


Infinite Dimensional Dynamical Systems / Systèmes dynamiques en
dimensions infinies (Org: Xiaqiang Zhao and/et Thomas Hillen)
 STEVE CANTRELL, Department of Mathematics, University of Miami,
Coral Gables, Florida 33124, USA
Brucellosis, botflies and brainworms: the impact of edge
habitats on pathogen transmission and species extinction

Ecological interactions between species that prefer different habitat
types but come into contact in edge regions at the interfaces between
habitat types are modeled via reactiondiffusion systems. The primary
sort of interaction described by the models is competiton mediated by
pathogen transmission. The models are somewhat novel because the
spatial domains for the variables describing the population densities
of the interacting species overlap but do not coincide. Conditions
implying coexistence of the two species or the extinction of one
species are derived. The conditions involve the principal eigenvalues
of elliptic operators arising from linearizations of the model system
around equlibria with only one species present. The conditions for
persistence or extinction are made explicit in terms of the parameters
of the system and the geometry of the underlying spatial domains via
estimates of the principal eigenvalues. The implications of the models
with respect to conservation and refuge design are discussed.
 YUMING CHEN, Wilfrid Laurier University
Pattern formation in isolated neurons

We consider delay differential equations describing the dynamics of isolated
neurons. First, we review some existing results on the existence and stability
of equilibria and periodic solutions, which are closely related to pattern
formation. Then, one result on pattern formation in periodic environment is
improved by using the theory of coincidence degree.
 CHRIS COSNER, University of Miami, Coral Gables, Florida 33124, USA
Multiple competitive reversals via changing boundary
conditions in a reactiondiffusion model

This research is in collaboration with Robert Stephen Cantrell and Yuan
Lou. We consider a diffusive LotkaVolterra competition model on a
bounded domain and ask how varying the boundary conditions from Neumann
through Robin to Dirichlet affects the dynamics of the model. We show
that by changing the boundary conditions the predictions of the model
can be changed from dominance by the first competitor and extinction of
the second to coexistence, then to dominance by the second competitor
and extinction of the first, then back to coexistence, then back to
dominance by the first competitor, and this switching can occur several
times. To construct examples of this phenomenon we start with a
situation where the two species are identical and the coefficients are
constant and then perturb the system with an appropriate spatially
heterogeneous term in one of the growth rates. The mathematical
methods underlying the analysis include bifurcation/continuation theory
and various more or less classical ideas from the theory of
differential equations.
 O. DIEKMANN, Utrecht University Mathematisch Instituut, NL 3508
TA Utrecht, The Netherlands
Quasilinear population models

Whenever the development of an individual is affected by a quantity
such as food which, in turn, is affected by the individuals
(consumption!), the resulting population model is quasilinear. If we
cut the feedback loop, solve a parameterized nonautonomous linear
problem and then restore the loop, we obtain a fixed point problem that
can be solved under appropriate assumptions. This methodology yields a
constructive definition of an infinitedimensional dynamical system.
The qualitative theory of such systems is still very much in its
infancy and accordingly there are many open problems. The lecture is
based on joint work (over a large number of years) with Mats
Gyllenberg, Haiyang Huang, Markus Kirkilionis, Hans Metz and Horst
Thieme.
 YIHONG DU, University of New England, School of Mathematics, Armidale, New South
Wales 2351, Australia
Realization of prescribed patterns in certain population models

We show that for certain population models, heterogeneous environments
depending on a small parameter can be designed so that as the small
parameter goes to zero, the distributions of the population exhibit
clear prescribed spatial patterns in the environment. This will be
demonstrated in the single species logistic model and the two species
LotkaVolterra competition model.
 MAREK FILA, Comenius University, Bratislava, Slovakia
Growup on the boundary

We study the asymptotic behavior of positive solutions of a semilinear
parabolic equation with a nonlinear boundary condition. This problem
admits a unique stationary solution which is not bounded and attracts
all positive solutions. We find their growth rate at the singular
point on the boundary. This is a joint work with J. J. L. Velazquez
and M. Winkler.
 K.P. HADELER, University of Tuebingen, Germany
Moving populations with sedentary states

Traditionally epidemic spread in space is modeled either by diffusion
equations or by contact distributions. Evidently contacts between
individuals at different positions require that individuals move. Here
it is shown that contact models can be derived as limiting cases from
diffusion models with two levels of spatial mixing by appropriate
scaling of the parameters. This approach can be used in a much wider
range of problems: Many populations, human and animal, show two rather
distinct migration patterns, not only humans move around in their
neighborhood and occasionally travel over long distances. In an
appropriate scaling the neighborhood shrinks to a point, individuals
switch between a sedentary and a migrating state. A key example showing
the effects of the migration pattern together with the nonlinearity
is Fisher's equation with a resting state, i.e., a scalar
reaction diffusion equation coupled to an ordinary differential
equation, formally similar to the FitzhughNagumo system but with very
different properties.
 MARK LEWIS, Department of Mathematical and Statistical
Sciences, University of Alberta, Edmonton, Alberta T6G 2G1
Pattern formation via scentmarking

In this talk I will propose a model for aggregating individuals through
a random walk with bias toward scent marks which are made the
individuals. The probability density function for an individual is
given as the solution to a coupled system of partial/ordinary
differential equations. This model differs from chemotaxis equations,
where the scent marks can move. The analysis of the system, via
application of an energy method, leads to distinct aggregations with
abrupt edges. Applications will be made to the formation of
territories, and connections will be made to ecological models for
aggregating populations (e.g. Turchin 1989, Journal of Animal
Ecology 58: 75100).
 FRITHJOF LUTSCHER, University of Alberta, Edmonton, Alberta T6G 2G1
From Langevin's equation to biological application

The Langevin equation for individual movement and the corresponding PDE
for the (probability) density, known as the Kramers equation, have been
used widely in the realm of physics. In biological modeling, however,
the location jump process and the corresponding diffusion equation are
the most common tools. In this talk, we briefly discuss the underlying
assumptions of both approaches to conclude that the Langevin approach
might be more appropriate in many cases in biological modeling. We then
use scalings to find conditions under which the simpler diffusion
equation approximates the more complicated Kramers equation. We then
prove an approximation theorem using the moments of the Kramers
equation. Next, we introduce a birthdeath process into Kramers
equation to describe population growth and spread. We show that the
resulting reaction Kramers equation (analogous to reaction diffusion
equation) has a unique global solution under some conditions on the
reaction term. Finally, we apply the modeling framework to chemotaxis.
We show that a certain class of chemotaxis equation has the same form
as the well known VlasovPoissonFokkerPlanck system from statistical
mechanics, and that the moment closure procedure introduced above leads
to the classical PatlakKellerSegel model. This is joint work with
K.P. Hadeler and T. Hillen.
 PETER POLACIK, University of Minnesota, Minneapolis, Minnesota 55455, USA
On global solutions of semilinear parabolic equations on
R^{N}

We exhibit several interesting properties of solutions of semilinear
parabolic equations on R^{N}. Even though the solutions we examine
have compact trajectories and decay to zero at spatial infinity
uniformly with respect to time, they behave much differently from
solutions of the Dirichlet problem on bounded domains. Issues to be
discussed include quasiconvergence (convergence to a set of equilibria)
and examples without asymptotic symmetrization.
 MARY PUGH, Department of Mathematics, University of Toronto,
Toronto, Ontario M5S 3G3
Blowingup exact solutions of longwave unstable thin
film equations

Longwave unstable thin film equations
h_{t} = (h^{n} h_{xxx})_{x}  B (h^{m} h_{x})_{x} 

are a fourthorder analogue of the the semilinear heat equation. A
"reaction" term destabilizes a "diffusion" term, allowing for a
competition between effects. For n=1, Bertozzi and Pugh proved for
the critical and supercritical cases of the equation that the initial
value problem can blow up in finite time. Witelski, Bertozzi, and
Bernoff have done extensive computational and asymptotics work
suggesting this blowup is selfsimilar. Here, I present exact
solutions for the critical case that blow up in a selfsimilar manner.
This is joint work with Dejan Slepcev (University of Toronto).
 BRIAN SLEEMAN, University of Leeds, Leeds LS2 9JT, United Kingdom
A reinforced random walk model of tumour angiogenesis

It is well accepted that the growth of a tumour is dependent on its
ability to induce the growth of new blood vessels; a process called
angiogenesis. Hopes have been raised that an antiangiogenic
treatment may be effective in the fight against cancer. We formulate,
using the theory of reinforced random walks, an individual cellbased
model of angiogenesis. The antiangiogenic potential of angiostatin, a
known inhibitor of angiogenesis, is also examined.The capillary
networks predicted by the model are in good qualitative agreement with
experimental observations.
 MOXUN TANG, Michigan State University, East Lansing, Michigan 48824, USA
Turing patterns in the CIMA reaction diffusion system

Though Alan Turing predicted in 1952 that chemicals can react and
diffuse in such a way to destabilize a homogeneous stationary state,
and result in inhomogeneous spatial patterns, the first experimental
evidence for Turing structures was only observed in 1990 in the
chloriteiodidemalonic acidstarch (CIMA) reaction. In this talk I
will describe some fundamental properties of Turing patterns, through
our mathematical analysis for the LengyelEpstein model, a twovariable
reaction diffusion system which captures the crucial feature of the
reaction.
 HORST THIEME, Arizona State University
Compact attractors for a class of nonlinear evolution equations

In the present paper we consider the nonlinear evolution equation
u¢+Au ' G(u), where A: D(A) Í X® X is maccretive
with (I+lA)^{1} compact for some l > 0, and
G:[`(D(A))]® X is continuous, and we prove that the orbit
{u(t); t Î R_{+}} is relatively compact if and only if u
is uniformly continuous, and both u and G °u are bounded on
R_{+}. In the same spirit, we derive conditions for orbits of
bounded sets to have compact attractors. Some consequences and an
example from agestructured population dynamics illustrate the
effectiveness of the abstract result.
 JUN CHENG WEI, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Ground state solutions for the GiererMeinhardt system

In this talk, I will report recent progress towards understanding
the ground state solutions for the following GiererMeinhardt system in
R^{N}, N ³ 2:

ì ï ï í
ï ï î

e^{2} Du u+ 
u^{p}
v^{q}

=0 
 
 
u, v > 0 ; u, v ® 0 as x ® +¥, 




where p, q,m satisfy the following condition:
p > 1, q > 0, m > 1, 
qm
p1

> 1. 

When N=2, we construct multibump solutions concentrating on regular
kpolygons, or concentric polygons, or honeycomb. The distance
between bumps are of eloglog[ 1/(e)]. When N ³ 2, we construct solutions concentrating on an (N1)sphere. (Joint
work with M. del Pino, M. Kowalczyk, W.M. Ni)
 YINGFEI YI, Georgia Institute of Technology
On almost automorphic pattern formation and spatial
chaos

Almost automorphy is a notion first introduced by S. Bochner in 1955 to
generalize the almost periodic one. It is proven to be a fundamental
notion in characterizing multifrequency phenomena and their generating
dynamical complexity. This lecture will discuss the existence of
almost automorphic spatial dynamics in lattice differential equations
and their role played in the onset of the pattern formation and spatial
chaos.
 XIAOQIANG ZHAO, Memorial University of Newfoundland
Saddle point behavior for monotone semiflows and
reactiondiffusion models

The saddle point behavior is established for monotone semiflows with
weak bistability structure and then these results are applied to three
reactiondiffusion systems modelling manenvironmentman epidemics,
single loop positive feedback and two species competition,
respectively.
 XINGFU ZOU, Memorial University, St. John's, Newfoundland A1C 5S7
Periodic solutions of neutral functional differential equations

We consider periodic neutral functional differential equations. By
combining the theory of monotone semiflows generated by neutral
functional differentail equations with Krasnosel'skii's fixed point
theorem, we establish sufficient conditions for existence, uniqueness
and global attractivity of a periodic solution of the equation. We also
apply the results to a concrete neutral equation that models
singlespecies growth, the spread of epidemics, and the dynamics of
capital stocks in peirodic environment.

