


Mathematics Inspired by Biological Models
Org: Fred Brauer (UBC) and Pauline van den Driessche (Victoria) [PDF]
 JULIEN ARINO, University of Manitoba
An alternative formulation for a delayed logistic equation
[PDF] 
The logistic equation with time delay is closely linked to the
evolution of the theory of delay differential equations (DDE). Known
in the mathematical community as Wright's equation, it is a standard
example of the richness of behaviors exhibited by DDE, but also of
the problems that arise in their analysis. However, the delayed
logistic equation is seldom used in theoretical ecology. After an
introduction to the delayed form of the logistic equation, I will
discuss the reasons that lead to its shunning by the theoretical
ecology community. I will then propose an alternative formulation of
the equation taking into account survival through the maturation
process. This alternative form, of which a global analysis has been
conducted, has a totally different behavior. I will describe this
behavior, pointing out situations in which our equation seems better
suited for the description of single species dynamics with delay than
the classical DDE logistic.
This is joint work with Lin Wang and Gail Wolkowicz.
 CAROLINE BAMPFYLDE, University of Alberta, 632 Central Academic Building,
Edmonton, Alberta T6G 2G1
Perturbing a littoralzone lake community to release a
biological
[PDF] 
Rusty crayfish (Orconectes rusticus) are aggressive invaders of
the Great Lakes ecosystem. When introduced into new lakes, they drive
down native crayfish populations, disturb macrophytes, interfere with
fish recruitment, and cause the overgrazing of algae and snails.
Recently, the population density of rusty crayfish in some lakes has
far exceeded previously recorded levels. Management of this nuisance
species is necessary.
The interaction between rusty crayfish and indigenous smallmouth bass
(Micropterus dolomieu) involves a mixture of competitive and
predatorprey relationships. Juvenile smallmouth bass compete with
all life stages of the invasive rusty crayfish. However, mature
smallmouth bass are major predators of rusty crayfish. Intraspecific
interactions for rusty crayfish also include cannibalism and resource
competition.
We used mathematical and computer models to investigate the influence
of biological control of rusty crayfish by smallmouth bass. The
method is to apply perturbations to shift the dominance in a
competitive bottleneck from rusty crayfish to smallmouth bass. The
perturbations include crayfish trapping, trawling and changes to lake
fishing regulations.
Our model was developed and parameterised using long term field data
and laboratory experiments. The analysis suggests methods for
effective control. Model validation will be carried out by use of a
controlled experiment in Lake Ottawa, Michigan. We will test the
hypothesis that trawling for crayfish is sufficient for control
without changing fishing regulations. Our long term goal is to
implement the control methods in selected lakes.
 DANIEL COOMBS, University of British Columbia, Dept. of Mathematics,
Vancouver, BC V6T 1Z2
Virus competition at multiple scales
[PDF] 
Viruses compete and are subject to natural selection at multiple
levels: withincell, withinhost and withinpopulation (of hosts). We
looked at how viruses can optimally exploit their hosts and how this
behaviour may influence the most successful strategy at the
betweenhost, or epidemiological level. I will present a fairly
general way to consistently combine models of disease process and
disease spread with the goal of understanding the net selection
pressure on a model virus. The method is illustrated using a popular
model for HIV dynamics nested within a simple epidemiological model.
This is joint work with Mike Gilchrist (Tennessee).
 ERIC CYTRYNBAUM, University of British Columbia, Department of Mathematics
Finding the centerhow to solve simple geometry problems at
the cellular scale
[PDF] 
Fragments of fish melanophore cells can form and center aggregates of
pigment granules by dyneinmotordriven transport along a
selforganized radial array of microtubules (MTs). In this talk, I
will present a system of integrodifferential equations that model
pigment aggregation and MTaster selforganization and the subsequent
centering of both structures. The model is based on the observations
that MTs are immobile and treadmill, while dyneinmotorcovered
granules have the ability to nucleate MTs. Scaling arguments and
perturbation theory allow for analysis in limiting cases. This
analysis explains the mechanism of aster selforganization as a
positive feedback loop between motor aggregation at the MT minus ends
and MT nucleation by motors. Furthermore, the centering mechanism is
explained as a global geometric bias in the cell established by
spontaneously nucleated microtubules. Numerical simulations lend
additional supports to the analysis. The model sheds light on role of
polymer dynamics and polymermotor interactions in cytoskeletal
organization.
 LEAH EDELSTEINKESHET, University of British Columbia, Dept. of Mathematics
Models of actin dynamics and cell motility
[PDF] 
Actin is a biopolymer that forms a major part of the
cytoskeletonthe structure that endows shape and motility to animal
cells. I will describe the work of Ph.D. student, Adriana T. Dawes
(joint with Bard Ermentrout, and Eric Cytrynbaum) on the dynamics of
the actin cytoskeleton and its relationship to the speed of motion of
a cell. We derive a 1D model describing the density of actin
filaments and their tips. In this model, we assume that actin tips
push out the leading edge of the cell by the polymerization thermal
ratchet mechanism (proposed by Mogilner and Oster, 1996). We include
the effects of nucleation of new filaments, capping of their tips, as
well as polymerization and disassembly of the filaments to arrive at a
set of PDEs. In 1D, the model can be partly analysed in closed form
to determine when travelling wave solutions (depicting steady state
motion of a cell) exist, and how their speed depends on rate constants
and biochemical parameters. Numerical simulations extend the results
where analysis is cumbersome. We use the model to investigate the
effects of three types distinct mechanisms of filament nucleation, and
conclude that side branching best describes experimentally observed
actin density distributions.
 ROD EDWARDS, University of Victoria, Dept. of Math & Stats, P.O. Box 3045 STN CSC, Victoria, BC V8W 3P4
Stochastic feedback and beats: a generic model for circadian
rhythms
[PDF] 
Most organisms undergo circadian rhythms: at the cellular level,
protein concentrations go through 24hour cycles. These are intrinsic
(they run in the absence of light) but respond to the diurnal cycle of
sunlight. These cycles are thought to involve genetic regulatory
processestranscription and translation of proteins that affect the
expression of other genes and produce oscillations through feedback.
However, all such known `transcriptionaltranslational oscillators'
have periods of no more than 3 hours. So an important question is how
such fast, `ultradian' oscillations can produce slow `circadian' ones.
Another problem is that the particular genes and regulatory processes
involved vary from organism to organism. This poses the theoretical
question: How did circadian oscillations develop independently using
different components in different organisms? We propose a
biochemically realistic model that offers possible solutions to both
of these questions as well as allowing entrainment by light. The
mathematics is elementary but the mechanism is elegant, and some more
difficult questions arise when the inherently stochastic nature of the
gene regulation is taken into account.
 MEREDITH GREER, Bates College, Lewiston, Maine, USA
Interaction of Infectious and Noninfectious Proteins in Prion
Disease: Models, Simulations, and Steady State Study
[PDF] 
A prion is an infectious form of protein that differs from a naturally
produced protein only in its folding. Prions are thought to cause
several diseases, with BSE (Bovine Spongiform Encephalopathy) perhaps
the most widely known example. Diseases associated with prions have
very long incubation periods, are difficult to detect in all but the
latest stages, and are highly fatal. These characteristics alone make
study of prions interesting, but even more so, there is the question
of prion replication. Proteins do not possess any nucleic acid.
Without DNA or RNA, how does the structure copy itself and spread?
There is evidence that prions form polymers or aggregates, most likely
with additional stability. Some or all of these polymers attach to
the similar naturally produced protein and convert it to the
infectious variety. Polymers also split. Altogether, both the
overall quantity of infectious proteins, and the number of polymer
strands, increase. To model these phenomena, we represent prion
polymer length as a continuous structure variable. We obtain a system
of two partial differential equations modeling interaction of the
infectious and noninfectious conformations of prion protein within an
infected individual. We use this system to create numerical
simulations of disease progress within such an individual. Under some
circumstances, we can simplify to a system of three ordinary
differential equations. In the ODE case, we discuss steady states,
their stability, and relative parameter changes that affect their
viability.
 ABBA GUMEL, University of Manitoba, Dept. of Mathematics, Winnipeg,
Manitoba, R3T 2N2
Modeling the Impact of an Imperfect Vaccine and ART in
Curtailing HIV Spread
[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 HIV would require a vaccine. This talk aims at using
mathematical modelling to assess the potential impact of using an
imperfect antiHIV vaccine and ART in combatting HIV. Deterministic
models, which incorporate many of the essential biological features of
HIV (such as stagedprogression and differential infectivity) and
anticipated vaccine characteristics (e.g., "take", "degree",
"duration" and offering some therapeutic benefits) as well as the
ARTinduced evolution and transmission of drugresistant HIV strain,
would be presented and analyzed to determine thresholds conditions for
effective control of HIV within a community.
 THOMAS HILLEN, University of Alberta
Mathematical Models for Mesenchymal Motion
[PDF] 
Mesenchymal motion is a form of cellular movement that occurs in
threedimensions through tissues formed from fibre networks, for
example the invasion of tumor metastases through collagen networks.
The movement of cells is guided by the directionality of the network
and in addition, the network is degraded by proteases. I derive
mathematical models for mesenchymal motion in a timely varying network
tissue. The models are based on transport equations and their
driftdiffusion limits. It turns out that the mean drift velocity is
given by the mean orientation of the tissue and the diffusion tensor
is given by the variancecovariance matrix of the tissue orientations.
I will discuss relations to existing models and future applications.
 LEV IDELS, Malaspina UniversityCollege
Time delays in periodic harvesting
[PDF] 
To model a fish population in periodic environment we introduce a Getz
delay differential equation with a parameter which describes
population outbreaks.
This dynamical system then analyzed and the global existence of a
solution is established.
We study some harvesting problems that were posed by F. Brauer,
e.g., whether periodic variations in the model transform a
stable equilibrium to the stable periodic solution, will the stability
of equilibrium be preserved and result in stable periodic solution?
We illustrate numerically that the resulting system has a very rich
dynamic.
 MARK LEWIS, University of Alberta
Spread and persistence of competitive species in advective
flows
[PDF] 
My talk will focus on mathematics inspired by biological problems
involving multispecies competitive spread and persistence in advective
flows. The biological problems are: movement of vegetation in
response to climate change, and persistence of populations in rivers.
In the analysis of these problems, I will connect the classical
critical domain size problem with the theory of spread rates and
travelling waves. I will finish with some recent work on the role of
disease in the historical spread of competing species.
Some of the research is in collaboration with Frithjof Lutscher, Paul
Moorcroft and Alex Potapov.
 CONNELL MCCLUSKEY, Wilfrid Laurier University
Stability for a Class of Epidemic Models with Mass Action
Incidence
[PDF] 
Many epidemic models with mass action incidence can be written as a
sum of constant, linear and quadratic terms:
x¢ = K + N x + Q x_{1} x_{2} 

where K and Q are constant vectors and N is a matrix. Under
very reasonable assumptions on K, N and Q, it will be shown that
this ndimensional system has a globally asymptotically stable
equilibrium. The result resolves the asymptotic behaviour of several
models in the literature for which the global dynamics had not been
determined. The main results are obtained by the use of a Lyapunov
function.
 REBECCA TYSON, University of British Columbia Okanagan, 3333 University Way,
Kelowna, BC V1V 1V7
Recolonization of Harvested Forest stands by Tamiasciurus
hudsonicus
[PDF] 
We present a model for the population dynamics of Tamiasciurus
hudsonicus within a forest environment subject to harvesting and
regrowth. The forest is represented as a mosaic of patches at various
levels of development from harvested to mature forest. Each patch
grows over time, eventually becoming mature forest unless it is
harvested again. At the same time, each harvested patch is gradually
recolonized by squirrels. We find that the time it takes for a second
growth forest patch to be recolonized at the mature forest level is
much longer than expected. In particular, it is much longer than the
time it takes for the second growth patch to produce as many cones as
an equivalent mature forest patch. We also report that recolonization
pressure decreases squirrel populations in neighbouring patches. We
discuss reasons for these behaviours and predict how squirrel
populations are affected by different harvesting geometries.
 LIN WANG, University of Victoria
Transient Oscillations in Chemostat Models
[PDF] 
Despite the fact that the competitive exclusion principle holds in
most classical chemostat models, transient oscillations are frequently
observed in actual experiments, which cannot be explained from those
classical models. What could be the source hiding behind this
phenomenon? In this talk we give two possible sources, namely,
nonnegligible speciesspecific death rate and delay in growth. We
show how each source can give rise mathematically to transient
oscillations.
This talk is based on joint work with G. S. K. Wolkowicz and H. Xia.
 JAMES WATMOUGH, University of New Brunswick, Fredericton, NB
Disease transmission at home and abroad
[PDF] 
Most models of disease transmission make very simple assumptions about
the incidence of infection. In differential equation models these are
usually bilinear and assume a wellmixed population. Many models have
been proposed to study heterogeneities arising from age structure,
behavioural groups, stages of infection and spatial variation. More
recently, network models, in various forms, have been used to model
heterogeneities in the transmission setting. For example,
transmission may occur in a household, a hospital, a workplace, or on
a transit system. In this talk I present a simple ordinary
differential equation model for disease transmission with multiple
groups and multiple settings and formulate conditions for the spread
of the disease through a population. The assumptions lead to a model
that has no explicit spatial variable, yet still account for spatial
variation through the various transmission settings.
 JIANHONG WU, York University
Modeling Eradicating Vectorborne Diseases via Structured
Culling
[PDF] 
We derive appropriate mathematical models to assess the effectiveness
of culling as a tool to eradicate vectorborne diseases. The model,
focused on the culling strategies determined by the stages during the
development of the vector, becomes either a system of autonomous delay
differential equations with impulses (in the case where the adult
vector is subject to culling) or a system of nonautonomous delay
differential equations where the timevarying coefficients are
determined by the culling times and rates (in the case where only the
immature vector is subject to culling). Sufficient conditions are
derived to ensure eradication of the disease, and simulations are
provided to compare the effectiveness of larvicides and insecticide
sprays for the control of West Nile virus. We show that eradication
of vectorborne diseases is possible by culling the vector at either
the immature or the mature phase, even though the size of the vector
is oscillating and above a certain level.
This is a joint work with S. A. Gourley and R. S. Liu.
 XINGFU ZOU, University of Western Ontario
Existence and global attractivity of posititve periodic
solution in LotkaVolterra competition systems with deviating
argument
[PDF] 
In this talk, we consider the periodic LotkaVolterra competition
systems with deviating arguments. We present some sufficient
conditions, and sufficent and neccessary conditions for a special
case, for existence of a positive periodic solution. We also
establish some 3/2 type criteria for the global attractivity of the
positive periodic solution.
This is joint work with Xianhua Tang.

