143 research outputs found
Competitive exclusion for chemostat equations with variable yields
In this paper, we study the global dynamics of a chemostat model with a
single nutrient and several competing species. Growth rates are not required to
be proportional to food uptakes. The model was studied by Fiedler and Hsu [J.
Math. Biol. (2009) 59:233-253]. These authors prove the nonexistence of
periodic orbits, by means of a multi-dimensional Bendixon-Dulac criterion. Our
approach is based on the construction of Lyapunov functions. The Lyapunov
functions extend those used by Hsu [SIAM J. Appl. Math. (1978) 34:760-763] and
by Wolkowicz and Lu [SIAM J. Appl. Math. (1997) 57:1019-1043] in the case when
growth rates are proportional to food uptakes
Global dynamics of the chemostat with variable yields
In this paper, we consider a competition model between species in a
chemostat including both monotone and non-monotone response functions, distinct
removal rates and variable yields. We show that only the species with the
lowest break-even concentration survives, provided that additional technical
conditions on the growth functions and yields are satisfied. LaSalle's
extension theorem of the Lyapunov stability theory is the main tool.Comment: 7 page
The Operating Diagram for a Two-Step Anaerobic Digestion Model
The Anaerobic Digestion Model No. 1 (ADM1) is a complex model which is widely
accepted as a common platform for anaerobic process modeling and simulation.
However, it has a large number of parameters and states that hinder its
analytical study. Here, we consider the two-step reduced model of anaerobic
digestion (AM2) which is a four-dimensional system of ordinary differential
equations. The AM2 model is able to adequately capture the main dynamical
behavior of the full anaerobic digestion model ADM1 and has the advantage that
a complete analysis for the existence and local stability of its steady states
is available. We describe its operating diagram, which is the bifurcation
diagram which gives the behavior of the system with respect to the operating
parameters represented by the dilution rate and the input concentrations of the
substrates. This diagram, is very useful to understand the model from both the
mathematical and biological points of view
Averaging in Hamiltonian systems with slowly varying parameters
The aim of this paper is to describe the general averaging principle and to discuss the particular case of single-frequency systems, the case of systems with constant frequencies and the case of Hamiltonian systems. We show how the stroboscopic method, which is a method of the nonstandard perturbation theory of differential equations, can be used in this kind of problems. We give various examples which illustrate the simplicity and the effectiveness of the method
Predicting coexistence of plants subject to a tolerance-competition trade-off
Ecological trade-offs between species are often invoked to explain species
coexistence in ecological communities. However, few mathematical models have
been proposed for which coexistence conditions can be characterized explicitly
in terms of a trade-off. Here we present a model of a plant community which
allows such a characterization. In the model plant species compete for sites
where each site has a fixed stress condition. Species differ both in stress
tolerance and competitive ability. Stress tolerance is quantified as the
fraction of sites with stress conditions low enough to allow establishment.
Competitive ability is quantified as the propensity to win the competition for
empty sites. We derive the deterministic, discrete-time dynamical system for
the species abundances. We prove the conditions under which plant species can
coexist in a stable equilibrium. We show that the coexistence conditions can be
characterized graphically, clearly illustrating the trade-off between stress
tolerance and competitive ability. We compare our model with a recently
proposed, continuous-time dynamical system for a tolerance-fecundity trade-off
in plant communities, and we show that this model is a special case of the
continuous-time version of our model.Comment: To be published in Journal of Mathematical Biology. 30 pages, 5
figures, 5 appendice
The peaking Phenomenon and Singular Perturbations : An Extension of Tikhonov's Theorem
We study the asymptotic behaviour, when the parameter tends to infinity, of a class of singularly perturbed triangular systems , . The first equation may be considered as a control system recieving the inputs from the states of the second equation. With zero input, the origin of the first equation is globally asymptotically stable. We assume that all solutions of the second equation tend to zero arbitrarily fast when tends to infinity. Some states of the second equation may peak to very large values, before they rapidly decay to zero. Such peaking states can destabilize the first equation. The paper introduces the concept of \em instantaneous stability, to measure the fast decay to zero of the solutions of the second equation, and the concept of uniform infinitesimal boundedness to measure the effects of peaking on the first equation. Whe show that all the solutions of the triangular system tend to zero when and tend to infinity. Our results are a generalization of the classical Tikhonov's theorem of singular perturbation theory, concerning the asymptotic behaviour of the solutions in the particular case where the second equation is of the form . Our results are formulated in both classical mathematics and nonstandard analysis
Operating diagram of a flocculation model in the chemostat
The objective of this study is to analyze a model of the chemostat involving the attachment and detachment dynamics of planktonic and aggregated biomass in the presence of a single resource. Considering the mortality of species, we give a complete analysis for the existence and local stability of all steady states for general monotonic growth rates. The model exhibits a rich set of behaviors with a multiplicity of coexistence steady states, bi-stability, and occurrence of stable limit cycles. Moreover, we determine the operating diagram which depicts the asymptotic behavior of the system with respect to control parameters. It shows the emergence of a bi-stability region through a saddle-node bifurcation and the occurrence of coexistence region through a transcritical bifurcation. Finally, we illustrate the importance of the mortality on the destabilization of the microbial ecosystem by promoting the washout of species.L'objectif de cette étude est d'analyser un modèle du chémostat impliquant la dynamique d'attachement et de détachement de la biomasse planctonique et agrégée en présence d'une seule ressource. En considérant la mortalité des espèces, nous donnons une analyse complète de l'existence et de la stabilité locale de tous les équilibres pour des taux de croissance monotones. Le modèle pré-sente un ensemble riche de comportements avec multiplicité d'équilibres de coexistence, bi-stabilité et apparition des cycles limites stables. De plus, nous déterminons le diagramme opératoire qui dé-crit le comportement asymptotique du système par rapport aux paramètres de contrôle. Il montre l'émergence d'une région de bi-stabilité via une bifurcation noeud col et l'occurrence d'une région de coexistence via une bifurcation transcritique. Enfin, nous illustrons l'importance de la mortalité sur la déstabilisation de l'écosystème microbien en favorisant le lessivage des espèces
Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation
The standard model for the dynamics of a fragmented density-dependent population is built from several local logistic models coupled by migrations. First introduced in the 1970s and used in innumerable articles, this standard model applied to a two-patch situation has never been completely analysed. Here, we complete this analysis and we delineate the conditions under which fragmentation associated to dispersal is either beneficial or detrimental to total population abundance. Therefore, this is a contribution to the SLOSS question. Importantly, we also show that, depending on the underlying mechanism, there is no unique way to generalize the logistic model to a patchy situation. In many cases, the standard model is not the correct generalization. We analyse several alternative models and compare their predictions. Finally, we emphasize the shortcomings of the logistic model when written in the rr-KK parameterization and we explain why Verhulst’s original polynomial expression is to be preferred
Asymmetric dispersal in the multi-patch logistic equation
The standard model for the dynamics of a fragmented density-dependent population is built from several local logistic models coupled by migrations. First introduced in the 1970s and used in innumerable articles, this standard model applied to a two-patch situation has never been fully analyzed. Here, we complete this analysis and we delineate the conditions under which fragmentation associated with dispersal is either favorable or unfavorable to total population abundance. We pay special attention to the case of asymmetric dispersal, i.e., the situation in which the dispersal rate from patch 1 to patch 2 is not equal to the dispersal rate from patch 2 to patch 1. We show that this asymmetry can have a crucial quantitative influence on the effect of dispersal
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