Global Production Increased by Spatial Heterogeneity in a Population Dynamics Model

Spatial and temporal heterogeneity are often described as important factors having a strong impact on biodiversity. The effect of heterogeneity is in most cases analyzed by the response of biotic interactions such as competition of predation. It may also modify intrinsic population properties such as growth rate. Most of the studies are theoretic since it is often difficult to manipulate spatial heterogeneity in practice. Despite the large number of studies dealing with this topics, it is still difficult to understand how the heterogeneity affects populations dynamics. On the basis of a very simple model, this paper aims to explicitly provide a simple mechanism which can explain why spatial heterogeneity may be a favorable factor for production.We consider a two patch model and a logistic growth is assumed on each patch. A general condition on the migration rates and the local subpopulation growth rates is provided under which the total carrying capacity is higher than the sum of the local carrying capacities, which is not intuitive. As we illustrate, this result is robust under stochastic perturbations

Aggregation methods in dynamical systems and applications in population and community dynamics

Approximate aggregation techniques allow one to transform a complex system involving many coupled variables into a simpler reduced model with a lesser number of global variables in such a way that the dynamics of the former can be approximated by that of the latter. In ecology, as a paradigmatic example, we are faced with modelling complex systems involving many variables corresponding to various interacting organization levels. This review is devoted to approximate aggregation methods that are based on the existence of different time scales, which is the case in many real systems as ecological ones where the different organization levels (individual, population, community and ecosystem) possess a different characteristic time scale. Two main goals of variables aggregation are dealt with in this work. The first one is to reduce the dimension of the mathematical model to be handled analytically and the second one is to understand how different organization levels interact and which properties of a given level emerge at other levels. The review is organized in three sections devoted to aggregation methods associated to different mathematical formalisms: ordinary differential equations, infinite-dimensional evolution equations and difference equations

Towards methodological approaches to implement the zooplankton component in “end to end” food-web models

The modelling of marine zooplankton has made great progress over the two last decades covering a large range of representations from detailed individual processes to functional groups. A new challenge is to dynamically represent zooplankton within marine food webs coupling lower trophic levels to fish and to thereby further our understanding of the role of zooplankton in global change. In this respect, the “rhomboid strategy” (deYoung et al., 2004) has been suggested as a generic approach to model the various trophic levels of pelagic ecosystems and is deemed to be adaptable to different spatial and temporal frames of applications. The present paper identifies directions to develop zooplankton modelling by combining the skills of modellers, experimentalists, observers and theoreticians. In the first part, we present the main types of existing models, specifying the scientific issues, their characteristic time and space scales, across the ecological organization levels. In the second part, we focus on the strengths and weaknesses of parameterizations for the different processes. Finally in the last part, we make suggestions for improving these parameterizations by combining experiments and observations, using modelling techniques to transfer information across scales and testing theories which can themselves help to organize experimental and modelling research

Modelling, singular perturbation and bifurcation analyses of bitrophic food chains

International audienceTwo predator-prey model formulations are studied: for the classical Rosenzweig-MacArthur (RM) model and the Mass Balance (MB) chemostat model. When the growth and loss rate of the predator is much smaller than that of the prey these models are slow-fast systems leading mathematically to singular perturbation problem. In contradiction to the RM-model, the resource for the prey are modelled explicitly in the MB-model but this comes with additional parameters. These parameter values are chosen such that the two models become easy to compare. In both models a transcritical bifurcation, a threshold above which invasion of predator into prey-only system occurs, and the Hopf bifurcation where the interior equilibrium becomes unstable leading to a stable limit cycle. The fast-slow limit cycles are called relaxation oscillations which for increasing differences in time scales leads to the well known degenerated trajectories being concatenations of slow parts of the trajectory and fast parts of the trajectory. In the fast-slow version of the RM-model a canard explosion of the stable limit cycles occurs in the oscillatory region of the parameter space. To our knowledge this type of dynamics has not been observed for the RM-model and not even for more complex ecosystem models. When a bifurcation parameter crosses the Hopf bifurcation point the amplitude of the emerging stable limit cycles increases. However, depending of the perturbation parameter the shape of this limit cycle changes abruptly from one consisting of two concatenated slow and fast episodes with small amplitude of the limit cycle, to a shape with large amplitude of which the shape is similar to the relaxation oscillation, the well known degenerated phase trajectories consisting of four episodes (concatenation of two slow and two fast). The canard explosion point is accurately predicted by using an extended asymptotic expansion technique in the perturbation and bifurcation parameter simultaneously where the small amplitude stable limit cycles exist. The predicted dynamics of the MB-model is in a large part of the parameter space similar to that of the RM-model. However, the fast-slow version of MB-model does not predict a canard explosion phenomenon

A kinetic inhibition mechanism for the maintenance process.

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. © 2006 Elsevier Ltd. All rights reserved

Aggregation Methods in Food Chains.

The aim of this paper is to apply aggregation methods to food chains under batch and chemostat conditions. These predator-prey systems are modelled using ODEs, one for each trophic level. Because the models are based on mass conservation laws, they are conservative and this allows perfect aggregation. Furthermore, it is assumed that the ingestion rate of the predator is smaller than that of the prey. On this assumption, approximate aggregation can be performed, yielding further reduction of the dimension of the system. We will study a food chain often found in wastewater treatment plants. This food chain consists of sewage, bacteria and worms. In order to show the feasibility of the aggregation methods, we will compare simulated results for the reduced and the full model of this food chain under chemostat conditions

Consumption and release of dissolved organic carbon by marine bacteria in a pulsed-substrate environment: from experiments to modelling.

To investigate the effects of episodic occurrence of dissolved organic carbon(DOC) in the natural environment, bacterial degradation of labile DOC was studied under laboratory-controlled conditions followed by modelling. A single labile DOC compound was periodically added to the experimental culture and its degradation by a monospecific marine bacterial strain was followed. The measured variables were DOC and bacterial biomass determined from the particulate organic carbon values. Experimental dynamics showed a repetition of 2 successive patterns after each DOC pulse:(1) substrate consumption and bacterial growth in the first few hours after substrate addition, followed by(2) bacterial reduction(organic carbon-related) and associated non-labile DOC release within the next few hours. Based on these experimental results, the Dynamic Energy Budget theory was applied for the first time to such conditions to develop a mechanistic model that comprised 7 parameters and 4 state variables in which bacterial biomass was fractionated into reserve and structure compartments. The model was constructed by accounting for a constant specific maintenance rate and comprised 2 different cell maintenance fluxes, one fuelled from cell reserves when substrate was abundant and one from reserves and cell structures when starvation occurred. This new model of bacterial degradation adequately matched experimental measurements and accurately reproduced the accumulation of non-labile DOC in the culture during the experiment. This model can easily be implemented in an aquatic biogeochemical model and could provide better understanding of the role of bacteria in carbon cycling in fluctuating environments. © Inter-Research 2009

Structural sensitivity and resilience in a predator-prey model with density-dependent mortality

International audienceNumerous formulations with the same mathematical properties can be relevant to model a biological process. Different formulations can predict different model dynamics like equilibrium vs. oscillations even if they are quantitatively close (structural sensitivity). The question we address in this paper is: does the choice of a formulation affect predictions on the number of stable states? We focus on a predator–prey model with predator competition that exhibits multiple stable states. A bifurcation analysis is realized with respect to prey carrying capacity and species body mass ratio within range of values found in food web models. Bifurcation diagrams built for two type-II functional responses are different in two ways. First, the kind of stable state (equilibrium vs. oscillations) is different for 26.0– 49.4% of the parameter values, depending on the parameter space investigated. Using generalized modelling, we highlight the role of functional response slope in this difference. Secondly, the number of stable states is higher with Ivlev’s functional response for 0.1–14.3% of the parameter values. These two changes interact to create differentmodel predictions if a parameter value or a state variable is altered. In these two examples of disturbance, Holling’s disc equation predicts a higher system resilience. Indeed, Ivlev’s functional response predicts that disturbance may trap the system into an alternative stable state that can be escaped from only by a larger alteration (hysteresis phenomena). Two questions arise from this work: (i) how much complex ecological models can be affected by this sensitivity to model formulation? and (ii) how to deal with these uncertainties in model predictions

Aggregation methods in food chains with nutrient recycling.

This paper is devoted to the study of food chain models under batch and chemostat conditions where nutrient recycling is taken into account. The food chain is formed by a nutrient and two populations, prey and predator (producers and consumers). Species at both trophic levels digest their food source only partly. The unusable part of the food is ejected in the reactor as faeces together with metabolic products. The excreted material together with death material, detritus, is decomposed and this gives the recycling of the nutrient. In closed (batch-type environment) systems the elemental matter needed by producers must be provided through recycling where light energy from the environment supplies the necessary energy that fuels the life processes. In open (chemostat-type environment) systems this energy is added to the system via the chemical energy stored in the organic compounds in the inflow. Bifurcation analysis is used to study the effects of material recycling on the long-term dynamic behaviour of these simple food chains. An aggregation method is developed for situations in which each trophic level is characterized by differing time scales. This allows us to reduce the dimension of the model which gives good approximations after the fast transient. We will show that first-order approximations are needed in order to get the same qualitative long-term dynamics for both the full and the reduced model. © 2002 Elsevier Science B.V. All rights reserved

Relations Between Bacterial Biomass and Carbon Cycle in Marine Sediments: An Early Diagenetic Model

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