Abstract We present a dynamical model of a multi-site fishery. The fish stock is located on a discrete set of fish habitats where it is catched by the fishing fleet. We assume that fishes remain on fishing habitats while the fishing vessels can move at a fast time scale to visit the different fishing sites. We use the existence of two time scales to reduce the dimension of the model : we build an aggregated model considering the habitat fish densities (...) and the total fishing effort. We explore a regulation procedure, which imposes an average residence time in patches. Several equilibria exist, a Fishery Free Equilibria (FFEs) as well as a Sustainable Fishery Equilibria (SFEs). We show that the dynamics depends on a threshold which is similar to a basic reproduction ratio for the fishery. When the basic reproduction ratio is less or equal to 1, one of the FFEs is globally asymptotically stable (GAS), otherwise one of the SFEs is GAS. Content Type Journal Article Category Regular Article Pages 1-22 DOI 10.1007/s10441-012-9155-3 Authors Pierre Auger, UMI IRD 209, UMMISCO, & université Pierre et Marie Curie, Paris VI. IRD France Nord, 93143 Bondy, France Ali Moussaoui, Département de Mathématiques, Université Aboubekr Belkaid, Tlemcen, Algeria Gauthier Sallet, INRIA project team: MASAIE, INRIA-Nancy Grand Est, Nancy, France Journal Acta Biotheoretica Online ISSN 1572-8358 Print ISSN 0001-5342. (shrink)
We consider a two-patch epidemiological system where individuals can move from one patch to another, and local interactions between the individuals within a patch are governed by the classical SIRS model. When the time-scale associated with migration is much smaller than the time-scale associated with infection, aggregation methods can be used to simplify the initial complete model formulated as a system of ordinary differential equations. Analysis of the aggregated model then shows that the two-patch basic reproduction rate is smaller than (...) the 1 patch one. We extend this result to a linear chain of P patches (P > 2). These results are illustrated by some examples for which numerical integration of the system of ordinary differential equations is performed. Simulations of an individual based model implemented with a multi-agent system are also carried out. (shrink)
As the result of the complexity inherent in nature, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often deal with models for structured populations in which individuals are classified regarding their age, size, activity or location, and this structuring of the population leads to high dimensional systems. In many instances, the dynamics of the system is controlled by processes whose time scales are very different from (...) each other. Aggregation techniques take advantage of this situation to build a low dimensional reduced system from which behavior we can approximate the dynamics of the complex original system.In this work we extend aggregation techniques to the case of time dependent discrete population models with two time scales where both the fast and the slow processes are allowed to change at their own characteristic time scale, generalizing the results of previous studies. We propose a non-autonomous model with two time scales, construct an aggregated model and give relationship between the variables governing the original and the reduced systems. We also explore how the properties of strong and weak ergodicity, regarding the capacity of the system to forget initial conditions, of the original system can be studied in terms of the reduced system. (shrink)
In this work we consider a structured population with groups and subgroups of individuals. The intra-group dynamics is assumed to be fast in comparison with the inter-group dynamics. We study linear discrete models where the slow dynamics is represented by a single matrix and the fast dynamics is described by means of the first k terms of a converging sequence of different matrices. The number k can be interpreted as the ratio between the two time scales.The aim of this work (...) is to extend aggregation techniques to the case of fast changing environments. The main idea of aggregation is to build up a new system, with lower dimension, that summarizes the information concerning the fast process. This "aggregated" system provides essential information on the original one. It is shown that the asymptotic behavior of the original system can be approximated by the asymptotic behavior of the aggregated system when the ratio between the two time scales is large enough. (shrink)
A compartmental model is described for the spread of Gambian sleeping sickness in a spatially heterogeneous environment in which vector and human populations migrate between two "patches": the village and the plantations. The number of equilibrium points depends on two "summary parameters": gr the proportion removed among human infectives, and R0, the basic reproduction number. The origin is stable for R0 1. Control strategies are assessed by studying the mix of vector control between the two patches that bring R0 below (...) 1. The results demonstrate the importance of vector control in the plantations. For example if 20 percent of flies are in the village and the blood meal rate in the village is 10 percent, then a 20 percent added vector mortality in the village must be combined with a 9 percent added mortality in the plantations in order to bring R0 below 1. The results are quite insentive to the blood meal rate in the village. Optimal strategies (that minimize the total number of flies trapped in both patches) are briefly discussed. (shrink)
Limitations of antiarrhythmic drugs on cardiac sudden death prevention appeared since the early 80's. The "Cardiac Arrhythmia Suppression Trial"(CAST) showed more recently that mortality was significantly higher inpatients treated with some particular antiarrhythmic drugs than in non-treated patients. In this field, our group recently demonstrated that a bolus of a Class 1B antiarrhythmic drug was able to trigger a ventricular fibrillation due to transient blocks induction. The aim of the present work was to systematically study, by use of the van (...) Capelle and Durrer (VCD) model which allows to simulate ventricular activation wave propagation, the link between arrhythmogenic effects and the ability of transient blocks to possibly degenerate in severe arrhythmias. A fragment of the ventricular wall is represented by an array of 16384elements electrically coupled. Effects of induction of one or several transient blocks, as the effects of their size and duration on possible induction of reentries have been studied. Results obtained show that various combinations between these different parameters may trigger reentries, ventricular tachycardia and/or more complex patterns assimilable to ventricular fibrillation. These results clearly evidence the fact that possible induction of transient blocks may directly be related to risk factor associated to arrhythmogenic effects of antiarrhythmic drugs. (shrink)
Two populations are subdivided into two categories of individuals (hawks and doves). Individuals fight to have access to a resource which is necessary for their survival. Conflicts occur between individuals belonging to the same population and to different populations. We investigate the long term effects of the conflicts on the stability of the community. The modelis a set of ODE's with four variables corresponding to hawk and dove individuals of the two populations. Two time scales are considered. A fast time (...) scale is used to describe frequent encounters and fightings between individuals trying to monopolize the resource. A slow time scale is used for the demography and the long term effects of encounters. We use aggregation methods in order to reduce this model into a system of two ODE's only for the total densities of the two populations which is found to be a classical Lotka-Volterra competition model. We study different cases of proportions of hawks and doves in both populations on the global coexistence and the mutal exclusion of the two populations. Pure dove tactics in both populations are unstable. In cases of mixed hawk and dove in both populations, there is coexistence. Pure dove or mixed hawk-dove tactics in one population can coexist with pure hawks in the other one when the costs of fightings between hawks are large enough. (shrink)
The aim of this work is to present aggregation methods of hierarchically organized systems allowing one to replace the initial micro-system by a macro-system described by a few global variables. We also study the relations between the fast micro-dynamics and the slow macro-dynamics which can produce global properties. Emergence corresponds to a bottom-up coupling that is the result effected by a micro-level at a macro-level. As an example, we present prey-predator models with different time scales in an heterogeneous environment. A (...) fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to growth and interactions between the populations. Preys must go on spatial patches where resources are located and where predators can attack them. The efficiency of the predators to catch preys is patch dependent. Perturbation methods allow us to aggregate the initial system of differential equations for the patch sub-populations into a macro-system of two differential equations governing the total population densities. We study the case of density independent and density dependent migrations. In the latter case, we show that different functional responses can emerge in the macro prey-predator model as a result of the coupling between the slow and fast systems. (shrink)
Aggregation methods allow one to replace a large scale dynamical system (micro-system) by a reduced dynamical system (macro-system) governing a small number of global variables. This aggregation of variables can be performed when two time scales exist, a fast time scale and a slow time scale. Perturbation theory allows to obtain an approximated aggregated dynamical system which describes the behaviour of a few number of slow time varying variables which are constants of motion of the fast part of the micro-system. (...) Aggregation methods are applied to the case of the devastation of the great barrier reef by the starfishes. We recall the Antonelli/Kazarinoff model which implies a stable limit cycle for the corals and starfish populations. This prey-predator model describes the interactions between two species of corals and the starfish. Then, we generalize the Antonelli/Kazarinoff model to the case of two spatial patches with a fast part describing the starfish migration on the patches and the human manipulation of the communities by divers and, a slow part describing the growth and the interactions between the populations. We obtain an aggregated model governing the total coral densities on the patches and the total starfish population. This model can exhibit stable limit cycle oscillations and a Hopf bifurcation. The critical value of the bifurcation parameter is expressed in terms of the proportions of coral species and starfish on the two patches. This implies for example that rather than random killing of starfish by the Australian military, it may be better to send teams of divers to outbreaking reefs when they first occur who will then manipulate the community structure to increase protection. (shrink)
Mutual exclusion between congeneric species has been observed such as the case of the grey and red squirrels in Great Britain and the case of the twoHippolais warbler speciesHippolais icterina andH. polyglotta in Europe. This process can lead to the formation of an extinction wave which propagates. Two main assumptions are tested, competition and selective predation. The aim of this work is to present spatial models of these two processes. The animals of two species are assumed to move on a (...) two dimensional array of spatial patches with local interactions of competition or of selective predation between them. We focus on the case of mutual exclusion. Initially, the two competing species occupy complementary areas in an array of spatial patches with a small common zone. Numerical simulations show that under particular conditions, one species gets extinct and the other invades the whole set of spatial patches. These simulations show that with time, the length of the overlapping zone stabilizes and moves at a constant velocity. The limit length of the overlapping band and the velocity of the extinction wave are found to be functions of the parameters of the models. We relate this general model to the case of two sibling species of birds:H. icterina andH. polyglotta. (shrink)
The present work is aimed at investigating the effects of myocardial infarction and ischemia on induction of ventricular fibrillation. Electrophysiologic effects of global and local ischemia (variation of the dispersion of refractory periods as well as conduction velocity) on initiation of reentry mechanisms was studied by means of computer simulations based on a cellular automata model of propagation of activation wave through a ventricular surface element. A local area of ischemia where effects of the dispersion of refractory periods are investigated (...) is then simulated. This is made using a Gaussian distribution characterized by its mean and standard deviation. These simulations show that ischemia is capable of initiating reentry phenomena which propagate through the whole ventricle; they are responsible for ventricular fibrillation which causes sudden cardiac death, even when ischemia only involves limited parts of the myocardium. Statistical study of the probability of reentries as a function of both of the size of ischemic zones and the rate of dispersion of refractory periods shows that the latter parameter is of primary importance in triggering cardiac reentries. (shrink)
The complexity and the variability of parameters occurring in ecological dynamical systems imply a large number of equations.Different methods, more or less successful, have been described to reduce this number of equations. For instance, in the paper of Auger and Roussarie (1993), the authors describe how to obtain a reduction by considering different time-scales. They consider a system which can be sub-divided into sub-systems such that the strengths of the intra-sub-systems interactions are much larger than those of the inter-sub-systems interactions. (...) Using the Central Manifold Theorem, they obtain on the slow-manifold, a perturbation of the external dynamics by the internal dynamics. (shrink)
Aggregation of variables allows to approximate a large scale dynamical system (the micro-system) involving many variables into a reduced system (the macro-system) described by a few number of global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the micro-system. Perturbation methods enable to approximate the large micro-system by a macro-system going on at a slow time scale. Aggregation has been performed for systems of ordinary differential equations in which time is a continuous (...) variable. In this contribution, we extend aggregation methods to time-discrete models of population dynamics. Time discrete micro-models with two time scales are presented. We use perturbation methods to obtain a slow macro-model. The asymptotic behaviours of the micro and macro-systems are characterized by the main eigenvalues and the associated eigenvectors. We compare the asymptotic behaviours of both systems which are shown to be similar to a certain order. (shrink)
The aim of this work is to study complex ecological models exhibiting simple dynamics. We consider large scale systems which can be decomposed into weakly coupled subsystems. Perturbation Theory is used in order to get a reduced set of differential equations governing slow time varying global variables. As examples, we study the influence of the individual behaviour of animals in competition and predator-prey models. The animals are assumed to do many activities all day long such as searching for food of (...) different types. The degree of competition as well as the predation pressure are dependent upon these activities. Preys are more vulnerable when doing some activities during which they are very exposed to predators attacks rather than for others during which they are hidden. We study the effect of a change in the average individual behaviour of the animals on interspecific relationships. Computer simulations of the whole sets of equations are compared to simulations of the reduced sets of equations. (shrink)
We study the influence of the individual behaviour of animals on predator-prey models. Populations of preys and predators are divided into sub-populations corresponding to different activity classes. The animals are assumed to do many activities all day long such as searching for food of different types. The preys are more vulnerable when doing some activities during which they are very exposed to predators attacks rather than for others during which they are hidden. We study activity sequences of the animals and (...) also the effect of a change in the average individual behaviour of the animals on Lotka-Volterra prey-predator interactions. Numerical simulations are realized for the whole sets of equations (governing the subpopulations) and are compared to the simulations of the reduced sets of equation (governing the populations). We look for the validity of the method with respect to a scaling factor which measures the differences between the two time scales associated to the fast-varying variables and to the slow-time varying global variables. It is shown that when the two time scales differ of about two orders of magnitude, the approximation is satisfying. (shrink)
We study the case of two sibling species ofHippolais(Aves). Very little differences can be observed in the morphology of both species. The breeding area of these species are complementary. Roughly, one species breeds North and East of Europe (Hippolais icterina) while the other breeds South and West of Europe (Hippolais polyglotta). There exitst a narrow zone of sympatry passing through Burgundy. Since several years, it has been observed that this area of sympatry was moving in the North-East direction at a (...) European scale. This means that progressivelyH. icterina is declining and is replaced byH. polyglotta. Some assumptions can be made in order to explain this evolution, for instance competition or predation. Series of observations concerning the diets of nestlings of both species have been realized. These observations show some differences in the diet compositions. The breeding success of the two species has been studied. Numerical simulations of a competition model taking into account the observed differences between the food types eaten by the two species are presented. These simulations do not explain the regression ofH. icterina. Then, we present numerical simulations of a predation model with one predator attacking the nestlings of both species. These simulations show that with time one of the two preys must extinct. Predation rather than competition seems to be the right explanation. (shrink)
We assume the existence of a specific G1 protein which is an initiator of DNA replication. This initiator is supposed to be synthesized according to Michaelis-Menten kinetics. In order to start DNA replication, it is assumed that this G1 specific protein must be produced in a required amount. Intra-cellular growth inhibitors and extra-cellular growth factors control the production of the initiator. This model allows to calculate the average G1 phase time as a function of the various chemical concentrations of nutrients, (...) enzymes, growth inhibitors and growth factors. This model is compared to cell kinetics experiments on a leukemic cell line responding to Interleukin 3 deprivation. The curves giving the experimental average G1 phase times with respect to Interleukin-3 concentrations are fitted by the mathematical model with a quite good agreement. (shrink)
