Stabilizing effect of cannibalism in a two stages population model

International audienceIn this paper we build a prey-predator model with discrete weight structure for the predator. This model will conserve the number of individuals and the biomass and both growth and reproduction of the predator will depend on the food ingested. Moreover the model allows cannibalism which means that the predator can eat the prey but also other predators. We will focus on a simple version with two weight classes or stage (larvae and adults) and present some general mathematical results. In the last part, we will assume that the dynamics of the prey is fast compared to the predator's one to go further in the results and eventually conclude that under some conditions, cannibalism can stabilize the system: more precisely, an unstable equilibrium without cannibalism will become almost globally stable with some cannibalism. Some numerical simulations are done to illustrate this result

Workshop on Raising Data using the RDBES and TAF (WKRDBESRaiseTAF; outputs from 2022 meeting)

41 páginasThe Workshop on Raising Data using the RDBES and TAF (WKRDBES-Raise&TAF) met online (26–30 of September 2022) to evaluate the use of the Regional Database and Estimation System (RDBES) format to reproduce the 2022 InterCatch input and output, identifying a Transparent Assessment Framework (TAF) structure to organize the intermediate steps and to propose standardized output formats. The main outcomes of WKRDBES-Raise&TAF were: · RDBES provides sufficient support for current national estimation protocols. However, some minor issues were reported that hampered an exact reproduction of the estimates. Therefore, adaptations of the data model should not be excluded completely. · All the input to stock assessment that InterCatch currently provides, could be reproduced. The participants started from the current stock extracts that can be downloaded from InterCatch. · A workflow was proposed with a national TAF repository for each country, a stock estimation repository and a stock assessment repository. The intermediate output of those repositories will be stored in an ‘intermediate output database’ and depending on the user role, you will get access to the relevant stages in this workflow. · The following requirements for the standard output formats were defined: they cannot be more restrictive than the InterCatch input and output format; they should present measures of uncertainty and sample sizes (for national estimates) and should have a configurable domain definition (for national estimates). Despite those successful outcomes, the current plan for transition to an operational system was concluded to be too optimistic. WKRDBES-Raise&TAF therefore recommends to the Working Group on Governance of the Regional Database and Estimation System (WGRDBESGOV) to revise the roadmap and allow RDBES to be in a test phase also for 2023. WKRDBES-Raise&TAF felt the need to test the proposed workflow on a small scale and therefore recommends to the WGRDBESGOV to arrange a workshop where two stocks (pok.27.3a46 (Saithe (Pollachius virens) in Subareas 4, 6 and Division 3.a (North Sea, Rockall and West of Scotland, Skagerrak and Kattegat) and wit.27.3a47d (Witch (Glyptocephalus cynoglossus) in Subarea 4 and Divisions 3.a and 7.d (North Sea, Skagerrak and Kattegat, eastern English Channel)) will be set up to go through the whole flow.Peer reviewe

Modélisation mathématique structurée en taille du zooplancton

version non finaleThe aim of this thesis is the formulation and the study of some prey-predator models to describe phytoplankton and zooplankton population dynamics, with size-structure of the zooplankton. This study was mainly motivaded by the data collected by the Laboratoire d'Oceanographie de Villefranche-Sur-Mer, and especially the evolution since 1966 of the zooplankton size-spectra. In a first part we show data and build different models with either a continuous size-structure, leading to a partial differential equation coupled to an ordinary differential equation, or a discrete structure corresponding to different develop- ment stages and leading to a system of ordinary differential equations. A mathematical study is done for some particular cases of these models (equilibria and their stability, stabilization of an equilibrium using a positive control). Cannibalism being common in zooplankton community, we also study it, particularly in a two stages model. We show that cannibalism can stabilize the dynamics or even can be an evolutionary stable strategy. Finally we try to compare numerically models to data and observe some qualitative similaritiesL'objet de cette thèse est la formulation et l'étude de modèles proie-prédateur avec une structure en taille du prédateur, afin de décrire les populations de phytoplancton et zooplancton. Cette étude a été motivée par les données collectées par le Laboratoire d'Océanographie de Villefranche-Sur-Mer dont l'évolution depuis 1966 du spectre de taille du zooplancton dans la baie de Villefranche. Dans une première partie nous présentons les diverses données collectées et proposons différents modèles dans un cadre assez général, ayant soit une structure continue donnant lieu à une équation aux dérivées partielles couplée avec une équation différentielle ordinaire, ou soit une structure discrète, pouvant correspondre à des stades de développement et donnant lieu à un système d'équations différentielles ordinaires. Ensuite une étude mathématique de ces modèles est faite pour certains cas particuliers (stabilité des équilibres, stabilisation d'un équilibre par un contrôle positif). Le cannibalisme étant présent au sein du zooplancton, nous mettons l'accent sur son étude, notamment sur un modèle comprenant deux classes de taille de prédateurs. Nous montrons que le cannibalisme peut stabiliser la dynamique ou encore être une stratégie évolutionnairement stable. Finalement nous tentons de confronter numériquement ces modèles aux données : les simulations donnent des résultats qualitativement proches des observations

Modélisation mathématique structurée en taille du zooplancton

version non finaleThe aim of this thesis is the formulation and the study of some prey-predator models to describe phytoplankton and zooplankton population dynamics, with size-structure of the zooplankton. This study was mainly motivaded by the data collected by the Laboratoire d'Oceanographie de Villefranche-Sur-Mer, and especially the evolution since 1966 of the zooplankton size-spectra. In a first part we show data and build different models with either a continuous size-structure, leading to a partial differential equation coupled to an ordinary differential equation, or a discrete structure corresponding to different develop- ment stages and leading to a system of ordinary differential equations. A mathematical study is done for some particular cases of these models (equilibria and their stability, stabilization of an equilibrium using a positive control). Cannibalism being common in zooplankton community, we also study it, particularly in a two stages model. We show that cannibalism can stabilize the dynamics or even can be an evolutionary stable strategy. Finally we try to compare numerically models to data and observe some qualitative similaritiesL'objet de cette thèse est la formulation et l'étude de modèles proie-prédateur avec une structure en taille du prédateur, afin de décrire les populations de phytoplancton et zooplancton. Cette étude a été motivée par les données collectées par le Laboratoire d'Océanographie de Villefranche-Sur-Mer dont l'évolution depuis 1966 du spectre de taille du zooplancton dans la baie de Villefranche. Dans une première partie nous présentons les diverses données collectées et proposons différents modèles dans un cadre assez général, ayant soit une structure continue donnant lieu à une équation aux dérivées partielles couplée avec une équation différentielle ordinaire, ou soit une structure discrète, pouvant correspondre à des stades de développement et donnant lieu à un système d'équations différentielles ordinaires. Ensuite une étude mathématique de ces modèles est faite pour certains cas particuliers (stabilité des équilibres, stabilisation d'un équilibre par un contrôle positif). Le cannibalisme étant présent au sein du zooplancton, nous mettons l'accent sur son étude, notamment sur un modèle comprenant deux classes de taille de prédateurs. Nous montrons que le cannibalisme peut stabiliser la dynamique ou encore être une stratégie évolutionnairement stable. Finalement nous tentons de confronter numériquement ces modèles aux données : les simulations donnent des résultats qualitativement proches des observations

Stability results on some zooplankton size-structured models

International audienceStructured models are increasingly used in biological modelling, particularly to describe marine ecosystems, where the behaviour of individuals is strongly depen- dant of their size. To modelize zooplankton community, we first have to describe how an individual of some size feeds, and then how it uses the acquired food to grow and reproduce (according to some dynamic energy budget in order to guarantee mass conservation). Since the model includes cannibalism throughout zooplank- ton population, we obtain a variant of the well-known McKendrick-von Foerster equation with integral terms which appear in growth, mortality and reproduction. Such models are often hard to analyse mathematically. Nevertheless, with some more hypotheses on the cannibalism behavior, we can find equilibria of the model as fixed points of a function in a finite dimensional space. The linearized system around the equilibrium provides us, thanks to the use of linear semigroup theory, some local (un)stability results about these equilibria. Results obtained will be applied to a simple version of the model, which allows us to go further into the mathematical analysi

Stabilization of an oscillating n-dimensional structured population model

International audienceWe consider a model of a predator with q size-classes feeding on one resource. The resource is also modelled by a logistic function. This model may have sustained oscillations in dimension (q+1) around an unstable equilibrium. A possible control consists in harvesting every class. We show that (under good assumptions) this model can be reduced to a two-dimensional predator-prey model having sustained oscillations. We design positive controls to suppress the oscillations and to stabilize the equilibrium of the reduced and full model

A Continuous size-structured model applied to zooplankton comunity

International audienceSize appears to be a very good indicator of physiological behaviour in the zooplankton community, like predation or even detritus production. In order to get a better understanding of planktonic ecosystem dynamics, we build a continuous sizestructured model. This approach is very popular in fish studies (Benoit and Rochet,2004; Andersen and Pedersen, 2009). Our model incorporates predation on phytoplankton and cannibalism within the zooplankton community, and various predation behaviours are tested. Mathematical results are in the most cases hard to get in such models. Nevertheless for some choice of predation function, and external mortality, we can obtain mathematically equilibria. Numerical simulations show that these equilibria are not always stable, and Hopf bifurcations appear when model parameters vary

Positive control for global stabilization of predator-prey systems

International audienceIn this paper we propose a positive linear control law for the stabilization of positive equilibria in predator-prey systems. This problem is motivated by the introduction, which is by essence positive, of predators in biological control applications, to prevent excessive levels of prey (pests). We build a linear controller that we saturate at zero and prove, under some conditions, the global asymptotic stability of the equilibrium. Without one of these conditions, the point is shown to be globally attractive but may be unstable

Documentation of the Apecosm Python Package

PDF documentation of the Apecosm Python package. The code is available on https://github.com/apecosm/python-apecos