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This symposium is financially supported by NWO
Theories and Methods in Nonlinear Population Dynamics: Aggregation-, Dynamic Energy Budget-, Game- and Adaptive Dynamics theory, Perturbation Methods and Deterministic and Stochastic Modelling, VU, W&N building, Room C121, De Boelelaan 1087, NL 1081 HV Amsterdam, 3 and 4 September 2004 How to get to the VU.
The main aim of this meeting is to encourage and stimulate the exchange of researchers working on the analysis of population dynamic models, for instance those resulting from the Dynamics Energy Budget theory. The discussed mathematical analysis tools include bifurcation, aggregation and singular perturbation theory. In addition to the analysis of deterministic models, methods to analyse stochastic models will be discussed.
You are cordially invited to contribute to this symposium by giving a talk on Friday 3 September (20-30 min) about one of the subjects mentioned above. Please contact Bob Kooi, email: kooi@bio.vu.nl by sending the title and abstract.
Further details with exact place, time schedule and the final program will be given in a comming announcement.
The organizers: Pierre Auger, Johan Grasman, Bob Kooi, Bas Kooijman, Jean-Christophe Poggiale.
Rafael Bravo de la Parra, Universidad de Alcala, Spain
homepage
Email: rafael.bravo@uah.es
Title: Approximate aggregation of discrete dynamical systems
Abstract: When modeling biological systems, in particular ecological ones, we usually find very complex systems that we should try to manage to get some insights. A first approach to do this consists in building an abstract model describing the real system in detail. This leads to a family of models involving a very large number of variables. The complexity of the system is included in the model and computer simulation becomes the only available tool to carry out its study. At the other extreme we can find those models avoiding almost every detail in order to be mathematically tractable. These models of ecological communities only deal with a few variables, assuming that the internal structure of the population has no important effect and so can be neglected. This assumption corresponds to an approximation of the total system by means of a reduced one that should be checked. However, in most cases, simplified models are used and few arguments are given to justify them. A tool trying to fill the gap between these two approaches is approximate aggregation. Approximate aggregation consists in describing some features of the dynamics of a general system in terms of the dynamics of a reduced system governed by a few number of global variables. We think of a hierarchically organized population, thus subdivided into subpopulations, which allows distinguishing two processes of a general nature and whose corresponding time scales are very different from each other. Results will be presented showing that, under quite general conditions, the asymptotic behavior of the reduced system can be known in terms of the corresponding behavior for the reduced system. We present aggregation methods for different kinds of discrete dynamical systems linear and non-linear, deterministic and stochastic.
Fabio Dercole, Politecnico di Milano, Italy
homepage
Email: dercole@elet.polimi.it
Title: Border collision bifurcations in the evolution of mutualistic interactions
Abstract: The slow evolution of two adaptive traits that regulate the interactions between two mutualistic populations (e.g. plants and their impollinator insects) is described. For frozen values of the traits, the two populations can either coexist or go extinct. The values of the traits for which populations extinction is guaranteed are therefore of no interest from an evolutionary point of view. In other words, the evolutionary dynamics must be studied only in a viable subset of the two-dimensional trait space, which is bounded due to the physiological cost of extreme trait values. Thus, evolutionary dynamics experience so-called border collision bifurcations, when a system invariant in trait space hits the border of the viable subset. Physically speaking, these bifurcations correspond to the extinction of the two populations driven by their coevolution (coevolutionary suicide). The unfolding of standard and border collision bifurcations with respect to two parameters of biological interest is presented. The algebraic and boundary-value problems characterizing the border collision bifurcations are described together with some details concerning their computation.
Tiago Domingos, Joao Rodrigues and Rui Mota
Instituto Superior Tecnico, Lisboa, Portugal
homepage
Email: tdomingos@ist.utl.pt
Title: Logistic Model with Variable Carrying Capacity: Analysis with Bifurcation and Optimal Control Theories
Abstract: We create an ecological logistic model with variable carrying capacity, adapted from Cohen's (1995) model for the human population (Rodrigues and Domingos, 2004; Mota et al., 2004). The variables are N, the usual variable in the logistic model, and K, the carrying capacity. Carrying capacity increases due to increases in the value of the population. The model is applicable to multiple ecological phenomena, at several time scales. It can be used to model the process of niche construction, as a given population improves its environment, e.g. beavers building dams (Odling-Smee et al., 2003). It can be used to model ecological succession, with N being community biomass and K increasing along time due to improvements in the physical environment, namely soil creation and climate regulation (Walker and Moral, 2003). It can be used to model biodiversity dynamics after mass extinctions (Erwin, 2001). N can also be interpreted as natural capital, in the context of the ecological economic approach to sustainability (Rodrigues et al., 2004; Mota and Domingos, 2004). We show that the two-dimensional model can be reduced to a single dimensional model, with the variable N and an additional parameter (Mota et al., 2004). For certain parameter values, this single dimensional model has an Allee effect. The feedbacks on carrying capacity, the strength of the effect on K of increasing N, may have a deleterious effect on N. Increasing this parameter may lead to extinction. However, increasing it even further may bring again the model out of extinction. We study the optimal harvesting of a system described by this model (Mota and Domingos, 2004). Harvesting along time is chosen such that is maximizes a given functional, which expresses the current economic value of the chosen harvesting program. The solution is obtained by using optimal control theory, in the form of Pontryagin's Maximum Principle. A two-dimensional dynamical system (where the state variables are N and the harvesting rate) is obtained as a necessary condition for the optimal solution, and this dynamical system is then analysed using the techniques of bifurcation theory. The resulting dynamical system has a very rich behaviour. Phase space may have two interior equilibria, one unstable and one saddle, with the existence of heteroclinic bifurcations. There are two major possibilities for optimal trajectories: going to extinction, or going to a non-zero steady state. One of these trajectories may be optimal for any initial value of N, but, for some parameter values, the optimal trajectory will depend on the initial value of N. For the latter case, Pontryagin's Maximum Principle provides only a local optimum, so the global optimum has to be found numerically using the Hamilton-Jacobi formulation.
References:
Ulrike Feudel, Thilo Gross, Wolfgang Ebenhoh
Institut fur Chemie und Biologie des Meeres, Oldenburg, Germany.
Email: u.feudel@icbm.de
homepage
Title: Towards a general stability theory of population dynamical models
Abstract: Depending on environmental parameters population dynamical models may exhibit different long-term behaviors such as stationary points, periodic orbits, quasiperiodic and chaotic motion. When environmental parameters are varied transitions from one long-term behavior to another can occur. The knowledge of such transitions, called bifurcations, provides the regions in parameter space where a certain behavior can be expected and the location of the threshold values for instabilities leading to possibly undesired states of the ecosystem. We present a general model for food chains and study its stability properties with respect to perturbations. This general model includes predator-prey response functions which may depend on prey as well as on the predator. Computing the loss of the stability of the steady-state where all interacting species are present in the system. Since we do not specify the predator-prey functional response explicitly we are able to discuss the role of the exact form of predator-prey interaction with respect to the stability properties of the system. Furthermore we can show how prevalent chaotic behavior is in ecosystems. Finally, we extend our stability analysis to food webs and discuss the role of interference between different species.
Johan Grasman
Biometris, Wageningen UR
Email: Johan.Grasman@wur.nl homepage
Title: Stochastic modelling and extinction in food chains
Abstract: In a substrate-bacterium-worm model external fluctuations affecting the intrinsic growth rate are being incorporated. It is analyzed how under chemastat conditions the system persists: a measure for this persistence is the expected extinction time of bacterium or worm. Formulating the Fokker-Planck equation for this problem we arrive at an elliptic singular perturbation problem for the extinction time. This analytical result is compared with the outcome of a Monte Carlo simulation.
References :
Rachid Mchich, Pierre Auger and Christophe Lett
UMR CNRS 5558, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France.
Email: mchich@biomserv.univ-lyon1.fr
Title: Effects of aggregative and solitary individual behaviors on the dynamics of a predator-prey system
Abstract: We present a predator prey model taking into account the individual behaviors of preys and predators. The preys (respectively the predators) can be grouped or solitary. When the preys are grouped, they face less predation but also have less access to the resource, that has to be shared with other animals within the group. On the other hand, predators in a group capture a prey more easily but must to share it with other predators within the group. We present a theoretical game model describing the interactions between the different strategies that can be adopted by preys and predators. This game occurs at a fast time scale and is coupled with a predator-prey model at a slow time scale. We obtain a reduced model governing the total preys and predators densities. We present the analysis of the aggregated model by studying the effects of the different behavioral strategies on the total dynamics of the predator-prey model.
References:
Caroline Tolla, J.C. Poggiale, S. A. L. M. Kooijman and P. Auger
Centre d'Ocanologie de Marseille Laboratoire
d'Ocanographie et de Biogochimie UMR CNRS 6535 Campus de Luminy - Case
901 13288 Marseille Cedex 09, France
Email: caroline.tolla@com.univ-mrs.fr
Title: A new mathematical formulation of preferential use of a substrate above another, based on enzymatic kinetics
Abstract: The organic matter fate is conditioned by physical, biological and biogeochemical properties of the environment which all interact. Due to this complexity and to understand the environment changes, its crucial to model the microbial communities dynamic responsible for biogeochemical processes. At the moment, classical models (Monod, 1958, Marr-Pirt, 1963-65, Droop, 1975) allow us to analyse and quantify the bacterial communities dynamic. They formulate, however, simplistic assumptions and offer a non appropriate description of bacterial dynamic in environmental conditions variations. My aim is to improve the present models analysing the microbial dynamic in sediments by taking explicitly into account the functionality of bacterial communities. To do that, we construct a mechanistic model of bacterial communities dynamic in sediments based on the DEB theory (Kooijman S.A.L.M., 2000). In this lecture, I will describe the substrates interactions and more precisely the inhibition process of denitrification by the presence of oxygen. I will present a new mathematical formulation based on enzymatic kinetics describing suppression of the uptake of a substrate (nitrate) by other substrates (dioxygen), which involves a mechanism, where a substrate drives out another substrate that is bound in a SU-complex. Finally, we will compare this new formulation with the switch model proposed by Kooijman (2000), the unilateral binding inhibition proposed by Brandt (2002) and a particular case.
References :
Discussion after introduction lectures by (9.00h-16.00 h):
Pierre Auger, Bob Kooi, Rafael Bravo de la Parra and Jean-Christophe Poggiale
Title: Bifurcation analysis of a predator-prey model with predators using hawk and dove tactics
Abstract:
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 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 intraspecific fighting between the predators and the interspecific handling and searching for the prey by 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 complete bifurcation analysis of the model and the effects of the different predator tactics on persistence and stability of the prey-predator community are discussed.
References :
Pierre Auger
Title: Aggregation methods in population dynamics
Abstract: Ecological systems are composed with different levels of organisation. Usually, one considers the individual, the population, the community and ecosystem levels. These levels of organisation correspond to different levels of observation of the system, i.e., to different time and space scales. The dynamics of the complete system is the result of the coupled dynamical processes that take place in each of its levels of organisation at different time scales. Variables aggregation methods take advantage of these different time scales to reduce the dimension of mathematical models and make possible to build a reduced model that governs a few global variables at the slow time scale. I shall present applications of aggregation methods in population dynamics. I shall show spatial aggregation methods for predator-prey and host-parasitoid models in a network of spatial patches connected by fast migrations. I shall also present a predator-prey model taking into account the interactions between individual predators at a fast time scale.
References :
Jean-Christophe Poggiale
Title: Geometrical Singular Pertubations theory in the framework of stochastic differential equations
Abstract:
Singular perturbations theory proposes very powerfull tools to analyse systems with different time scales. For instance, in a food chain where each trophics level is associated to a caracteristic time scale, it is possible to simplify the whole complex dynamics in different successive one dimensional systems. This approach permitted for instance to prove the existence of chaotic dynamics in Kolmogorov three dimensional systems. In this lecture, we discuss about an extension of the usual theorems of Geometrical Singular Pertubations theory, namely Fenichel theorems, in the framework of stochastic differential equations. We present some results concerning the effects of noise on dynamics bifurcations and give some examples. We finally discuss about the potential applications of these results in population dynamics.
References :
Tania Sousa, R. Mota, T. Domingos and S. A. L. M. Kooijman.
Title: Thermodynamic formalism of heterotrophic organisms in the DEB theory
Abstract: We formalise and extend a thermodynamic analysis for a heterotrophic organism that follows the rules for uptake and use of substrate as specified the Dynamic Energy Budget (DEB) theory. The incoming and outgoing fluxes of matter through the outer surface of the organism are made explicit, as well as the internal processes, to get some insight into internal entropy production. We replace the original strong assumption in the DEB theory that the entropy of biomass is negligibly small by the weaker assumption that the temperature times the change in entropy is very small compared to the change in enthalpy. This is supported by empirical evidence. We demonstrate that this change in assumptions does not affect the energy balance that is used in DEB theory. We also show that the internal irreversibility term can be specified by the net fluxes of chemical compounds in and out of the organism and is equal to the total heat released in the reactions inside the organism. The processes of assimilation, dissipation and growth are no longer necessarily exothermic; this is one of the most important consequences of relaxing the assumptions on entropy. A more direct justification is presented for the method of indirect calorimetry method, which equates the total dissipating heat to a weighted sum of consumed dioxygen, and produced carbon dioxide and nitrogen waste. We show, for the first time, how specific enthalpies and specific entropies for the reserve and the structure of an organism can be obtained from empirical data. We use published data on Klebsiella pneumoniae growing aerobically in a continuous culture. As expected, the specific entropy of structure is lower than that of reserve.
Bas Kooijman
Title: Recent developments in Dynamic Energy Budget (DEB) theory
Abstract:
Recent developments in Dynamic Energy Budget (DEB) theory frequently involve new dynamics of Synthesizing Units (SU), e.g.
These developments introduce behavioural and chemical traits in cellular and population dynamics, that have some features in common (e.g. time budgetting). A systematic analysis of the set of possible and realistic extensions could open a new field in population dynamics that links it more tightly to ecosystem dynamics.
References :