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 - Si mi. ÑM +f æè M,C(s) öø + 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 (electro-magnetic, 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. n-switches, which are after a change of variable purely gradient), and also for population dynamics systems like the historical Lotka-Volterra (in dimension 2) model, which is purely Hamiltonian. Other systems, namely the Liénard family of ODEs, are mixed potential-Hamiltonian 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.

JEAN-LUC GOUZÉ, INRIA Sophia-Antipolis
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 non-linear 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 criss-cross 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 criss-cross 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 long-term survival of some or all interacting species in an ecosystem. There have been extensive investigations on uniform persistence for discrete and continuous-time 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 continuous-time 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), 388-395.

[2]
P. Magal and X-Q. Zhao, Global Attractors and Steady States for Uniformly Persistent Dynamical Systems. Submitted, 2003.

[3]
G. R. Sell and Y. You, Dynamics of Evolutionary Equations. Springer-Verlag, New York, 2002.

[4]
X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications. Canad. Appl. Math. Quart. 3(1995), 473-495.

JEAN-CHRISTOPHE POGGIALE, Université de la Méditerranée, UMR 6117, Campus de Luminy, Case 901, 13288 Marseille Cedex 9
Surge-Uptake 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 non-constant 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 surge-uptake. 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 non-resistant and resistant strains of bacterial infection. The model connects two population levels-bacteria and patients. The bacteria population is divided into non-resistant 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 non-resistant 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 non-resistant 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 fluid-immersed 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 85287-1804, 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 (continuous-time 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

Self-cycling 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.