Applied Dynamical Systems Seminar (Winter 2012, MUN)
Organizer: Dr. Yuan Yuan
Rongsong Liu (University of Wyoming, USA)
``Spatiotemporal Mutualistic Model of Mistletoes and
Birds'' (Jan. 6, 1:00pm, HH-3017,
Departmental Colloquium)
Abstract:
A mathematical model which incorporates the spatial dispersal and interaction
dynamics of mistletoes and birds is derived and studied to gain insights of
the spatial heterogeneity in abundance of mistletoes. Fickian diffusion and
chemotaxis are used to model the random movement of birds and the aggregation
of birds due to the attraction of mistletoes respectively. The spread of
mistletoes by birds is expressed by a convolution integral with a dispersal
kernel. Two different types of kernel functions are used to study the model,
one is Dirac delta function which reflects one extreme case that the spread
behavior is local, and the other one is a general non-negative symmetric
function which describes the nonlocal spread of mistletoes. When the kernel
function is taken as the Dirac delta function, the threshold condition for the
existence of mistletoes is given and explored in term of parameters. For the
general non-negative symmetric kernel case, we prove the existence and
stability of non-constant equilibrium solutions. Numerical simulations are
conducted by taking specific forms of kernel functions. Our study shows that
the spatial heterogeneous patterns of the mistletoes are related to the
specific dispersal pattern of the birds which carry mistletoe seeds.
Etienne Farcot (INRIA Sophia Antipolis Mediterranee in the Virtual Plants team, France)
``
Patterns Induced by Active Transport in Cellular Tissues'' (Jan. 11, 4:00pm, HH-3013)
Abstract:
In this work, we consider a typical model of transport of a substance in a cellular tissue, where transporter molecules accumulate on the boundaries of cells
in function of the local flux of the substance between adjacent cells. The
origin of such models is the study of organ arrangements in plants, also called phyllotaxis. It is known that the regular patterns of phyllotaxis depend
strongly on the active transport of auxin, a plant hormone. Computer simulations have shown that several variants of this active transport mechanism can lead
to realistic phyllotactic patterns. However, very few is known about the general patterning abilities of this model. As a first step, we will present some results on existence and stability of steady states with homogeneous auxin
distribution. A particular focus will be put on tissues having the shape of
a ring of cells.
Xiaoqiang Zhao (Memorial University)
``Some Remarks on Periodic Solutions for Functional
Differential Equations''
(Jan. 27, 1:00pm, HH-3017)
Abstract:
In this talk, I will show how the theory of global attractors and
steady states for uniformly persistent dynamical systems can be
used to obtain the existence of positive periodic solutions for
dissipative periodic functional differential equations of retarded
and neutral types. This result is then applied to a multi-species
competition system and some epidemic models. A note on the weak
compactness of solution maps will also be given.
Zhen Wang (Memorial University)
``Global Dynamics of A Time-Delayed Dengue Transmission Model''
(Feb. 3, 1:00pm, HH-3017)
Abstract:
In this talk, I will report our recent research on a time-delayed
dengue transmission model. We first introduce the basic reproduction number for
this model and show that it serves as a threshold parameter that predicts
whether the disease will persist or not. We further establish a set of
sufficient conditions for the existence and global attractivity of the endemic
equilibrium by the method of fluctuations. This talk is based on the joint work
with Dr. Xiaoqiang Zhao.
Yi Zhang (Memorial University)
``Traveling wavefronts of non-local reaction-diffusion models for cell adhesion and cancer invasion
''
(Feb. 10, 1:00pm, HH-3017)
Abstract:
In this talk, we will consider non-local reaction-diffusion models with integro-partial differential equation for cell adhesion and cancer invasion. We provide a valid approach to establishing the existence of traveling wavefronts via the Banach fixed point theorem when the adhesion coefficient is relatively small. Numerical simulations are presented to illustrate the main results, and comparisons of wave patterns in different parameters are demonstrated. This talk is based on the joint work with Dr. Chunhua Ou.
Jinyong Ying (Memorial University)
``Equivariant bifurcation in a symmetrical ring network with delay''
(Feb. 17, 1:00pm, HH-3017)
Abstract:
In this talk, we investigate the bifurcation of a symmetrical ring network model with delay. Firstly, we present the linear stability
by analyzing the distribution of a transcendental equation. Then, applying the center manifold theory, normal form approach and symmetrical bifurcation theory, we study the one- and two- dimensional equivariant pitchfork and Hopf bifurcations. Numerical simulation are given
to verify the theoretical results. This is a joint work with Dr. Yuan.
Hermann Brunner (MUN, Hong Kong Baptist University)
``Finite-time blow-up of solutions to semilinear parabolic
integro-differential equations''
(Mar. 2, 1:00pm, HH-3017)
Abstract:
The first part of this talk will be dedicated to a brief review of the theory of
finite-time blow-up of solutions to semilinear parabolic PDEs on bounded or unbounded
spatial domains $\Omega $. The second part of the talk will focus on semilinear
parabolic integro-differential equations where the reaction term is now nonlocal
and given a by Volterra-type memory term. I will describe the recently obtained
(jointly with Lizao Li) extension of Fujita's fundamental 1966 result for PDEs to such
nonlocal problems on $\Omega = \RR^{N}$. However, for other unbounded spatial
domains $\Omega \varsubsetneq \RR^{N}$ many key questions remain to be answered.
The talk will conclude with a short discussion of current and future work on
the computational solution of semilinear parabolic integro-differential equations.
Yuan Yuan (MUN)
``A non-nitrogen-fixing/nitrogen-fixing phytoplankton growth model with nutrient and light``
(Mar. 9, 1:00pm, HH-3017)
Abstract:
We propose a mathematical model with the growth of two kinds of phytoplankton:
non-nitrogen-fixing and nitrogen-fixing phytoplankton,
competing for light and nutrient.
We introduce the two general functional groups F and U to present the nonlinear interactions among the two types of phytoplankton and nutrient through the implicit effect of the light.
We give some sufficient conditions for the
existence of the single-species survival and the coexistence of two species survival steady states,
discuss the influence of
nutrient source, light intensity and growth, loss rates on the dynamics of the ecosystem.
Numerical results are given to illustrate the qualitative behavior of the system under different environmental conditions.
The temporal dynamical behavior have been compared with the constant and changeable water depths in an annual cycle, showing the impact of the water depth.
Yuxiang Zhang (MUN)
`` A reaction-diffusion Lyme disease model with
seasonality''
(Mar. 17, 1:00pm, HH-3017)
Abstract:
In this talk, I will report our recent research on a
reaction-diffusion Lyme disease model with seasonality. In the case
of a bounded habitat, we obtain a threshold result on the global
stability of either disease-free or positive periodic state. In the
case of an unbounded habitat, we establish the existence of the
spreading speed of the disease and its coincidence with the minimal
wave speed for the time-periodic traveling wave fronts. We also
estimate parameter values based on some published data, and use them
to study the Lyme disease transmission in Port Dove, Ontario. Our
numerical simulations are well consistent with our analytic results.
This talk is based on the joint work with Dr. Xiaoqiang Zhao.
Rui Peng (MUN)
`` Effects of Diffusion and Advection on the Principal Eigenvalue of a
Periodic-parabolic Problem with Applications ''
(Mar. 23, 1:00pm, HH-3017)
Abstract:
The principal eigenvalue is a basic concept in the field of
reaction-diffusion partial differential equations. In recent decades,
there has been a large amount of research work devoted to the
investigation of the qualitative properties of the principal eigenvalue
and its eigenfunction for second-order linear elliptic operators. As far
as the periodic-parabolic operator is concerned, however, to our best
knowledge, much less has been known for the associated principal
eigenvalue, especially when the advection term appears. The principal
eigenvalue for linear periodic-parabolic operators is important because
it not only contains the autonomous elliptic case but is interesting in
its own right when a time periodic environment is involved.
In my talk, we are concerned with the one-dimensional periodic-parabolic
eigenvalue problem. The dependence of the principal eigenvalue on the
diffusion and advection coefficients is investigated. In particular,
the asymptotic behaviors of the principal eigenvalue as the diffusion
and advection coefficients go to zero or infinity are derived. These
qualitative results are then applied to a nonlocal reaction-diffusion-
advection equation modelling the spatiotemporal evolution of a single
phytoplankton species with periodic incident light intensity.
This is joint work with Dr. Xiaoqiang Zhao.
Binguo Wnag (Lanzhou University)
``Limit Set Trichotomy and Dichotomy for Skew-Product Semiflows with
Application" (Mar. 30, 1:00pm, HH-3017)
Abstract:
In this talk, I will report Limit Set Trichotomy and Dichotomy for Skew-
Product Semiflows.
Under suitable monotone and concave assumptions, the limit set
trichotomy and dichotomy for skew-product semiflows are established . As an
application, a delayed Hopfild-type neural network model is considered.
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