# MODEMIC Seminar

. Thursday 3th December 2015 at 11h30, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Title: Prédateur généraliste et espèce invasive

Summary:

Nous étudierons un système de réaction-diffusion modélisant l’interaction entre un prédateur généraliste et une proie invasive. Contrairement aux systèmes proie-prédateur classiques, le caractère généraliste du prédateur signifie que ce dernier peut consommer des proies alternatives. Ceci lui permet de survivre en l’absence de la proie invasive ce qui lui permet de stopper et même d’inverser la propagation de cette dernière. En étudiant la dynamique asymptotique du modèle, nous verrons les différents phénomènes mis en œuvre dans ce processus.

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. Monday 21th September 2015 to Friday 25th September 2015, Supagro, 2, place Pierre Viala, Montpellier. Building 11, room 203.

Week: Ecole chercheurs “Modèles ressources consommateurs”

Summary:

see the web page (in french)

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. Friday 3th July 2015 at 14h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Tewfik Sari  (IRSTEA, UMR ITAP, Montpellier)

Title: Is patchiness always beneficial to carrying capacity ?

Summary:

For modeling the dynamics of the same species in a patchy environment, it is widely accepted to assume that each subpopulation in each patch follows a local logistic law and that the various patches are coupled by migrations. Taking the case of two patches as a simple example, a well-known result is that, in the presence of migration, the total equilibrium population can be greater than the sum of the carrying capacities. This spectacular result, somewhat paradoxical, has been widely discussed and has led to speculations about the general virtues of patchiness and dispersal, for example in the context of conservation ecology and the SLOSS question.

In this talk we will bring two contributions. Firstly, we will determine the exact conditions under which the total equilibrium population can be greater than the sum of the carrying capacities.

Secondly, we will question the legitimacy of modeling a patchy logistic population as in the standard system of two logistic equations coupled by migrations. The logistic model is often justified on phenomenological grounds. However, it can also be derived from mechanistic considerations. Depending on the mechanism being considered, we will show that the correct generalization to a patchy situation is not necessarily represented by the standard system and that the equilibrium total population can be different from that predicted by this model.

Joint work with Roger Arditi (Ecology & Evolution, Department of Biology University of Fribourg) et Claude Lobry (Equipe INRA-INRIA Modemic et Université de Nice).

The talk will be given in french.

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. Friday 3th July 2015 at 15h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Claude Lobry  (Univ. Nice & MODEMIC team)

Title: Bifurcation de cycles dans certaines équations “lentes-rapide”

Summary:

Les équations de van der Pol, de Fitzhugh-Nagumo, Rosenzweig-Mac Arthur, dans leurs versions lentes-rapides, présentent de “canards” pour certaines valeurs des paramètres. Beaucoup a été dit sur ce qui se passe lorsque la valeur à canard est associée à une bifurcation de Hopf : si \epsilon est le petit paramètre, si a est le paramètre de bifurcation , si pour a < 0 = 0  on a un foyer stable  il y a apparition d’un “petit cycle” dont la taille croit comme \sqrt(a) jusqu’à une valeur de l’ordre de \varepsilon où le cycle “explose” vers un grand cycle (de l’ordre de l’unité). A contrario il me semble que peu a été dit sur les scénario suivant : Un grand cycle limite stable apparait spontanément en entourant un cycle instable qui décroit d’abord brusquement via des solutions canard puis disparait en un foyer. Je montre que ce scénario existe pour des valeur réalistes des paramètres des équations ci dessus.

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. Wednesday 10th june 2015 at 14h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Peter Kloeden  (Goethe-Universitat, Frankfurt am Main, Germany)

Title: Numerical approximation of random and stochastic (partial) differential equations

Summary:

Higher order numerical schemes for stochastic differential equations (SODEs) can be derived systematically using stochastic Taylor expansions based on iterated applications of the Ito formula. For stochastic partial differential equations (SPDEs) there is no general Ito formula that can be used in this way.  Nevertheless higher order temporal expansions for mild solutions of SPDEs are possible using Taylor-like expansions with an idea that was first used for pathwise random ordinary differential equations (RODEs). This will be illustrated first for RODEs and then extended to SPDEs. The same  relationship between RODEs and SODES as well as RPDEs and SPDEs will be indicated as well as other issues that arise in their discretization.

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. Tuesday 9th june 2015 at 11h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Tomas Caraballo  (Dep. de Ecuaciones Diferenciales y Analisis Numerico, Univ. de Sevilla, Spain)

Title: Pullback attractors for random and non-antonomous dynamical systems: an introduction with applications

Summary:

In this talk we will introduce the basic tools for the theory of pullback attractors in order to handle both non-autonomous and random dynamical systems. We will show how these problems can be analyzed in a unified formulation thanks to the concept of cocycle. We will also emphasize on the different effects that different kind of noise can produce on the asymptotic behaviour of the solutions. Our results will be illustrated with some basic academic examples as well as some others which can be interesting from the applications point of view (such as, e.g. reaction-diffusion equations).

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. Tuesday 21th april 2015 at 16h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

M. Soledad Aronna  (IMPA, Rio de Janeiro, Brazil)

Title: Limit solutions for control systems

Summary:

For a control Cauchy problem $\dot x = f(t,x,u,v) + \sum_{\alpha=1}^m g_\alpha(x) \dot u_\alpha, \quad x(a)=\bar x$, on an interval [a,b], we propose in [2] the notion of limit solution} x that is defined  for L1 impulsive inputs  u  and for standard, bounded measurable, controls  v. Here L1 denotes the space of everywhere defined Lebesgue integrable functions. The limit solution corresponding to a control u in  L1 is itself a (everywhere defined) function of L1 and, loosely speaking, it is the limit (in some sense) of standard Carathéodory solutions associated to absolutely  continuous controls approximating u.

We investigate existence and uniqueness of the limit solutions, and  prove consistency with the classical Carathéodory solutions when u and x are absolutely continuous. We prove that, in the commutative case (i.e. when the Lie brackets $[g_{\alpha},g_{\beta}]\equiv 0$, for all $\alpha,\beta=1,\dots,m$), $x$ coincides with the  solution constructed via multiple fields’ rectification (see [4,1]).
For the generic, noncommutative case, we show that x subsumes  former concepts of solution.  In particular,  when $u$ has  bounded variation, we investigate the relation between limit solutions and graph completion solutions (see [5,3]).

Even though some specific problems are better addressed by means of special representations of the solutions,  we believe that various theoretical and practical issues call  for a unified notion of trajectory.  For instance, this is the case of optimal control problems, possibly with state and endpoint constraints, for which no extra assumptions (like e.g. coercivity, boundedness, commutativity) are made in advance.

References:

[1] M.S. Aronna and F. Rampazzo. A note on systems with ordinary and impulsive controls. IMA J. Mathem. Control Inform., 2014.
[2] M.S. Aronna and F. Rampazzo. L1 limit solutions for control systems. J. Diff erential Equations, 258(3):954-979, 2015.
[3] A. Bressan and F. Rampazzo. On di fferential systems with vector-valued impulsive controls. Boll. Un. Mat. Ital. B (7), 2(3):641-656, 1988.
[4] A. Bressan and F. Rampazzo. Impulsive control systems with commutative vector fi elds. J. Optim. Theory Appl., 71(1):67-83, 1991.
[5] R.W. Rishel. An extended Pontryagin principle for control systems whose control laws contain measures. J. Soc. Indust. Appl. Math. Ser. A Control, 3:191-205, 1965.

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Friday 27th february 2015 at 14h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Camila Romero  (CMM, Universidad de Chile, Santigo, Chile)

Title: Numerical study of the time crisis problem for prey-predator model

Summary:

In this presentation, I will study numerically the minimization of the time crisis function. This function represents the total time spent by a solution of a control system outside of a given subset of the state space. One essential feature of this problem is that the functional to be minimized is non-smooth.  I will show numerical simulations on the controlled Lotka-Volterra system for both the regularized problem and the original one. In this problem, the crisis represents a threshold above which the number of preys will typically damage the crop  (hence the control acts on the number of predators only). Simulations have been performed using a direct method (Bocop).

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. Tuesday 17th february 2015 at 16h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Title: Consideration of inhomogeneity in continuous bioreactors for the bioremeditation of water resources

Summary:

We study optimal and suboptimal control strategies for the treatment of a polluted water resource by using aside a continuous bioreactor. The control consists in choosing the inlet volumetric flow rate for filling the bioreactor with contaminated water from a considered resource (lake, reservoir, water-table…). We tackle an optimization problem which aims to minimize the time needed to reach a prescribed minimal value of contamination in the resource. Next, we study the influence of inhomogeneities of concentrations in the bioreactor, considering a system based onpartial differential equations which describe its behavior, and using an Hybrid Genetic Algorithm. We also  show that applying the optimal feedback derived for perfectly mixed bioreactor does not allow to reach the target with small diffusion parameters as it derives the bioreactor to washout (the bioreactor polluted water equilibrium with no biomass).

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. Tuesday 3th february 2015 at 16h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Thomas Koffel  (UMR Eco & Sols, Montpellier)

Title: The envelope method: merging resource competition theory with adaptive dynamics

Summary:

Classical resource competition theory is a central framework in theoretical ecology. It allows representing graphically the different competitive outcomes of a consumers resources system, among which coexistence, competitive exclusion and priority effects as functions of the external resource supplies. Its general formulation, which includes other kind of limiting factors such as parasitism or predation, led to cornerstone concepts such as the competitive exclusion principle.

Yet, it is unclear how this picture translates when the consumers can evolve by means of natural selection. Indeed, some recent studies showed that evolution can either destabilize coexistence or lead to sympatric diversification. Those works implemented evolution using adaptive dynamics, a convenient and general approach based on invasion of a rare mutant in a resident population. However, the focus on evolution was usually associated with the discarding of the main advantage of the theory, its graphical approach.

The goal of this presentation is to transpose the graphical concepts of the classical resource competition theory in the eco-evolutionary case. For the sake of clarity, those ideas will be applied to the example of the tri-trophic food chain throughout the presentation. We will first present how the notion of geometrical envelope of a family of curves enables to work with a continuum of competitors. Then, we will show the fundamental link between local and global properties of this envelope and a central concept of adaptive dynamics, the Evolutionary Stable Strategy (ESS). This will allow summarizing the different possible evolutionary outcomes of the system as functions of the external resource supplies in an evolutionary bifurcation diagram.  Each region of this diagram can be associated with its corresponding Pairwise Invasibility Plot (PIP), synthetizing their evolutionary characteristics: adaptation, stable coexistence or evolutionary branching among others. In the end, we will address the question of the role of the invasion range considered, from a local one with infinitesimal mutations to a global one associated with the “everything is everywhere” paradigm.

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. Tuesday 20th january 2015 at 16h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Pierre-Alexandre Bliman  (Inria, Rocquencourt, France and Fundação Getulio Vargas, Rio de Janeiro, Brazil)

Title: Ensuring the success of biological control of mosquito-borne diseases by bacteria Wolbachia, through almost global stabilizing feedback law

Summary:

The use of Wolbachia bacteria is a promising method currently considered to block transmission of diseases such as dengue fever, chicungunha and yellow fever, whose area is currently spreading. While full scale experiments are presently organized in Australia and Brazil, the practical aspects of the introduction of mosquitoes infected by the bacteria in a healthy population, are still a subject to be studied. It is an important issue, linked both to cost and effectiveness.
The goal of this talk is the presentation of a method aimed at reducing the number of released mosquitos, and thus the treatment cost, without jeopardizing the success of the introduction of the bacteria — something which could happen e.g. in case of underestimation of the initial population size. We provide a model describing the interactions between healthy and infected populations. Taking advantage of the fact that measurements are carried out during the whole release period, the use of control theory techniques is possible. We propose a simple feedback control law that stipulates the amount of mosquitoes to be introduced at every time instant and analyze the asymptotic behavior of the closed-loop system thus obtained. According to the information available to us, it is the first attempt to use feedback for the introduction of Wolbachia in a population of arthropods. Monotonicity techniques are key ingredients of the proofs.

Work in common with

Maria Soledad Aronna (IMPA, Rio de Janeiro, Brazil and Fundação Getulio Vargas, Rio de Janeiro, Brazil)
Flávio C. Coelho (Fundação Getulio Vargas, Rio de Janeiro, Brazil)
Moacyr A.H.B. da Silva (Fundação Getulio Vargas, Rio de Janeiro, Brazil)
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. Tuesday 6th january 2015 at 16h, Supagro, 2, place Pierre Viala, Montpellier. Building 21, room Ferguson, Level 0.

Matthew Wade (School of  Civil  Engineering & Geosciences, Univ. Newcastle, U.K.)

Title: BIOMETHANE COMMUNITY MODELLING: UNDERSTANDING THE DYNAMICS, STABILITY AND SENSITIVITY OF ANAEROBIC DIGESTION

Summary:

Anaerobic digestion as an engineered biological process has received renewed scientific interest in the last 20 years given  concerns about energy security  and climate change. Anaerobic systems are challenging to study due to their complex nature, with a biological component  that operates generally in stages under different  physicochemical conditions, giving rise to different microbial groups with their own specific characteristics and functions.

Understanding the behaviour of anaerobic digestion is reliant on finding the relationship between process inputs, system states (microbial and process characteristics) and their response to the inputs. Mathematical models that aggregate microbiological community form and function have been widely utilised to characterise  these systems. A theoretical basis for describing the process, coupled with empirical observation, allows for the development of systematic knowledge and the possibility for diagnosing real  world problems, process optimisation and control. Existing models, such as ADM1 are complex and intractable. In this work, several simplified models characterising typical substrate biomass relationships were developed based on well understood stoichiometric principles.

The goal of the work was to understand the relationship between these dynamical sys- tems and parameters, which drive their response to perturbations in the input.  Stability analysis of these simple models allows for the exploration of the limits of the system in relation to both the input or control parameters,  such as dilution rate and substrate input concentration, but also the stoichiometric parameters that are generally fixed for a specific microorganism. Although analysis of simple models, such as a two species, four dimension system, allow for a rigorous analytical approach that can determine a more general description of stability, when the system is increased to three species and six dimensions, only partial analytical solutions can be found and numerical analysis is required.

Global sensitivity analysis was employed to look for the parameters that were most critical to driving changes in the system response. This not only indicates the parameters of interest but can highlight the variables (substrate or biomass concentrations),  the location in time of their sensitivity and the nature of the change, for example a shift in the amplitude and phase of response.  Whilst all variables in the system are dependent to some extent on the model parameters, the sensitivity analysis highlights those most important in determining steady state stability.   Coupling this knowledge with bifurcation analysis can result in identifying the nature of these shifts, which may be positive (e.g. achieving enhanced performance)  or negative  (e.g.  washout of organisms from the reactor).  This ultimately leads to the question of whether the organisms themselves can be modified or evolved for better performance.

Although the work presented considers only simplified models of real-world systems, their structure allows these questions to be investigated to a greater extent. The generic nature of the analysis means that the motifs can be expanded and coupled to form more complex and descriptive models of the anaerobic digestion process, providing a viable framework to address real-world problems, applicable to biomethane processes.

Past seminars: 2014, 201320122011, 2010, 2009