著作名稱：  Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries 
年度：  2018 
類別：  期刊論文 
摘要：  We investigate the spreading behavior of two invasive species modeled by a LotkaVolterra diffusive competition system with two free boundaries in a spherically symmetric setting. We show that, for the weak–strong competition case, under suitable assumptions, both species in the system can successfully spread into the available environment, but their spreading speeds are different, and their population masses tend to segregate, with the slower spreading competitor having its population concentrating on an expanding ball, say Bt, and the faster spreading competitor concentrating on a spherical shell outside Bt that disappears to infinity as time goes to infinity. 
關鍵字：  free boundary problem, LotkaVolterra competition model, asymptotic spreading speed. 
著作名稱：  Biased movement and the ideal free distribution in some free boundary problems 
年度：  2018 
類別：  期刊論文 
摘要：  Some invasive species can persist in their habitat but eventually spread very slow in a nonlinear fashion to expand their habitat range. In order to capture this phenomenon, we consider reaction–diffusion–advection models with a free boundary modeling the spreading and the biased movement of species in onedimensional spatially heterogeneous environments. Under a condition of low resource quality, we find that large advection can lead to the spreading of the species but the spreading speed goes asymptotically to zero. Moreover, we investigate the effect of the resource on the dynamics of the current problem. Finally, we bring the notion of an ideal free distribution (IFD) into free boundary problems to understand the mechanism such that the species can eventually match the environmental quality perfectly. Under the current problem setting, the IFD may not hold even if the population plays an ideal free strategy. We then provide a sufficient condition for the IFD to be reached when using an ideal free strategy. 
關鍵字：  Free boundary problem; Biased movement; Ideal free distribution; Spreading speed 
著作名稱：  (with Hirokazu Ninomiya)Traveling curved waves in two dimensional excitable media 
年度：  2017 
類別：  期刊論文 
摘要：  This paper treats a free boundary problem in twodimensional excitable media arising from a singular limiting problem of a FitzHughNagumotype reactiondiffusion system. The existence and uniqueness up to translations of twodimensional traveling curved waves solutions is shown. To study the stability of the waves, the local and global existence and uniqueness of solutions to the free boundary problem near the waves under certain assumptions is established. The notion of the arrival time is introduced to estimate the propagation speed of solutions to the free boundary problem, which allows us to establish the asymptotic stability of traveling curved waves by using the comparison principle. It is also pointed out that the gradient blowup can take place if the initial data are far from the traveling curved waves, which means the interface may not always be represented by a graph.

關鍵字：  traveling curved waves, Vshaped traveling waves, front, back, excitable media 
著作名稱：  (with YinLiang Huang) Positive steady states of reactiondiffusionadvection competition models in periodic environment 
年度：  2017 
類別：  期刊論文 
摘要：  In this paper, we consider the positive steady states for reaction–diffusion–advection competition models in the whole space with a spatially periodic structure. Under the spatially periodic setting, we establish sufficient conditions for the existence of positive steady states of this model, respectively, by investigating the sign of the principal eigenvalue for some linearized eigenvalue problems. As an application, a Lotka–Volterra reaction–diffusion–advection model for two competing species in a spatially periodic environment is considered. Finally, some numerical simulations are presented to seek dynamical behaviors. 
關鍵字：  Positive steady states, Reaction–diffusion–advection, Population dynamics, Periodic environment 
著作名稱：  A general approach to the asymptotic behavior of traveling waves in a class of threecomponent lattice dynamical systems 
年度：  2016 
類別：  期刊論文 
摘要：  In this paper, we develop a general approach to deal with the asymptotic behavior of traveling wave solutions in a class of threecomponent lattice dynamical systems. Then we demonstrate an application of these results to construct entire solutions which behave as two traveling wave fronts moving towards each other from both sides of xaxis for a threespecies competition system with Lotka–Volterra type nonlinearity in a lattice. 
關鍵字：  Lattice dynamical systems, Traveling wave solutions, Asymptotic behavior, Entire solution, Lotka–Volterra model 
著作名稱：  (with ChiuYen Kao, ChihWen Shih) Absolute stability and synchronization in neural field models with transmission delays 
年度：  2016 
類別：  期刊論文 
摘要：  Neural fields model macroscopic parts of the cortex which involve several populations of neurons. We consider a class of neural field models which are represented by integrodifferential equations with transmission time delays which are spacedependent. The considered domains underlying the systems can be bounded or unbounded. A new approach, called sequential contracting, instead of the conventional Lyapunov functional technique, is employed to investigate the global dynamics of such systems. Sufficient conditions for the absolute stability and synchronization of the systems are established. Several numerical examples are presented to demonstrate the theoretical results. 
關鍵字：  Absolute stability, Synchronization, Neural field models, Delay equations 
著作名稱：  (with Harunori Monobe) On a free boundary problem for a reactiondiffusionadvection logistic model in heterogeneous environment, 
年度：  2016 
類別：  期刊論文 
摘要：  In this paper, we investigate a reaction–diffusion–advection equation with a free boundary which models the spreading of an invasive species in onedimensional heterogeneous environments. We assume that the species has a tendency to move upward along the resource gradient in addition to random dispersal, and the spreading mechanism of species is determined by a Stefantype condition. Investigating the sign of the principal eigenvalue of the associated linearized eigenvalue problem, under certain conditions we obtain the sharp criteria for spreading and vanishing via system parameters. Also, we establish the longtime behavior of the solution and the asymptotic spreading speed. Finally, some biological implications are discussed. 
關鍵字：  Free boundary problem; Reaction–diffusion–advection equation; Heterogeneous environments; Population dynamics 
著作名稱：  (with JongShenq Guo, Hiroshi Matano and Masahiko Shimojo) On a free boundary problem for the curvature flow with driving force 
年度：  2016 
類別：  期刊論文 
摘要：  We study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles. Our first main result concerns the classification of solutions; every solution falls into one of the three categories, namely, area expanding, area bounded and area shrinking types. We then study in detail the asymptotic behavior of solutions in each category. Among other things we show that solutions are asymptotically selfsimilar both in the area expanding and the area shrinking cases, while solutions converge to either a stationary solution or a traveling wave in the area bounded case. We also prove results on the concavity properties of solutions. One of the main tools of this paper is the intersection number principle, however in order to deal with solutions with free boundaries, we introduce what we call “the extended intersection number principle”, which turns out to be exceedingly useful in handling curves with moving endpoints. 
關鍵字：  curvature flow, free boundary problem, motion of planar curves, asymptotically selfsimilar, convergence, isoperimetric ratio, singularity, extended intersection number principle 
著作名稱：  (with JongShenq Guo) Dynamics for a twospecies competitiondiffusion model with two free boundaries 
年度：  2015 
類別：  期刊論文 
摘要：  To understand the spreading and interaction of twocompeting species, we study the dynamics for a twospecies competition–diffusion model with two free boundaries. Here, the two free boundaries which describe the spreading fronts of two competing species, respectively, may intersect each other. Our result shows there exists a critical value such that the superior competitor always spreads successfully if its territory size is above this constant at some time. Otherwise, the superior competitor can be wiped out by the inferior competitor. Moreover, if the inferior competitor does not spread fast enough such that the superior competitor can catch up with it, the inferior competitor will be wiped out eventually and then a spreading–vanishing trichotomy is established. We also provide some characterization of the spreading–vanishing trichotomy via some parameters of the model. On the other hand, when the superior competitor spreads successfully but with a sufficiently low speed, the inferior competitor can also spread successfully even the superior species is much stronger than the weaker one. It means that the inferior competitor can survive if the superior species cannot catch up with it. 
關鍵字：  competition–diffusion model, free boundary problem, spreading– vanishing trichotomy, population dynamics 
著作名稱：  The minimal habitat size for spreading in a weak competition system with two free boundaries 
年度：  2015 
類別：  期刊論文 
摘要：  In this paper, we focus on the dynamics for a Lotka–Volterra type weak competition system with two free boundaries, where free boundaries which may intersect each other as time evolves are used to describe the spreading of two competing species, respectively. In the weak competition case, the dynamics of this model can be classified into four cases, which forms a spreading–vanishing quartering. The notion of the minimal habitat size for spreading is introduced to determine if species can always spread. Some sufficient conditions for spreading and vanishing are established. Also, when spreading occurs, some rough estimates for spreading speed and the longtime behavior of solutions are established 
關鍵字：  Competition–diffusion model; Free boundary problem; Spreading–vanishing quartering; The minimal habitat size 
著作名稱：  (with JongShenq Guo, Yi Wang and ChinChin Wu) The minimal speed of traveling wave solutions for a diffusive three species competition system 
年度：  2015 
類別：  期刊論文 
摘要：  In this paper, we study the minimal speed of traveling wave solutions for a diffusive three species competition system. Our main concern is the linear determinacy for the minimal speed. We provide some conditions on the parameters of the competition system such that the linear determinacy is assured. The main idea is by studying the linear determinacy of the corresponding approximated lattice dynamical systems and using the discrete Fourier transform. 
關鍵字：  Three species competition system, Lattice dynamical system, Traveling wave, Minimal speed, Linear determinacy 
著作名稱：  Spreading speed and traveling waves for a twospecies weak competition system with free boundary 
年度：  2013 
類別：  期刊論文 
摘要：  In this paper, we will focus on the spreading speed for a
LotkaVolterra type weak competition model with free boundary in onedimensional habitat. Based on the comparison principle for free boundary problems, we provide some estimates of the spreading speed. Also, we deal with traveling wave solutions for the same model and show that there exists a traveling wave solution with monotone profile using a shooting method and the Schauders fixed point theorem. 
關鍵字：  LotkaVolterra model, Free boundary, Spreading speed, Traveling wave, Wave speed 
著作名稱：  (with JongShenq Guo) Traveling wave front fora twocomponent latticedynamical system arisingin competition models 
年度：  2012 
類別：  期刊論文 
摘要：  We study traveling front solutions for a twocomponent system on
a onedimensional lattice. This system arises in the study of the
competition between two species with diffusion (or migration), if
we divide the habitat into discrete regions or niches. We consider
the case when the nonlinear source terms are of Lotka–Volterra type and of monostable case. We first show that there is a
positive constant (the minimal wave speed) such that a traveling
front exists if and only if its speed is above this minimal wave
speed. Then we show that any wave profile is strictly monotone.
Moreover, under some conditions, we show that the wave profile
is unique (up to translations) for a given wave speed. Finally, we
characterize the minimal wave speed by the parameters in the
system. 
關鍵字：  Traveling front, Lattice dynamical system, Competition model, Monostable, Minimal wave speed, Wave profile 
著作名稱：  (with JongShenq Guo) Recent developments onwave propagation in 2species competitionsystems 
年度：  2012 
類別：  期刊論文 
摘要：  In this paper, we shall survey some recent results on the wave
propagation in 2species competition systems with LotkaVolterra
type nonlinearity. This includes systems with continuous and
discrete diffusion (or migration). We are interested in both
monostable case and bistable with strong competition case. Questions
on minimal speed for the monostable case, uniqueness of wave speed
and propagation failure in the bistable case, monotonicity and
uniqueness of wave profile for both cases are addressed. Finally, we
give some open problems on wave propagation in 2species competition
systems. 
關鍵字：  Lattice dynamical system, traveling wavefront, bistable, monostable, wave speed. 
著作名稱：  (with JongShenq Guo) On a free boundary problem for atwospecies weak competition system 
年度：  2012 
類別：  期刊論文 
摘要：  We study a LotkaVolterra type weak competition model with a free boundary
in a onedimensional habitat. The main objective is to understand the asymptotic behavior of two competing species spreading via
a free boundary. We also provide some sufficient conditions for spreading success and spreading failure, respectively.
Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists)
cannot be faster than the minimal speed of traveling wavefront solutions for the competition model on the whole real line without a free boundary. 
關鍵字：  LotkaVolterra model, free boundary, spreadingvanishing dichotomy, spreading speed 
著作名稱：  (with JongShenq Guo) Wave propagation for a twocomponent lattice dynamical system arising in strong competition models 
年度：  2011 
類別：  期刊論文 
摘要：  We study a LoktaVolterra type competition system with bistable nonlinearity
in which the habitat is divided into discrete niches.
We show that there exist nonmonotone stationary solutions when the migration coefficients
are sufficiently small.
Also, we prove that the propagation failure phenomenon occurs.
Finally, we focus on the traveling wave with nonzero wave speed.
By investigating the asymptotic behavior of tails of wave profiles,
we show that nonzero speed wave profiles are monotone.
Moreover, the nonzero wave speed is unique in the sense that
the wave cannot propagate with two different nonzero wave speeds. 
關鍵字：  Lattice dynamical system, stationary solution, propagation failure, traveling wave, wave speed. 
著作名稱：  (with Yuan Lou) Global dynamics of atritrophic model for twopatches with cost of dispersal 
年度：  2011 
類別：  期刊論文 
摘要：  To understand how cost of dispersal affects population dynamics, we
study a threetrophic level food chain model, proposed by DeAngelis
et al. \cite{DL}, in two patches. The system consists of one
resource species, two consumers and a top predator. The top predator
feeds on two consumers and both consumers feed on the resource.
Only consumers move between the patches, possibly with a fraction of
loss in population during the movement.
The two competing consumers are identical in every aspect except their dispersal rates
between two patches.
If two consumers have the same dispersal rate
from patch 1 to patch 2,
we completely determine the global dynamics of the model and show that
there exists an ``optimal"
dispersal rate from patch 2 to patch 1 for the consumer
such that, in terms of the theory of Adaptive Dynamics,
it is a globally evolutionarily stable strategy and also a convergent stable
strategy.
If there is a minimum dispersal speed from patch 1 to patch 2,
we are able to completely determine the evolutionarily stable strategy for dispersal between
two patches.
Our results offer insights into the evolution of dispersal in muititrophic level food chains,
e.g., how the evolution of fast or slow dispersal for the consumer species
depend upon the variation of the predation risk in the habitat.
Our result suggests that even if most individuals die during the
movement, a positive dispersal rate
can still evolve. 
關鍵字：  Dispersal, patch dynamics, food chain, cost of dispersal, optimal dispersal strategy 
著作名稱：  (with JongShenq Guo) Front propagation for a two dimensional periodic monostable lattice dynamical system 
年度：  2010 
類別：  期刊論文 
摘要：  We study the traveling wave front solutions for a twodimensional periodic
lattice dynamical system with monostable nonlinearity. We first show
that there is a minimal speed such that a traveling wave solution
exists if and only if its speed is above this minimal speed.
Then we prove that any wave profile is strictly monotone.
Finally, we derive the convergence of discretized minimal speed to
the continuous minimal speed. 
關鍵字：  Lattice dynamical system, monostable, traveling wave, wave speed, wave profile. 
著作名稱：  (with JongShenq Guo) Entire solutions for a twocomponentcompetition system in a lattice 
年度：  2010 
類別：  期刊論文 
摘要：  We study entire solutions of a twocomponent competition system
with LotkaVolterra type nonlinearity in a lattice.
It is known that this system has traveling
wave front solutions and enjoys comparison principle.
Based on these solutions, we construct some new entire solutions
which behave as two traveling wave fronts moving towards each other
from both sides of xaxis. 
關鍵字：  Lattice dynamical system, entire solution, traveling wave front, wave speed 
著作名稱：  (with JongShenq Guo) Existence and uniqueness of traveling waves for a monostable 2D lattice dynamical system 
年度：  2008 
類別：  期刊論文 
摘要：  We study traveling waves for a twodimensional lattice dynamical system
with monostable nonlinearity. We prove that there is a minimal speed
such that a traveling wave exists if and only if its speed is above this
minimal speed. Then we show the uniqueness (up to translations)
of wave profile for each given speed. Moreover, any wave profile
is strictly monotone. 
關鍵字：  Lattice dynamical system, monostable, traveling wave, wave speed, wave profile. 
著作名稱：  The optimal dispersal strategy: a twopatch model with travel loss 
年度：  2016 
類別：  期刊論文 
摘要：  The dispersal of organisms plays an important role in determining the dynamics of ecological models. Ecologically, it is of interest in understanding how dispersal strategy influences the distribution of populations. An ideal free distribution (IFD) of populations has been used to predict the distribution of organisms among patches, where a key assumption is to assume that species can move freely between patches without paying any cost. If instead one assumes that there are losses when species moves from one patch to another, then ideal free distributions may not appear. In this note,
we examine a twopatch residentmutant model with travel loss and predict the optimal dispersal strategy for resident and mutant. Moreover, such strategy which produces a nonIFD is evolutionarily stable. Some same and different features of patch models with travel loss are discussed. 
關鍵字：  Optimal dispersal strategy, patch model, travel loss, ideal free distribution 