling pulse solutions for the minimal model. In this talk, we talk about the existence and instability of some Keller-Segel Equations with nonlinear chemical gradients and small diffusions by the geometric singular per-turbation theory and spectral analysis. This is a joint work with Y. S. Chen, John. M. Hong, and B. C. Huang.

**Keywords: Keller-Segel model, traveling wave solution**

### 36

### On The Dynamics of Shifts of Finite Type on Groups

### Jung-Chao Ban

### Department of Applied Mathematics National Dong Hwa University E-mail: changhong@mail.nutn.edu.tw

In this talk, we consider the dynamics of shifts of finite type on groups (G-SFTs) and focus on case that G is monoid or free group. By using the nonlinear recursive formula, the complete characterization for the entropy of GSFTs is established. Besides, we discuss various kinds of mixing GSFTs, and provide the connection between positive-entropy and mixing GSFTs.

### 37

### Continuation methods and numerical bifurcation analysis

### Te-Sheng Lin

### Department of Applied Mathematics National Chiao Tung University E-mail: tslin@math.nctu.edu.tw

A numerical continuation method is developed to follow time-periodic travelling-wave solutions of both local and non-local evolution partial differential equations (PDEs). It is found that the equation for the speed of the moving coordinate can be derived naturally from the governing equations together with a condi-tion that breaks the translacondi-tional symmetry. The derived system of equacondi-tions allows one to follow the branch of travelling-wave solutions as well as solutions that are time-periodic in a frame of reference travelling at a constant speed.

We then show examples of the bifurcation and stability analysis of long-wave models of electrified falling films as well as films on a rotating cylinder. Finally, we present our recent work on spontaneous autophoretic motion of colloidal particles in two-dimensional space.

**Keywords: numerical continuation method, time-periodic travelling-wave**
solution, numerical bifurcation analysis

### 38

### 3D Structure Prediction of Alpha-1,4 Fucosyltransferase

### Chi-Jen Wang and Ching-Ching Yu

### Department of Mathematics and Department of Chemistry and Biochemistry National Chung Cheng University

### cjwang@ccu.edu.tw

Fucosyltransferase (FucT) is an enzyme that transfers fucose sugar to a sugar
or protein. We are interested in one FucT from Helicobacter pylori which lives
*in gastric and duodenal. The function and structure of α-1,3 FucT NCTC11639*
*has been reported a decade ago, the dual functionality of α-1,3/4 FucT UA948*
*has been reported in 2017, however the structure of α-1,4 FucT has not. The*
*prior experiment shows that α-1,4 FucT DSM6709 also have α-1,3 linkage from.*

Further, its gene sequence also indicates this enzyme might have the same
func-tion as UA948. Our goal is to predict the structures of DSM6709, and compare
these structures to other FucTs to verify the possibility of the dual
*functionali-ties of α-1,4 FucT DSM6709.*

**Keywords: α-1,4 Fucosyltransferase, diverse linkages, structure prediction.**

### 39

### Backward bifurcation of a network-based SIS epidemic model with saturated treatment

### function

### Chun-Hsien Li

### Department of Mathematics National Kaohsiung Normal University

### E-mail: chli@nknu.edu.tw

In this talk, we present a study on a network-based SIS epidemic model
with a saturated treatment function to characterize the saturation phenomenon
of limited medical resources. In this model, we first o btain a t hreshold value
*R*0, which is the threshold condition for the stability of the disease-free
equi-librium. We show that a backward bifurcation occurs under certain conditions.

More precisely, the saturated treatment function leads to a such bifurcation. In
this* case, R*0*<* 1 is not sufficient to eradicate the disease from the population.

Numerical simulations are conducted to validate the theoretical results. This is a joint work with Yi-Jie Huang.

**Keywords:** complex networks, epidemic model, saturated treatment
func-tion, backward bifurcation.

### 40

### The influence of awareness on the spreading of infectious diseases

### Yu-Hao Liang

### Department of Applied Mathematics National University of Kaohsiung

### E-mail: yhliang@nuk.edu.tw

In this talk, I will discuss the influence of awareness on the epidemic spread-ing. We will propose a multiplex network where the spreading of the disease and information occurs, respectively, in two different layers of networks, namely the physical network and the virtual network. In addition, these two diffusive processes are assumed to interact and affect each other. Such concept would create a multiplex SIS-UAU epidemic model. Some recent results on these two models are to be introduced. This is a joint work with Jonq Juang.

### 41

### 偏微分方程

### Partial Differential Equations 地 點 ： M 2 1 0 數 學 館

TMS Annual Meeting

### 數 學 年 會

## 2 0 18

### 數 學 年 會 D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

### D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

### 演講摘要

Speech Abstracts

### D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

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1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

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*掃描QRCODE，點擊講者/標題查看

### 42

### On Well-posedness of Weak Solutions

### Tai-Ping Liu Institute of Mathematics

### Academia Sinica

### E-mail: liu@math.stanford.edu

There have been very substantial progresses on non-uniqueness of weak so-lutions for incompressible Navier-Stokes and Euler equations, and compressible Euler equations. The well-posedness problem is a fundamental problem in the theory of partial differential equations. This talk aims at proposing a different well-posedness theory. We illustrate this new theory with a recent study of the author with Shih-Hsien Yu on compressible Navier-Stokes equations, and also re-call the celebrated well-posedness theory for hyperbolic conservation laws. This talk gives concrete meaning to the author’s talk in annual differential equations meeting at Chung-San University few years ago on the same topic.

### 43

### Sharp regularizing estimates for the gain term of the Boltzmann collision operator

### Jin-Cheng Jiang Department of Mathematics National Tsing Hua University E-mail: jcjiang@math.nthu.edu.tw

We prove the sharp regularizing estimates for the gain term of the Boltzmann collision operator including hard sphere, hard potential and Maxwell molecule models. Our new estimates characterize both regularization and convolution properties of the gain term which were studied by Lions [4], Wennberg [6], Bouchut & Desvillettes [2], Mouhot & Villani [5] etc. and Duduchava & Kirsch

& Rjasanow [3], Alonso & Carneiro & Gamba [1] etc. respectively. The new
estimates have the following features. The regularizing exponent is sharp both
*in the L*^{2}based inhomogeneous and homogeneous Sobolev spaces which is exact
the exponent of the kinetic part of collision kernel. The functions in these
esti-mates belong to a wider scope of (weighted) Lebesgue spaces than the previous
regularizing estimates. Furthermore, for the estimates in homogeneous Sobolev
spaces, we only need functions lying in Lebesgue spaces instead of weighted
Lebesgue spaces, i.e., no loss of weight occurs in this case.

**Keywords: Boltzmann collision operator, Gain term, Regularizing, hard**
sphere, hard potential, Maxwell molecule, Fourier integral operator

**References**

*[1] R. Alonso, E. Carneiro and I.M. Gamba,Convolution inequalities*
*for the Boltzmann collision operator, Comm. Math. Physics, 298 (2010),*
pp. 293-322.

*[2] F. Bouchut and L. Desvillettes, A proof of the smoothing properties*
*of the positive part of Boltzmann’s kernel, Rev. Mat. Iberoamericana, 14*
(1998), pp. 47-61.

*[3] R. Duduchava, R. Kirsch and S. Rjasanow, On estimates of the*
*Boltzmann collision operator with cutoff, J. Math. Fluid Mech., 8 (2006),*
pp. 242-266.

### 44

*[4] P.-L. Lions, Compactness in Boltzmann’s equation via Fourier integral*
*operators and applications. I, II, J. Math. Kyoto Univ., 34 (1994), pp. *
391-427, pp. 429-461.

*[5] C. Mouhot and C.Villani, Regularity Theory for the Spatially *
*Homoge-neous Boltzmann Equation with Cut-Off, Arch. Rational Mech. Anal., 173*
(2004), pp. 169-212.

*[6] B. Wennberg, Regularity in the Boltzmann equation and the radon *
*trans-form, Comm. Partial Differential Equations, 19 (1994), pp. 2057-2074.*

### 45

### Concentration of source terms in generalized Glimm scheme for initial-boundary problem of

### nonlinear hyperbolic balance laws

### Ying-Chieh Lin

### Department of Applied Mathematics National University of Kaohsiung

### E-mail: linyj@nuk.edu.tw

In this talk, we consider the initial-boundary value problem for a nonlinear
hyperbolic* system of balance laws with sources a*^{x}*g and a*^{t}*h.* We assume that
the boundary data satisfy a inear or smooth nonlinear relation. Generalized
Riemann* and boundary Riemann problems are provided with the variation of a *
concentrated* on a thin T -shaped region of each grid. We generalize Goodman’s *
boundary interaction estimates and introduce a new version of Glimm scheme
to construct the approximation solutions and its stability is proved by
consid-ering* two types of conditions on a. The global existence of entropy solutions *
is established. Under some sampling condition, we find t he e ntropy solutions
converge* to its boundary values in L*^{1}loc as* x → 0*^{+} and the boundary values
satisfy* the boundary condition almost everywhere in t.*

**Keywords:** nonlinear balance laws, initial-boundary value problem,
Rie-mann problem, generalized Glimm scheme, concentration of source, wave
inter-action estimates, entropy solutions.

### 46

### Smoothing effect due to mixing in kinetic theorem and its application to the optical tomography

### I-Kun Chen

### Institute of Applied Mathematical Science National Taiwan University

### E-mail: ikun.chen@gmail.com

In kinetic theorem, it is known that the combination of collision or averaging and transport can result gaining of regularity, e.g., the celebrated Velocity Av-eraging Lemma by Golse, Perthame, and Sentis 1985 and the Mixture Lemma by Liu and Yu 2004. For stationary solution in a bounded convex domain, we find this effect can be realized by interplaying between velocity and space. We can decompose the solution to functions of different level of regularity due to different times of mixing and use it for optical tomography.

### 47

### Weak Interaction between Traveling Waves in the Three-species Competition-diffusion Systems

### Chueh-Hsin Chang

### Department of Applied Mathematics Tunghai University

### E-mail: changjuexin@thu.edu.tw

In this talk we consider the weakly interaction between two traveling wave so-lutions of the threes-species Lotka-Volterra competition-diffusion systems. Each of the two traveling wave solutions has one trivial component (called trivial waves). By the asymptotic behavior of the trivial waves and the existence re-sults of the two-species traveling waves, we can observe the dynamics of the distance between the two trivial waves. We proved that there exists an unsta-ble three-species traveling wave solution which is close to the two trivial waves.

This is a joint work with Prof. Chiun-Chuan Chen and Prof. Shin-Ichiro Ei.

**Keywords: Lotka-Volterra system, traveling wave solution**

### 48

### 離散數學 Discrete Mathematics 地 點 ： G 0 0 1 圖 書 館

TMS Annual Meeting

### 數 學 年 會

## 2018 ^{數 學 年 會}

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