L'écologie est une discipline quantitative dans laquelle les mathématiques sont présentes sous différentes formes depuis très longtemps. En conséquence, l'arrivée massive d'ordinateurs de plus en plus puissants dans les laboratoires dans les dernières décennies, a conduit à une explosion de la modélisation dans ce domaine, sous forme de calculs numériques mais également par l'analyse mathématique de modèles relativement simples. Cette croissance importante de l'activité de modélisation mathématique a été accompagnée par une augmentation de la complexité des modèles d'écologie qui tentent d'intégrer la plus grosse quantité de processus connus possible. Parallèlement, les moyens d'expérimentations et d'observation du milieu naturel n'ont pas cessé de s'améliorer, produisant ainsi des bases de données de plus en plus complètes dans la description du fonctionnement des écosystèmes. Paradoxalement, la formulation de base des processus utilisée dans les modèles complexes est toujours la même et fondée sur des expérimentations réalisées dans des conditions homogènes de laboratoire au cours du XXème siècle. Nous posons la question de l'intérêt d'une description adéquate d'un écosystème pour comprendre ses réponses à différentes perturbations. Une approche consiste à utiliser des formulations mécanistes des processus, c'est à dire des formulations fondées sur des détails expliquant la cause de la réalisation des processus, plutôt que des formulations empiriques acquises dans des conditions différentes du milieu dans lequel on les applique. Cette prise en compte des mécanismes induit encore un surcroit de complexité. Les mathématiques fournissent un ensemble d'idées et de méthodes permettant tout d'abord de produire des formulations adaptées à la prise en compte des mécanismes et également d'aborder cette complexité des modèles écosystémiques, voire dans certains cas de la réduire. Nous illustrerons cette démarche à travers des exemples d'applications variés.[-]

Oxygen is essential for burning food and generate energy, but may become limiting for aquatic organisms that rely on gas exchange under water. This is because breathing under water is challenging: the diffusion of oxygen is orders of magnitude lower in water than in air, while the higher density and viscosity of water greatly enhance the cost of breathing. Given that oxygen may be also be a limiting resource, respiration physiology may be relevant to understand energy budgets in aquatic ectotherms.
Traditionally, respiration physiology has focused on the benefits of extracting sufficient amounts of oxygen and thus prevent asphyxiation. However, breathing oxygen is intrinsically dangerous: while a shortage of oxygen quickly leads to asphyxiation, too much oxygen is toxic. Therefore, the ability to regulate oxygen consumption rates (i.e. respiratory control) is at a premium; good respiratory control will enable ectotherms to balance oxygen toxicity against the risk of asphyxiation across a wide range of temperatures.
In this presentation I will focus on the effects of body size and temperature on this balancing act with regard to oxygen uptake and consumption. Body size is intimately tied to oxygen budgets and hence energy budgets through size related changes in oxygen requirements and respiratory surfaces. Furthermore, a larger body size may represent a respiratory advantage that helps to overcome viscosity. Given that viscous forces are larger in cold water, this respiratory advantage represents a novel explanation for the pattern of larger body sizes in cold water, with polar gigantism as the extreme manifestation.
Temperature is also intimately tied to oxygen budgets and hence energy budgets through thermal controls on metabolism and temperature related changes in the availability of dissolved oxygen (notably diffusivity, viscosity and solubility). Thus, differences in temperatures may act more strongly on ectotherms that rely on aquatic rather than on aerial gas exchange. Comparing four different insect orders, I demonstrate that thermal tolerance is indeed affected more by the prevalent oxygen conditions in species with poor respiration control. In conclusion, the ability to regulate gas exchange (i.e. respiratory control) is thus a key attribute of species that helps to explain thermal responses from an oxygen perspective.[-]

Adaptive dynamics has shaped our understanding of evolution by demonstrating that, via the process of evolutionary branching, ecological interactions can promote diversification. The classical approach to study the adaptive dynamics of a system is to specify the ecological model including all trade-off functions and other functional relationships, and make predictions depending on the parameters of these functions. However, the choice of trade-offs and other functions is often the least well justified element of the model, and examples show that minor variations in these functions can lead to qualitative changes in the model predictions. In the first part of this talk, I shall revisit evolutionary branching and other evolutionary phenomena predicted by adaptive dynamics using an inverse approach: I investigate under which conditions a trade-off function exists that yields a given evolutionary outcome.
Evolutionary branching can amount to the birth of new species, but only if reproductive isolation evolves between the emerging branches. Recent studies show that mating is often assortative with respect to the very trait that is under ecological selection. Such "magic traits" can ensure reproductive isolation, yet they are by far not free tickets to speciation. In the second half of my talk, I discuss the consequences of sexual selection emerging from assortative mating, and show how a perfect female should search for mates.[-]

(Joint work with Gonçalo Jacinto and Patricia A. Filipe.) The effect of random fluctuations of internal and external environmental conditions on the growth dynamics of individual animals is not captured by the regression model typical approach. We use stochastic differential equation (SDE) versions of a general class of models that includes the classical growth curves as particular cases. Namely, we use models of the form $d Y_t=\beta\left(\alpha-Y_t\right) d t+\sigma d W_t$, with $X_t$ being the animal size at age $t$ and $Y_t=h\left(X_t\right)$ being the transformed size by a $C^1$ monotonous function $h$ specific of the appropriate underlying growth curve model. $\alpha$ is the average transfomed maturity size of the animal, $\beta>0$ is the rate of approach to it and $\sigma>0$ measures the intensity of the effect on the growth rate of $Y_t$ of environmental fluctuations. These models can be applied to the growth of wildlife animals and also to plant growth, particularly tree growth, but, due to data availability (data furnished by the Associação dos Produtores de Bovinos Mertolengos - ACBM) and economica interest, we have applied them to cattle growth.
We briefly mention the extensive work of this team on parameter simulation methods based on data from several animals, including alternatives to maximum likelihood to correct biases and improve confidence intervals when, as usually happens, there is shortage of data for animals at older ages. We also mention mixed SDE models, in which model parameters may vary randomly from animal to animal (due, for instance, to their different genetical values and other individual characteristics), including a new approximate parameter estimation method. The dependence on genetic values opens the possibility of evolutionary studies on the parameters.
In our application to mertolengo cattle growth, the issue of profit optimization in cattle raising is very important. For that, we have obtained expressions for the expected value and the standard deviation of the profit on raising an animal as a function of the selling age for quite complex and market realistic raising cost structures and selling prices. These results were used to determine the selling age that maximizes the expected profit. A user friendly and flexible computer app for the use of farmers was developed by Ruralbit based on our results.[-]

Large scale biodiversity patterns result from the historical processes of speciation and extinction. In particular, the balance between speciation and extinction rates determines how species richness varies through time, across species groups, and geographical regions. Phylogenetic diversification analyses, which rely on fitting stochastic birth-death processes to phylogenetic data, can be used to estimate these macroevolutionary rates from the phylogenies of extant species, potentially further informed by paleodata. I will present recent developments that model fine-scale variations in speciation rates and that can combine neontological and paleontological evidence. Applied to empirical data, these models reveal a wide variation in speciation rates across lineages. While several models have been developed to explain these variations by differences in specific traits or abiotic and biotic conditions, models that would help us better understand the actual processes that control diversification rates are lagging behind. Speciation research at the microscale has focused on understanding the establishment of reproductive barriers, but there is increasing evidence that variations in macroevolutionary speciation rates are poorly explained by variations in the rate at which populations acquire reproductive isolation. I will present recent developments that aim at understanding i) how variations in the rates at which reproductive isolation is initiated, at which populations acquire reproductive isolation, and at which incipient species go extinct combine to explain macroevolutionary speciation rate, and ii) how population sizes, mutation rates and the mode of speciation impact this latter relationship.[-]

Cyclic dynamics are one of the most notable phenomena in population biology and are known to occur in many communities both in the wild and the laboratory. Oscillations in biomass often exceed an order of magnitude, with period lengths ranging from days to decades, and may be spatially synchronized over continental scales. Their underlying causes, however, remain a long-standing enigma. In this presentation I will present modelling analysis from my own work for four paradigmatic case studies. These will present a journey from single species laboratory experiments to the world largest population oscillations, both in in period length and in absolute biomass. I will show that the mechanisms driving the cycles and their synchronization to external forcing may be very different in each case (e.g., predator-prey interactions or synchronized life cycles). But despite these idiosyncratic properties, notions from phase analysis and synchronization theory can be applied to capture observed population dynamics, providing a common theoretic framework for understanding these phenomena that have fascinated ecologists for centuries.[-]

Understanding the adaptation and evolution of populations is a huge challenge, in particular for microorganisms since it plays a main role in the virulence evolution or in bacterial antibiotics resistances. We propose a general stochastic model of population dynamics with clonal reproduction and mutations. Moreover the individuals compete for resources and exchange genes. We show that the horizontal gene transfer can have a major impact on the distribution of the successive mutational fixations, leading to dramatically different behaviors, from expected evolution scenarios to evolutionary suicide, including cyclic behaviours. We present different approaches to capture mathematically these scenarii.[-]

Evidence is increasing that large-scale abrupt changes in ecosystems, sheries, oceanic circulation patterns, or even human physiology are examples of catastrophic transitions between different system states. Such abrupt changes are typically referred to as tipping points. Recent theoretical findings suggest that distinct properties tend to rule system dynamics prior to tipping points. When quantified, these properties may be more generically used as indicators of resilience. As long-term data become increasingly available and experimental approaches are improving, the challenge has been to apply our theoretical metrics on ecological dynamics to anticipate, prepare, or navigate away from tipping points. In this talk, I will present how we can quantify resilience and detect tipping points highlighting examples from ecological and climate systems. Moreover, I will outline challenges and ideas on how we can operationalise such approaches and also how to better understand tipping point responses in a changing but evolving world.[-]

Dynamic Energy Budget (DEB) models describe how individual organisms acquire and use energy from food and have therefore been argued to consistently link different levels of biological organisation. Various types of DEB models, differing in the organisation and precedence of metabolic processes such as growth, maintenance and reproduction, have been proposed and investigated, although recently the term DEB theory has become more and more identified with the framework developed by Kooijman.
In this lecture I will address the question to what extent differences between DEB models affect the dynamics at the population and community level. I will show that maintenance costs, which are accounted for in all DEB models, have a crucial influence, but that metabolic organisation is of lesser importance. I will furthermore show that population and community dynamics are mostly determined by differences in the capacity of individuals with different body sizes or in different stages of their life history to transform food into new biomass. Such differences, which I refer to as ontogenetic asymmetry in energetics, are however influenced more by the types of food that individuals forage on in different stages of their life history than by their internal energetics. Ontogenetic shifts in resource use during life history are therefore likely to have a larger influence on population and community dynamics than the details of the individual energy budget.[-]