


Differential Equations and Dynamical Systems
Org: Elena Braverman (Calgary) and Michael Y. Li (Alberta) [PDF]
 MOHAMMADREZA ANVARI, University of Saskatchewan, Department of Mathematics,
106 Wiggins Road, Saskatoon, SK S7N 5E6
Relative Dynamics in Systems Biology
[PDF] 
The aim of this work is to design a method for developing mathematical
models in order to apply on natural dynamical systems. Discussions on
biological knowledge, in terms of objects, concepts and rules, suggest
a need for mathematical models and novel methodologies to contribute
to the conceptual or theoretical framework in studying dynamics on
organisms. Through this paper, we will describe our mathematical
method, "observational modelling", which is based on system theory,
factorspace theory, dynamical systems and fuzzy logic to provide a
relational description of dynamic on genomic systems, metabolic
systems and systems biology. In the other part, we are going to apply
Schopenhauer's philosophy to the specific dynamical systems, "relative
dynamical systems", through observational modelling. In these kind of
systems the Rule Bases could have their own dynamic, or some how
modification. However, investigating the conflict between this
dynamic and original dynamic of system is the aim.
 BARUCH CAHLON, Oakland University, Dept. of Math., Rochester, MI 48309,
USA
Stability Criteria for Certain High Even Order Delay
Differential Equations
[PDF] 
In this paper we study the asymptotic stability of the zero solution
of even order linear delay differential equations of the form
y^{(2m)}(t) = 
2m1 å
j=0

a_{j} y^{(j)}(t) + 
2m1 å
j=0

b_{j}y^{(j)} (tt) 

where a_{j} and b_{j} are certain constants and m ³ 1. Here
t > 0 is a constant delay. In proving our results we make use of
Pontryagin's theory for quasipolynomials.
It is clear that with 4m independent parameters in (1.1) one cannot
expect to get regions of stability. Our goal is to derive algorithmic
type stability criteria.
 ABBA GUMEL, University of Manitoba
Dynamics Analysis of HIV Vaccine Models
[PDF] 
Since its emergence in the 1980s, the human immunodeficiency syndrome
(HIV) continues to inflict major public health and socioeconomic
burdens globally. Currently, 3446 million people live with HIV and
over 20 million have so far died of the disease. Although the use of
antiretroviral therapy (ART) has been quite effective in slowing HIV
spread in some nations, it is generally believed that the global
control of the HIV pandemic would require a vaccine. Thankfully, a
number of candidate vaccines are currently undergoing various stages
of clinical trials ... and there is a need to qualitatively analyse
their potential impact. This talk will focus on designing and
analysing HIV vaccine models, which incorporate some of the key
biological features of HIV and expected vaccine characteristics, to
assess the potential impact of an imperfect vaccine in combatting the
HIV pandemic.
 HONGBIN GUO, University of Alberta
Global Dynamics of Multigroup SIR Epidemic Models
[PDF] 
For a class of multigroup SIR epidemic models with varying
subpopulation sizes, the global dynamics are completely determined by
the basic reproduction number R_{0}. More specifically, if R_{0} £ 1, then the diseasefree equilibrium is globally asymptotically
stable; if R_{0} > 1, then there exists a unique endemic equilibrium and
it is globally asymptotically stable in the interior of the feasible
region.
 SHAFIQUL ISLAM, University of Lethbridge
Piecewise linear approximation of absolutely continuous
invariant measures for random maps
[PDF] 
Fixed points of FrobeniousPerron Operators for random maps are the
probability density functions of absolutely continuous invariant
measures for the random maps. The FrobeniousPerron equation is a
functional equation and solving this equation explicitly is possible
only in very simple cases. In this talk we describe a method of
approximating fixed point of the FrobeniousPerron Operator for
random maps. We prove the convergence of our piecewise linear
approximation method.
 DAMIR KINZEBULATOV, University of Calgary, 2500 University Drive NW, Calgary,
Alberta T2N 1N4
On Nicholson's blowflies equation with a distributed delay
[PDF] 
The dynamics of the wellknown Nicholson's blowflies model,
N¢(t) = pN(tt) e^{aN(tt)}  dN(t), \tag1 
 (1) 
where N(t) is the size of the population at time t, coefficients p, d ³ 0 are the maximal daily egg production and adult death rates, respectively, 1/a is the population size providing maximal reproduction rate and t ³ 0 is the generation time, was intensively studied in the literature in the last decades.
It is believed that models with a distributed delay (generally, time
dependent) provide a more adequate description of population dynamics
than equations with a constant concentrated delay. In this talk, we
consider the Nicholson's blowflies model with a timedependent
distributed delay. Global and local behaviour of solutions is
investigated: positiveness and persistence, global attractivity and
oscillation.
 BILL LANGFORD, University of Guelph, Ontario
Models of CheyneStokes Respiration with Cardiovascular
Pathologies
[PDF] 
CheyneStokes respiration is a periodic breathing pattern,
characterized by short intervals of deep breathing each followed by an
interval of very little or no breathing (known as apnea). This work
improves on previous compartmental models of the human
cardiorespiratory system that simulate concentration of carbon
dioxide in compartments of the cardiovascular system and the lungs.
The parameter boundary on which Hopf bifurcation gives birth to a
period oscillation has been determined. The models predict that an
increase in either the ventilationperfusion ratio or feedback gain
can give rise to stable CheyneStokes oscillations. Physiologically,
it is observed that CheyneStokes respiration is more likely to occur
in people with pathologies such as chronic heart failure or
encephalitis, or in healthy humans during acclimatization to high
altitudes or after hyperventilation. Modifications of the model to
incorporate these conditions give good agreement with the
observations.
This paper is joint with Fang Dong.
 RONGSONG LIU, York University
Spatiotemporal Patterns of Vectorborne Disease Spread
[PDF] 
There are many factors contributing to the complicated and interesting
spatiotemporal spread patterns of vectorborne diseases. Here, we
focus on two major factors: the demographic and disease ages and the
spatial movement of the disease hosts. We derive from a structured
population model a system of delay differential equations describing
the interaction of five subpopulations for a vectorborn disease with
particular reference to West Nile virus, and we also incorporate the
spatial movements by considering the analogue reactiondiffusion
systems with nonlocal delayed terms. Specific conditions for the
disease eradication and sharp conditions for the local stability of
the diseasefree equilibrium are obtained using comparison techniques
coupled with the spectral theory of monotone linear semiflows. A
formal calculation of the minimal wave speed for the traveling waves
is given.
 JAMES MULDOWNEY, University of Alberta, Edmonton
Implications of the stability of linearizations
[PDF] 
Let x = f(t) be a bounded solution of the C^{1} autonomous system
x¢ = f(x) in R^{n}. It is an exercise to show that
the omega limit set of this solution is a stable hyperbolic
equilibrium if and only if the linearized system y¢ = [(¶f)/(¶x)] ( f(t) ) y is uniformly
asymptotically stable. This talk will present similar conditions for
the omega limit set to be a stable hyperbolic equilibrium or a
homoclinic or heteroclinic cycle with certain attraction properties.
Work is joint with Michael Li.
 GERGELEY RÖST, York University, Toronto / Univ. Szeged, Hungary
Bifurcation of periodic delay differential equations
[PDF] 
We study the bifurcation of timeperiodic scalar delay differential
equations, depending on a parameter. The complete bifurcation
analysis is performed explicitly, using Floquetmultipliers, spectral
projection and center manifold reduction. The case of strong
resonance is also discussed. Numerous examples are given to
illustrate our theorems: subcritical and supercritical bifurcations
into invariant tori can be observed for many notable equations.
 SAMIR SAKER, Calgary
Periodic solutions, oscillation and attractivity of nonlinear
delay discrete survival red blood cells model
[PDF] 
In this paper, we will consider the discrete nonlinear delay survival red
blood cells model
x(n+1)  x(n) = (n)x(n) + p(n) exp 
æ è

q(n) x^{m}(n) 
ö ø

, n = 1,2,..., 

where (n), p(n) and q(n) are positive sequences of period and
m is a positive constant. By using the continuation theorem in
conincidence degree theory as well as some priori estimates we prove
that the equation has a positive periodic solution x(n). We prove
that the solutions are permanence and establish some sufficent
conditions for the prevalence of the survival cells around the
periodic solution which are oscillation criteria of the positive
solutions about x(n). Also, we give an estimation of the lower and
uper bounds of the osillatory solution and establishe some sufficient
conditions for for the nonexistence of dynamical diseases on the
population which are the global attractivity results of x(n).
 PAULINE VAN DEN DRIESSCHE, University of Victoria, Victoria, BC
Modeling relapse in infectious diseases
[PDF] 
An integrodifferential equation is proposed to model a general
relapse phenomenon in infectious diseases including herpes. The basic
reproduction number R_{0} for the model is identified and
the threshold property of R_{0} established. For the case
of a constant relapse period (giving a delaydifferential equation),
this is achieved by conducting a linear stability analysis, and
employing the LyapunovRazumikhin technique and monotone dynamical
systems theory for global results. Numerical simulations, with
parameter values relevant for herpes, are presented to complement the
theoretical results, and no evidence of oscillatory solutions is
found.
Joint work with Xingfu Zou.
 QIAN WANG, University of Alberta
Some Results On A Diffusive SIR Model
[PDF] 
The model we are interested in is given by


= d_{1} S_{xx} + L  b(x) IS  b_{1} S, 
 

= d_{2} I_{xx} + b(x) IS  (b_{2} + g) I, 
 

= d_{3} R_{xx} + gI  b_{3} R, 
 



with homogeneous Neumann boundary conditions, where S, I, R
denote the number of susceptible, infectious and recovered population,
respectively. In this talk, I will present some results on the
existence and the structure of global attractor, and the existence of
positive steady states. We prove that there exists an threshold
parameter R_{0}, whose sign determines the structure of the global
attractor. If R_{0} £ 0, the diseasefree equilibrium P_{0} is
globally stable. If R_{0} > 0, P_{0} is unstable, and a positive,
xdependent endemic equilibrium P^{*} exists. In the special case
when b(x) = constant, P^{*} is independent of x and is
globally stable.
This is a joint work with Michael Y. Li.
 YUAN YUAN, Memorial University of Newfoundland
Synchronization and Desynchronization in a Delayed Discrete
Neural Network
[PDF] 
We consider a delayed discrete neural network of two identical neurons
with excitory interactions. After investigating the stability of the
given system, we establish a new scheme and use the scheme to analyze
the possible bifurcations occurring in the model. The process from
its stable equilibrium to its multiple periodic patterns is explored
clearly. A clarification for the asymptotically
synchronous/asynchronous regions of such a system with Z_{2}
symmetry is included.
 ZINGFU ZOU, University of Western Ontario
Rich dynamics in a nonlocal population model over three
patches
[PDF] 
We consider a system of nonlinear delay differential equations that
describes the mature population of a species with agestructure that
lives over three patches. We analyze existence of nonnegative
homogeneous equilibria, stability and Hopf bifurcation from the
equilibria. In particular, by employing both the standard Hopf
bifurcation theory and the symmetric bifurcation theory of functional
differential equations, we obtain very rich dynamics for the system,
such as transient oscillations, phaselocked oscillations,
mirrorreflecting waves and standing waves. In the standard Hopf
bifurcation case, we also derive formulas for determining the
stability and the direction of the Hopf bifurcation.

