


SS16  Biologie mathématique / SS16  Mathematical Biology Org: G. Wolkowicz (McMaster)
 MOSTAFA ADIMY, Université de Pau, Département de Mathématiques, 64000
Pau, France
Stability and Hopf Bifurcation in a Mathematical Model of
Pluripotent Stem Cell Dynamics

We study a mathematical model describing the dynamics of a pluripotent
stem cell population involved in the blood production process in the
bone marrow. This model is a differential equation with a time
delay. The delay describes the cell cycle duration. We show that the
delay can destabilize the system. In particularly, it is shown that
Hopf bifurcations can occur.
 BEDDR'EDDINE AINSEBA, Université Victor Segalen Bordeaux 2
Control of age structured population dynamics models

This talk is dedicated to the control of age dependent population
dynamics problems. We shall recall classical results on the subject
and will give new results concerning controllability when the controls
acts on small age classes and on a small part of the spatial domain.
 JULIEN ARINO, McMaster University, 1280 Main Street West, Hamilton,
Ontario L8S 4K1, Canada
Propagation of diseases in patch populations

We justify the introduction of metapopulation epidemic models by a
brief description of the various cases that lead to fragmented,
spatially heterogeneous populations. We then present recent models and
results, both by the author and by others, that describe such
populations.
 PIERRE AUGER, Université Claude Bernard Lyon 1, UMR CNRS 5558
Aggregation of variables in population dynamics models

Ecological systems are composed with different levels of
organization. Usually, one considers the individual, population,
community and ecosystem levels. These levels of organization
correspond to different levels of observation of the system with
different time and space scales. Therefore, the dynamics of the
complete system is the result of the coupled dynamical processes that
take place in each of its levels of organization at different time
scales. Aggregation methods take advantage of these different time
scales and are aimed to obtain a reduced model from a complete and
detailed model that governs a few global variables at the slow time
scale. We present the example of a host parasitoïd spatial model
with fast migration between patches. We use aggregation methods to
obtain a model governing the total host and parasitoïd densities of
the spatial network. We present numerical simulations of the complete
and aggregated models. When the migration is fast, we show that the
dynamics of the reduced model is qualitatively equivalent to the
dynamics of the complete model.
 JACQUES DEMONGEOT, IUF & UJF, Faculty of Medicine of Grenoble, 38700
La Tronche, France
Polynomial Hodge decomposition in modelling the
morphogenesis

After a presentation of morphogenetic problems involving morphogen
diffusion and cell differentiation, proliferation and migration
processes (gastrulation, tree growth, ...), we propose a general
operator taking into account these different mechanisms:
¶M/¶t = hDM  S_{i} m_{i}. ÑM +f 
æ è

M,C(s) 
ö ø

+ g(M), 

with Neumann conditions on the frontiers of the definition domain.
In the equation above, the proliferation term f ( M,C(s) )
is new and depends on the local curvature of the growing boundary.
The drift depends on external fields (electromagnetic, gravitational,
chemical, ...) and the diffusion is classical. The last reaction
term g(M) can be made explicit in terms of a polynomial Hodge
decomposition with a potential flow plus a Hamiltonian part.
Such a decomposition is afterwards proved for classical morphogenetic
systems (e.g. nswitches, which are after a change of variable
purely gradient), and also for population dynamics systems like the
historical LotkaVolterra (in dimension 2) model, which is purely
Hamiltonian. Other systems, namely the Liénard family of ODEs, are
mixed potentialHamiltonian and it is possible to give explicitly at
any order the polynomials of both the gradient and the conservative
parts. These polynomials serve to define algebraic closed curves
approximating the attractors of the systems.
We show that such a decomposition is practically very useful, the
parameters appearing in the gradient part being responsible for the
amplitude modulation of the asymptotic behavior (when it is a limit
cycle), those of the Hamiltonian part being responsible for the
frequency modulation.
Finally, we give two specific applications: one devoted to the plant
growth (normal and pathologic) and the other to the gastrulation
morphogenesis.
 JEANLUC GOUZÉ, INRIA SophiaAntipolis
Robust control for growth models in the chemostat

For different models of cell growth in the chemostat (constant yield
model, variable yield model, ...), we consider the problem of
stabilizing (with some adjustable rate) uncertain (unknown) models
around a given value of substrate. The dilution rate is the
controlled input. Errors in the measured outputs are also considered.
 HASSAN HBID, Département de Mathématiques, Faculté des Sciences,
Université Cadi Ayyad, Marrakech, Maroc
On some mathematical models arising in marine populations
dynamics

Mathematical models arising in marine populations will be presented.
The mathematical formulation of these models has the form of
nonlinear ordinary and partial differential equations, which involve
interesting mathematical questions. In my talk, I will address some
of these questions.
The first model describes the evolution of the distribution of larvae
of the sardine in the Moroccan Atlantic coast with respect to the
distribution of phytoplankton. The model consists of systems of
partial and ordinary differential equations describing at the same
time the evolution of the larvae population, which is structured by
the size, and the evolution of the phyto and zooplankton populations
taking into account the upwelling in the region.
The second model deals with the migration of larvae in the water
column. A mathematical analysis of this model will be discussed.
 MICHEL LANGLAIS, Université Victor Segalen Bordeaux 2, 146 rue Léo Saignat,
33076 Bordeaux Cedex, France
A mathematical model for crisscross and indirectly
transmitted disease

We study the global existence and large time behavior to a reaction
diffusion system coupled with an ordinary differential equation
modeling crisscross transmission between two species and indirect
transmission via a contaminated environment of an epidemic disease.
 PIERRE MAGAL, University of Le Havre
Global Attractors and Uniform Persistence

Uniform persistence is an important concept in population dynamics
since it characterizes the longterm survival of some or all
interacting species in an ecosystem. There have been extensive
investigations on uniform persistence for discrete and continuoustime
dynamical systems. A natural question is about the existence of
"interior" global attractors and "coexistence" steady states for
uniformly persistent dynamical systems. The existence of interior
global attractors was addressed by Hale and Waltman [1], and the
existence of coexistence steady states was investigated by Zhao [4].
In Magal and Zhao [2], we use a weaker notion of global attractors
than in [1] and [4]. We first obtain weaker sufficient conditions for
the existence of interior global attractors for uniformly persistent
dynamical systems, and hence generalize the earlier results on
coexistence steady states. We will also present two examples
(corresponding to discrete and continuoustime semiflows) that admit
global attractors in this weaker sense, but not in the usual sense of
[1] and [4]. By using these two examples, we also give an affirmative
answer to an open question presented by Sell and You [3].
References
 [1]

J. K. Hale and P. Waltman,
Persistence in infinite dimensional systems.
SIAM J. Math. Anal. 20(1989), 388395.
 [2]

P. Magal and XQ. Zhao,
Global Attractors and Steady States for Uniformly Persistent
Dynamical Systems.
Submitted, 2003.
 [3]

G. R. Sell and Y. You,
Dynamics of Evolutionary Equations.
SpringerVerlag, New York, 2002.
 [4]

X.Q. Zhao,
Uniform persistence and periodic coexistence states in
infinitedimensional periodic semiflows with applications.
Canad. Appl. Math. Quart. 3(1995), 473495.
 JEANCHRISTOPHE POGGIALE, Université de la Méditerranée, UMR 6117, Campus de
Luminy, Case 901, 13288 Marseille Cedex 9
SurgeUptake and Temporal Heterogeneity: a solution for the
Plankton Paradox

In the 1930s appeared the first empirical works highlighting the
competitive exclusion principle. This principle claims that, in a
homogeneous environment, the number of species cannot exceed the
number of resources. On the basis of this principle, Hutchinson, in
1961, defined the "Plankton Paradox". It means that the large
phytoplanktonic biodiversity encountered in a marine environment, for
example, exceeds the number of limiting resources. A large number of
theoretical works suggested various hypotheses to explain this
paradox. Among them is the temporal heterogeneity of the availability
of resources, which could be a consequence of the turbulence, for
instance. Some usual models (Monod, Droop), in a chemostat environment
with periodic input, have been used to study the effect of a
nonconstant input on the competition. The result is that the range of
parameters which lead to coexistence is so small that it is not
biologically relevant. We consider this problem with a model adapted
to the scale of phytoplanktonic cells in order to take account the
individual processes of surgeuptake. Then we discuss the effect of
these processes and temporal heterogeneity on the competition
dynamics.
 SHIGUI RUAN, Dalhousie University and University of Miami
Asymptotic Behavior in Nosocomial Epidemic Models with
Antibiotic Resistance

We analyze a model of an epidemic in a hospital setting that
incorporates antibiotic nonresistant and resistant strains of
bacterial infection. The model connects two population
levelsbacteria and patients. The bacteria population is divided
into nonresistant and resistant strains. The bacterial strains
satisfy ordinary differential equations describing the recombination
and reversion processes producing the two strains within each infected
individual. The patient population is divided into susceptibles,
infectives infected with the nonresistant bacterial strain, and
infectives infected with the resistant bacterial stain. The infective
classes satisfy partial differential equations for the infection age
densities of the two classes. We investigate the asymptotic behavior
of the solutions of the model with respect to three possible
equilibria:
(1) extinction of both infective classes,
(2) extinction of the resistant infectives and endemicity of
the nonresistant infectives, and
(3) endemicity of both infective classes.
Based on joint work with Erika D'Agata, Pierre Magal, and Glenn
Webb.
 HAL SMITH, Arizona State University
Gene transfer in Biofilms: mathematical models and implications

The talk will survey recent joint work with several collaborators on
modeling gene transfer in a fluidimmersed biofilm in both chemostat
and flow reactors. Such modeling is important for a number of
reasons. For example, exchange of plasmid among bacteria in the
mammalian gut may be a significant factor in the spread of
antibiotic resistance. Furthermore, it has been found that genetic
exchange can spread genes for enhanced biofilm forming capability.
 HORST THIEME, Arizona State University, Department of Mathematics and
Statistics, Tempe, AZ 852871804, USA
Spatially implicit metapopulations with discrete patch size
structure

A class of spatially implicit metapopulation models is presented which
are formulated in terms of the number of patches which carry a certain
number of individuals. This type of models which are structured
versions of the Levins metapopulation model allows populated patches
to become empty in finite time. Though the models are completely
deterministic, they share many features with stochastic processes
with continuous time and discrete state (continuoustime birth and
death chains), like being a system of infinitely many differential
equations. Results are presented concerning the existence of
solutions, extinction, persistence, and permanence of the
metapopulation, and the characterization of patch emigration
strategies which maximize the basic reproduction ratio of the
metapopulation.
This is joint work with Maia Martcheva.
 GAIL WOLKOWICZ, McMaster University, Department of Mathematics and
Statistics, 1280 Main Street West, Hamilton, Ontario L8S 4K1,
Canada
Nutrient Driven Self Cycling Fermentation in the Case of
Inhibition at High Concentrations

Selfcycling fermentation is a computer aided process used for
culturing microorganisms. There are a wide variety of potential
applications, including sewage treatment, toxic waste cleanup, the
production of antibiotics, and the examination of cell evolution.
After describing the process, a basic model of growth will be
formulated in terms of a system of impulsive differential equations.
The response function describing nutrient uptake and conversion of
nutrient to biomass will be assumed to be a unimodal function,
allowing for the nutrient to be growth limiting at low concentrations
and inhibitory at sufficiently high concentrations. Implications of
the analysis for operating the fermenter successfully as well as
efficiently will be discussed.
This represents joint work with Robert Smith and Guihong Fan.
 HUAIPING ZHU, York
Modelling the West Nile virus among birds and mosquitoes

We build a set of differential equations to model the transmission of
WNv among mosquitos and birds. Some analytical and numerical results
will be presented.

