Analysis of a predator-prey model with specific time scales: a geometrical approach proving the occurrence of canard solutions.

We study a predator-prey model with different characteristic time scales for the prey and predator populations, assuming that the predator dynamics is much slower than the prey one. Geometrical Singular Perturbation theory provides the mathematical framework for analyzing the dynamical properties of the model. This model exhibits a Hopf bifurcation and we prove that when this bifurcation occurs, a canard phenomenon arises. We provide an analytic expression to get an approximation of the bifurcation parameter value for which a maximal canard solution occurs. The model is the well-known Rosenzweig-MacArthur predator-prey differential system. An invariant manifold with a stable and an unstable branches occurs and a geometrical approach is used to explicitly determine a solution at the intersection of these branches. The method used to perform this analysis is based on Blow-up techniques. The analysis of the vector field on the blown-up object at an equilibrium point where a Hopf bifurcation occurs with zero perturbation parameter representing the time scales ratio, allows to prove the result. Numerical simulations illustrate the result and allow to see the canard explosion phenomenon.

Assessment of congruence between co-occurrence and functional networks: A new framework for revealing community assembly rules.

Describing how communities change over space and time is crucial to better understand and predict the functioning of ecosystems. We propose a new methodological framework, based on network theory and modularity concept, to determine which type of mechanisms (i.e. deterministic versus stochastic processes) has the strongest influence on structuring communities. This framework is based on the computation and comparison of two networks: the co-occurrence (based on species abundances) and the functional networks (based on the species traits values). In this way we can assess whether the species belonging to a given functional group also belong to the same co-occurrence group. We adapted the Dg index of Gauzens et al. (2015) to analyze congruence between both networks. This offers the opportunity to identify which assembly rule(s) play(s) the major role in structuring the community. We illustrate our framework with two datasets corresponding to different faunal groups and ecosystems, and characterized by different scales (spatial and temporal scales). By considering both species abundance and multiple functional traits, our framework improves significantly the ability to discriminate the main assembly rules structuring the communities. This point is critical not only to understand community structuring but also its response to global changes and other disturbances.

Is structural sensitivity a problem of oversimplified biological models? Insights from nested Dynamic Energy Budget models.

Many current issues in ecology require predictions made by mathematical models, which are built on somewhat arbitrary choices. Their consequences are quantified by sensitivity analysis to quantify how changes in model parameters propagate into an uncertainty in model predictions. An extension called structural sensitivity analysis deals with changes in the mathematical description of complex processes like predation. Such processes are described at the population scale by a specific mathematical function taken among similar ones, a choice that can strongly drive model predictions. However, it has only been studied in simple theoretical models. Here, we ask whether structural sensitivity is a problem of oversimplified models. We found in predator-prey models describing chemostat experiments that these models are less structurally sensitive to the choice of a specific functional response if they include mass balance resource dynamics and individual maintenance. Neglecting these processes in an ecological model (for instance by using the well-known logistic growth equation) is not only an inappropriate description of the ecological system, but also a source of more uncertain predictions.

An ecosystem-based approach to assess the status of Mediterranean algae-dominated shallow rocky reefs.

A conceptual model was constructed for the functioning the algae-dominated rocky reef ecosystem of the Mediterranean Sea. The Ecosystem-Based Quality Index (reef-EBQI) is based upon this model. This index meets the objectives of the EU Marine Strategy Framework Directive. It is based upon (i) the weighting of each compartment, according to its importance in the functioning of the ecosystem; (ii) biological parameters assessing the state of each compartment; (iii) the aggregation of these parameters, assessing the quality of the ecosystem functioning, for each site; (iv) and a Confidence Index measuring the reliability of the index, for each site. The reef-EBQI was used at 40 sites in the northwestern Mediterranean. It constitutes an efficient tool, because it is based upon a wide set of functional compartments, rather than upon just a few species; it is easy and inexpensive to implement, robust and not redundant with regard to already existing indices.

Consequence of a sudden wind event on the dynamics of a coastal phytoplankton community: an insight into specific population growth rates using a single cell high frequency approach.

Phytoplankton is a key component in marine ecosystems. It is responsible for most of the marine primary production, particularly in eutrophic lagoons, where it frequently blooms. Because they are very sensitive to their environment, the dynamics of these microbial communities has to be observed over different time scales, however, assessment of short term variability is often out of reach of traditional monitoring methods. To overcome these limitations, we set up a Cytosense automated flow cytometer (Cytobuoy b.v.), designed for high frequency monitoring of phytoplankton composition, abundance, cell size, and pigment content, in one of the largest Mediterranean lagoons, the Berre lagoon (South-Eastern France). During October 2011, it recorded the cell optical properties of 12 groups of pico-, nano-, and microphytoplankton. Daily variations in the cluster optical properties were consistent with individual changes observed using microscopic imaging, during the cell cycle. We therefore used an adaptation of the size-structured matrix population model, developed by Sosik et al. (2003) to process the single cell analysis of the clusters and estimate the division rates of 2 dinoflagellate populations before, during, and after a strong wind event. The increase in the estimated in situ daily cluster growth rates suggest that physiological changes in the cells can prevail over the response of abundance.

An ecosystem-based approach to assess the status of a Mediterranean ecosystem, the Posidonia oceanica seagrass meadow.

Biotic indices, which reflect the quality of the environment, are widely used in the marine realm. Sometimes, key species or ecosystem engineers are selected for this purpose. This is the case of the Mediterranean seagrass Posidonia oceanica, widely used as a biological quality element in the context of the European Union Water Framework Directive (WFD). The good quality of a water body and the apparent health of a species, whether or not an ecosystem engineer such as P. oceanica, is not always indicative of the good structure and functioning of the whole ecosystem. A key point of the recent Marine Strategy Framework Directive (MSFD) is the ecosystem-based approach. Here, on the basis of a simplified conceptual model of the P. oceanica ecosystem, we have proposed an ecosystem-based index of the quality of its functioning, compliant with the MSFD requirements. This index (EBQI) is based upon a set of representative functional compartments, the weighting of these compartments and the assessment of the quality of each compartment by comparison of a supposed baseline. The index well discriminated 17 sites in the north-western Mediterranean (French Riviera, Provence, Corsica, Catalonia and Balearic Islands) covering a wide range of human pressure levels. The strong points of the EBQI are that it is easy to implement, non-destructive, relatively robust, according to the selection of the compartments and to their weighting, and associated with confidence indices that indicate possible weakness and biases and therefore the need for further field data acquisition.

Scaling up the predator functional response in heterogeneous environment: when Holling type III can emerge?

Accurate parametrization of functional terms in model equations is of great importance for reproducing the dynamics of real food webs. Constructing models over large spatial and temporal scales using mathematical expressions obtained based on microcosm experiments can be erroneous. Here, using a generic spatial predator-prey model, we show that scaling up the microscale functional response of a predator can result in qualitative alterations of functional response on macroscales. In particular, a global functional response of sigmoid type (Holling type III) can emerge as a result of non-linear averaging of non-sigmoid local responses (Holling type I or II). We demonstrate that alteration between the local and the global response in the model is a result of the interplay between density-dependent dispersal of the predator across the habitat and heterogeneity of the environment. Using the method of aggregation of variables, we analytically derive the mathematical formulation of the global functional response as a function of the total amount of prey in the system, and reveal the key parameters which control the emergence of a Holling type III global response. We argue that this mechanism by which a global Holling type III emerges from a local Holling type II response has not been reported in the literature yet: in particular, Holling type III can emerge in the case of a fixed gradient of resource distribution across the habitat, which would be impossible in priorly suggested mechanisms. As a case study, we consider the interaction between phytoplankton and zooplankton grazers in the water column; and we show that the emergence of a Holling type III global response can allow for the efficient top-down regulation of primary producers and stabilization of planktonic ecosystems under eutrophic conditions.

From individuals to populations to communities: a dynamic energy budget model of marine ecosystem size-spectrum including life history diversity.

Individual metabolism, predator-prey relationships, and the role of biodiversity are major factors underlying the dynamics of food webs and their response to environmental variability. Despite their crucial, complementary and interacting influences, they are usually not considered simultaneously in current marine ecosystem models. In an attempt to fill this gap and determine if these factors and their interaction are sufficient to allow realistic community structure and dynamics to emerge, we formulate a mathematical model of the size-structured dynamics of marine communities which integrates mechanistically individual, population and community levels. The model represents the transfer of energy generated in both time and size by an infinite number of interacting fish species spanning from very small to very large species. It is based on standard individual level assumptions of the Dynamic Energy Budget theory (DEB) as well as important ecological processes such as opportunistic size-based predation and competition for food. Resting on the inter-specific body-size scaling relationships of the DEB theory, the diversity of life-history traits (i.e. biodiversity) is explicitly integrated. The stationary solutions of the model as well as the transient solutions arising when environmental signals (e.g. variability of primary production and temperature) propagate through the ecosystem are studied using numerical simulations. It is shown that in the absence of density-dependent feedback processes, the model exhibits unstable oscillations. Density-dependent schooling probability and schooling-dependent predatory and disease mortalities are proposed to be important stabilizing factors allowing stationary solutions to be reached. At the community level, the shape and slope of the obtained quasi-linear stationary spectrum matches well with empirical studies. When oscillations of primary production are simulated, the model predicts that the variability propagates along the spectrum in a given frequency-dependent size range before decreasing for larger sizes. At the species level, the simulations show that small and large species dominate the community successively (small species being more abundant at small sizes and large species being more abundant at large sizes) and that the total biomass of a species decreases with its maximal size which again corroborates empirical studies. Our results indicate that the simultaneous consideration of individual growth and reproduction, size-structured trophic interactions, the diversity of life-history traits and a density-dependent stabilizing process allow realistic community structure and dynamics to emerge without any arbitrary prescription. As a logical consequence of our model construction and a basis for future studies, we define the function Φ as the relative contribution of each species to the total biomass of the ecosystem, for any given size. We argue that this function is a measure of the functional role of biodiversity characterizing the impact of the structure of the community (its species composition) on its function (the relative proportions of losses, dissipation and biological work).

Structural sensitivity of biological models revisited.

Enhancing the predictive power of models in biology is a challenging issue. Among the major difficulties impeding model development and implementation are the sensitivity of outcomes to variations in model parameters, the problem of choosing of particular expressions for the parametrization of functional relations, and difficulties in validating models using laboratory data and/or field observations. In this paper, we revisit the phenomenon which is referred to as structural sensitivity of a model. Structural sensitivity arises as a result of the interplay between sensitivity of model outcomes to variations in parameters and sensitivity to the choice of model functions, and this can be somewhat of a bottleneck in improving the models predictive power. We provide a rigorous definition of structural sensitivity and we show how we can quantify the degree of sensitivity of a model based on the Hausdorff distance concept. We propose a simple semi-analytical test of structural sensitivity in an ODE modeling framework. Furthermore, we emphasize the importance of directly linking the variability of field/experimental data and model predictions, and we demonstrate a way of assessing the robustness of modeling predictions with respect to data sampling variability. As an insightful illustrative example, we test our sensitivity analysis methods on a chemostat predator-prey model, where we use laboratory data on the feeding of protozoa to parameterize the predator functional response.

Effects of chronic uranium exposure on life history and physiology of Daphnia magna over three successive generations.

Daphnia magna was exposed to waterborne uranium (U) at concentrations ranging from 10 to 75 microgL(-1) over three successive generations (F0, F1 and F2). Progeny was either exposed to the same concentration as mothers to test whether susceptibility to this radioelement might vary across generations or returned to a clean medium to examine their capacity to recover after parental exposure. Maximum body burdens of 17, 32 and 54 ng U daphnid(-1) were measured in the different exposure conditions and converted to corresponding internal alpha dose rates. Low values of 5, 12 and 20 microGy h(-1) suggested that radiotoxicity was negligible compared to chemotoxicity. An increasing sensitivity to toxicity was shown across exposed generations with significant effects observed on life history traits and physiology as low as 10 microgL(-1) and a capacity to recover partially in a clean medium after parental exposure to

Effect of predator density dependent dispersal of prey on stability of a predator-prey system.

This work presents a predator-prey Lotka-Volterra model in a two patch environment. The model is a set of four ordinary differential equations that govern the prey and predator population densities on each patch. Predators disperse with constant migration rates, while prey dispersal is predator density-dependent. When the predator density is large, the dispersal of prey is more likely to occur. We assume that prey and predator dispersal is faster than the local predator-prey interaction on each patch. Thus, we take advantage of two time scales in order to reduce the complete model to a system of two equations governing the total prey and predator densities. The stability analysis of the aggregated model shows that a unique strictly positive equilibrium exists. This equilibrium may be stable or unstable. A Hopf bifurcation may occur, leading the equilibrium to be a centre. If the two patches are similar, the predator density dependent dispersal of prey has a stabilizing effect on the predator-prey system.

A kinetic inhibition mechanism for maintenance.

To fulfil their maintenance costs, most species use mobile pools of metabolites (reserve) in favourable conditions, but can also use less mobile pools (structure) under food-limiting conditions. While some empirical models always pay maintenance costs from structure, the presence of reserve inhibits the use of structure for maintenance purposes. The standard dynamic energy budgets (DEB) model captures this by simply supplementing all costs that could not be paid from reserve with structure. This is less realistic at the biochemical level, and involves a sudden use of structure that can complicate the analysis of the model properties. We here propose a new inhibition formulation for the preferential use of reserve above structure in maintenance that avoids sudden changes in the metabolites use. It is based on the application of the theory for synthesizing units, which can easily become rather complex for demand processes, such as the maintenance. We found, however, a simple explicit expression for the use of reserve and structure for maintenance purposes and compared the numerical behaviour with that of a classical model in oscillating conditions, by using parameters values from a fit of the models to data on yeasts in a batch culture. We conclude that our model can better handle variable environments. This new inhibition formulation has a wide applicability in modelling metabolic processes.

Spatial synchrony in host-parasitoid models using aggregation of variables.

We consider a host-parasitoid system with individuals moving on a square grid of patches. We study the effects of increasing movement frequency of hosts and parasitoids on the spatial dynamics of the system. We show that there exists a threshold value of movement frequency above which spatial synchrony occurs and the dynamics of the system can be described by an aggregated model governing the total population densities on the grid. Numerical simulations show that this threshold value is usually small. This allows using the aggregated model to make valid predictions about global host-parasitoid spatial dynamics.

Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics.

Most classical prey-predator models do not take into account the behavioural structure of the population. Usually, the predator and the prey populations are assumed to be homogeneous, i.e. all individuals behave in the same way. In this work, we shall take into account different tactics that predators can use for exploiting a common self-reproducing resource, the prey population. Predators fight together in order to keep or to have access to captured prey individuals. Individual predators can use two behavioural tactics when they encounter to dispute a prey, the classical hawk and dove tactics. We assume two different time scales. The fast time scale corresponds to the inter-specific searching and handling for the prey by the predators and the intra-specific fighting between the predators. The slow time scale corresponds to the (logistic) growth of the prey population and mortality of the predator. We take advantage of the two time scales to reduce the dimension of the model and to obtain an aggregated model that describes the dynamics of the total predator and prey densities at the slow time scale. We present the bifurcation analysis of the model and the effects of the different predator tactics on persistence and stability of the prey-predator community are discussed.

Impact of spatial heterogeneity on a predator-prey system dynamics.

This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium.

Predator Migration Decisions, the Ideal Free Distribution, and Predator-Prey Dynamics.

The aim of the present work is to analyze the influence of optimal predator emigration decisions that can lead to the ideal free distribution (IFD) on the stability of predator-prey systems. The assumption of optimal decisions is then relaxed to analyze the possible influence of different degrees of deviation from the IFD. The first migration rule we analyze is based on the marginal-value theorem and assumes perfect knowledge of capture rate in the patch of residence and in the environment as a whole. When migration rates are high, this rule leads the predator population to the IFD. The results suggest that under these conditions predator migration plays no major role in the stability of the system. This is so because the systems naturally merge into a single patch. This result is independent of the particular functional response used. The other two rules we analyze take into account lower migration rates, the limitations in making optimal decisions by predators, and the possible constraints in the assessment of intake rate in the different patches. The results suggest that the processes that hinder the convergence of the populations to the IFD might make a major contribution to the stability of the system.