Mathematical Biology
Org: Gerda de Vries (Alberta) and Frithjof Lutscher (Ottawa) [PDF]
 CAROLINE BAMPFYLDE, University of Alberta, Department of Mathematical &
Statistical Sciences, 632 Central Academic Building,
Edmonton, AB, T6G 2G1, Canada
Biological control through intraguild predation: what is the
most efficient control method?
[PDF] 
Rusty crayfish (Orconectes rusticus) are aggressive invaders
of the Great Lakes ecosystem. They interact with indigenous
smallmouth bass (Micropterus dolomieu) through intraguild
predation. Mature bass are predators of rusty crayfish, but predation
is gapelimited, with the largest crayfish escaping predation. These
individuals are the most fecund and compete with juvenile bass,
causing a "juvenile competitive bottleneck". We use a
stagestructured model to explore the dynamics of the interacting
species and the possible biological control of rusty crayfish by
smallmouth bass. The modelling framework can also be used to evaluate
the most efficient method of mechanical control of crayfish using
either trapping or trawling or a combination of both.
Smallmouth bass is a sport fish in many of the invaded lakes, and
lakes with smallmouth bass have high recreational value. The impact
of different fishing regulations on crayfish control can also be
assessed. We use the framework to suggest methods for socially and
costeffective control.
 JACQUES BELAIR, Université de Montréal, CP 6128, Succ. centreville,
Montréal, QC, H3C 3J7
Stability in an Epidemiological Model with Two Transmission
Rates and Application to Antibiotic Resistance in Commensal
Bacteria
[PDF] 
Drugresistant bacteria often emerge when antibiotics are employed,
including for prophylactic use in livestock feed. We derive a
deterministic compartmental model to analyze the spread of resistance
in swine population of constant size. We separate the uninfected from
the infected, dividing the latter according to the consequences of the
infection on the individual behaviour, and thus on the transmission
rate of the disease. The specificity of the model resides in the
separation of the infected individuals depending on whether there
occurs a transformation in the behaviour of either the individual or
the disease itself, thus affecting the transmission of the disease. A
global stability result is obtained using Lyapunov techniques.
Convergence to the stable equilibrium is shown to depend on the value
of a parameter associated to a reproduction number. The model is
applied to the epidemiology of bacterial resistance in commensal
bacteria, relating colonized individuals to the infected class, with
antibiotics modifying the incidence rate of resistant mutants.
The objective of such formulation is to provide epidemiologists with a
tool to analyze and control the impact of the two transmission rates
on the dynamics of the disease.
 LUCIANO BUONO, University of Ontario Institute of Technology
Delays, Oscillations and Hopf Bifurcations in Drug Delivery
Systems
[PDF] 
Drug therapies are sometimes designed to reproduce the physiological
fluctuations in normal biological agents, such as hormones, which
entails the need for a periodic, yet sustained administration. The
generation of oscillatory variations in concentrations is thus
desirable in this context. We present such a system containing
explicit time delays, in which the oscillations are generated by the
delayed response of the permeability of a membrane at the boundary of
a reaction chamber. We completely analyse the stability of the
equilibrium, and present conditions under which Hopf bifurcations are
present. We also provide further conditions for multiple mode
instabilities to occur, by the occurence of double Hopf bifurcations,
and present evidence of highly nonsinusoidal oscillations.
 ERIC CYTRYNBAUM, Department of Mathematics, University of British Columbia,
1984 Mathematics Road, room 121
Connections between a 1Dspiral wave, fast and slow pulses
[PDF] 
In this talk, I will discuss the dynamic properties of a novel
unstable 1D spiral wave that sits on the boundary between the stable
rest solution and the stable traveling pulse solution to the FHN
equations. Some connections will be made to reflections, propagation
failure and formation of spiral waves in inhomogeneous excitable
media.
 RALUCA EFTIMIE, University of Alberta, Edmonton, AB, T6G 2G1
Modelling complex spatial animal group patterns: the role of
different communication mechanisms
[PDF] 
Signal reception is essential for the formation and movement of animal
groups. I will present a onedimensional hyperbolic model for group
formation that incorporates different mechanisms for the reception of
signals emitted by group members. Numerical simulations reveal a wide
range of spatial patterns that can form. Some of these are classical
patterns, such as traveling waves, or stationary pulses. There are
also new patterns, such as breathers, ripples zigzag pulses, or
semizigzag pulses.
 KEVIN HALL, NIH, Washington DC

 FRANK HILKER, University of Alberta, Centre for Mathematical Biology,
Mathematical & Statistical Sciences, 501 CAB, Edmonton, AB,
T6G 2G1
Dynamical behavior of a population with nonlinear growth and
fatal disease
[PDF] 
We consider a population with strong Allee effect (i.e., negative
population growth at small densities) that is subject to an infectious
disease. The disease is of SI type (susceptible  infected) and
induces an additional diseaserelated mortality. This is a reasonable
model for some animal diseases with an infected period that is long
compared to the life time. Mathematically, the model is composed of
two ordinary differential equations with cubic nonlinearity. We show
that the system can have six stationary states, three of which can be
locally stable simultaneously. The bifurcation behavior is
investigated numerically and exhibits the occurrence of Hopf, fold and
homoclinic bifurcations, all of them meeting in a codimensiontwo
BogdanovTakens bifurcation point. We discuss the implications of
this surprisingly rich dynamics for management in conservation biology
and biological control.
 YU JIN, Memorial University of Newfoundland
Spatial Dynamics of a Periodic Population Model with
Dispersal
[PDF] 
This paper is devoted to the study of spatial dynamics of a class of
periodic integrodifferential equations which describe the population
dispersal process via a dispersal kernel. By appealing to the theory
of asymptotic speeds of spread and traveling waves for periodic
semiflows, we establish the existence of the spreading speed
c^{*} and the nonexistence of traveling wave solutions with the
wave speed c < c^{*}. Then for the autonomous case we use the
method of upper and lower solutions to obtain the existence of
monotone traveling waves with the wave speed c ³ c^{*}. It
turns out that the spreading speed coincides with the minimal wave
speed for monotone traveling waves.
 ANMAR KHADRA, Department of Mathematics, The University of British
Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC,
Canada, V6T 1Z2
Robust Rhythmogenesis in Endocrine GnRH Neurons via
Autocrine Regulations in a Common Pool of Extracellular
Hormone
[PDF] 
Gonadotropinreleasing hormone (GnRH) is a decapeptide hormone
secreted by GnRH neurons located in the hypothalamus. It is
responsible for the onset of puberty and the regulation of hormone
release from the pituitary. There is a strong evidence suggesting
that these GnRH neurons are intrinsically capable of generating
pulsatile and episodic neurosecretion of this hormone. However, the
underlying mechanism for the GnRHpulse generator remains obscure.
The discovery of GnRH receptors allowing GnRH to exert autocrine
regulation on its own release, led several experimentalist in NIH to
propose in 2003 a mechanism underlying this effect. We developed in
2006 a mathematical model describing the proposed mechanism, then we
extended it to explain synchrony observed in GnRH neurons by
incorporating the idea of a common pool of GnRH hormone.
In this talk, we shall present this model and analyze several aspects
of it, especially robustness. We shall show that the coupling of a
heterogeneous family of GnRH neurons will not significantly alter the
general dynamics of the pulse generator. Indeed, we shall establish
that no more than 50% of these coupled neurons must be active
participants in the process to generate pulsatility. The effects of
requirement and averaging in parametervalues will be also discussed.
Several model predictions explaining the type of behaviour observed
experimentally upon the injection of exogenous GnRH will be stated.
These results will further demonstrate the essential properties of
synchrony observed and the robustness of the model proposed.
 FRITHJOF LUTSCHER, University of Ottawa
The effect of foraging patterns on population dynamics for
central place foragers
[PDF] 
Central place foragers are individuals living in a larger population
at a central place from which they emerge to forage and to which they
return to reproduce. Examples include ants, bats, colonial seabirds,
and cave crickets. Foraging area and foraging behavior may influence
population dynamics at the central place where reproduction occurs.
Typically, deterministic models for population dynamics consider
either a nonspatial setting (e.g. ODEs or difference equations), or a
spatial setting in which the species in question can reproduce
anywhere in the domain (e.g. PDEs and integrodifference equations).
Since neither of these two frameworks is suited for a central place
forager population, we introduce a system of two equations in discrete
time, one for the spatial distribution of resources and one for the
(nonspatial) density of consumers at the central place. The two
equations are connected via a `foraging kernel' that captures the
foraging behavior of individuals.
We study the resulting dynamics in two different cases.
(1) We assume a fixed foraging behavior in time and consider
the minimal patch size required to sustain a population. We show
how different foraging behaviors give qualitatively different
population dynamics.
(2) We assume that individuals forage in such a way that the
populationlevel food intake is maximized at each time step, i.e.,
the foraging kernel depends on the resource distribution. Several
new dynamical behaviors arise with this simplistic implementation of
optimal foraging: the minimal patch size becomes zero, and different
bifurcations occur.
Typically, optimal foraging has a stabilizing effect on the population
dynamics.
 FAHIMA NEKKA, Université de Montréal
Impact of Drug Intake Variability on Therapeutic Outputs
[PDF] 
Variability in drug intake is increasingly recognized as a major
source of variability in drug response. This topic, known in human
medicine as patient compliance with drugs is an old problem, dating
back to Hippocrates. This old interest has produced an overwhelming
literature on the subject, mainly from the behavioural aspect and
patient managements. However, until recently, the topic has been
mainly descriptive and suffered from a lack of real new ideas and
breakthroughs. This has been attributed to the absence of reliable
measurement techniques, methodological flaws of compliance and lack of
a conceptual rigor. With the aim to build a theoretical framework of
drug intake variability, we have conceptually formalized compliance in
different therapeutic contexts. We have used probabilistic and
stochastic approaches to reproduce the main characteristics and
attributes of drugintake patterns and to investigate their
responsibility in the pharmacokinetic/pharmacodynamic (PK/PD)
variability. Using these approaches, we have shown that inclusion of
random drugintake features can generate a dramatic influence on the
PK/PD variability that we properly characterized.
This work is in collaboration with Dr. Jun Li.
 ALEX POTAPOV, University of Alberta, Edmonton, AB, T6G 2G1
Modeling aquatic invasions and control in a lake system:
principles and approaches
[PDF] 
Bioeconomic approach to biological invasions requires accounting for
a number of processes and combining several analysis techniques:
models of establishment, population dynamics and control from
invasion biology; transportation models for invader flow between the
lakes; costbenefit analysis and discounting from economics; and
methods of optimization and optimal control. Each side of the
problem can be considered with a different degree of detailization,
therefore it is possible to build a collection of models of
different complexity. I will describe four models and most
important results about the lake invasions, which were obtained with
the help of these models.
 ROBERT SMITH, The University of Ottawa, 585 King Edward Ave
The basic reproductive ratio: does the emperor have no
clothes?
[PDF] 
The basic reproductive ratio, R_{0}, is defined as the number of
secondary infections arising from a single individual during his or
her entire infectious period, in a population of susceptibles. This
concept is fundamental to the study of epidemiology and withinhost
pathogen dynamics. Most importantly, R_{0} often serves as a
threshold parameter that predicts whether an infection will spread.
Related parameters which share this threshold behaviour, however, may
or may not give the true value of R_{0}. We give a brief overview of
common methods of formulating R_{0} and surrogate threshold parameters
from deterministic, nonstructured models. We also review common
means of estimating R_{0} from epidemiological data. Finally, we
survey the recent use of R_{0} in assessing emerging diseases such as
SARS and avian influenza, a number of recent livestock diseases, and
vectorborne diseases malaria, dengue and West Nile Virus.
 PHILIPPE TRACQUI, CNRS, Lab. TIMCIMAG/DynaCell, IN3S, Pavillon Taillefer,
Domaine de la Merci, 38706 La Tronche Cedex, France
Proteolytic control of the mechanical switch leading to in
vitro morphogenesis of capillarylike networks: a theoretical
analysis
[PDF] 
In vivo morphogenesis of capillary networks, or angiogenesis, is
closely mimicked by in vitro models of endothelial cells (ECs)
cultured on extracellular matrices, such as fibrin biogels with
tunable stiffness. Indeed, under specific microenvironmental
conditions, randomly seeded ECs selforganize into capillarylike
structures (CLS). Traction forces exerted by ECs affect the
initiation and progression of the biogel patterning and remodeling.
Considering the welldocumented mechanosensitivity of endothelial
cells, and especially the suggested role of secreted matrix
metalloproteinases, we develop and analyze a mathematical model of
this morphogenetic process which is able to reproduce several
qualitative and quantitative features of our in vitro experiments.
The results of the theoretical analysis show how CLS result from an
autobaricdriven instability and appear for a welldefined critical
traction force that is a function of the proteolytic ECs response to
extracellular stresses. This model also provides a basis for a
theoretical analysis of the anisotropic mechanical sensing of ECs and
its functional interdependence with ECs migration and CLS formation.
We additionally illustrate how the simulated model behaviors
contribute to define a modeldriven data acquisition framework that is
necessary to increase our understanding of angiogenesis both in
physiological and pathological contexts.
 REBECCA TYSON, UBC Okanagan, 3333 University Way, Kelowna, BC, V1V 1V7
Modelling the swimming behaviour of the nematode
[PDF] 
The swimming behaviour of biological organisms is a spatiotemporal
pattern which arises through the complex interactions of the swimmer's
nervous system and musculature, and the hydrodynamics of the
surrounding fluid. The entire system is enormously complex to
describe both mathematically and through simulations. Models of
swimming have traditionally focussed on one or perhaps two aspects of
the full swimsystem, and generally are restricted to one hydrodynamic
regime: either low or highReynolds number flow. We present a brief
overview of models of swimming, and our results using a promising
modelling approach based on the immersed boundary method.
 ALLAN WILLMS, University of Guelph, Guelph, ON, N1G 2W1
A Geometric Comparison of HodgkinHuxley and MultiState
Models
[PDF] 
Multistate models of ion channel gating have been used extensively,
but choosing optimally small yet sufficiently complex models to
describe particular experimental data remains a difficult task. In
order to provide some insight into appropriate model selection, we
present some basic results about the behaviour of solutions of
multistate models, particularly those arranged in a chain formation.
Some properties of the eigenvalues of constantrate multistate models
are presented, and an expression for the product of the eigenvalues of
a coupled chain is developed in terms of those of its constituent
chains. We look at a geometric description of a threestate chain and
in particular, analyze differences between a chain equivalent to a
HodgkinHuxley model and a chain with identical rates. One
distinguishing feature between these two types of chains is that decay
from the open state in the HodgkinHuxley model is dominated by the
most negative eigenvalue while the identical rate chain displays a mix
of modes over all eigenvalues.
