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Picoche C, Young WR, Barraquand F. Local intraspecific aggregation in phytoplankton model communities: spatial scales of occurrence and implications for coexistence. J Math Biol 2024; 88:68. [PMID: 38661851 DOI: 10.1007/s00285-024-02067-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2022] [Revised: 12/17/2023] [Accepted: 02/20/2024] [Indexed: 04/26/2024]
Abstract
The coexistence of multiple phytoplankton species despite their reliance on similar resources is often explained with mean-field models assuming mixed populations. In reality, observations of phytoplankton indicate spatial aggregation at all scales, including at the scale of a few individuals. Local spatial aggregation can hinder competitive exclusion since individuals then interact mostly with other individuals of their own species, rather than competitors from different species. To evaluate how microscale spatial aggregation might explain phytoplankton diversity maintenance, an individual-based, multispecies representation of cells in a hydrodynamic environment is required. We formulate a three-dimensional and multispecies individual-based model of phytoplankton population dynamics at the Kolmogorov scale. The model is studied through both simulations and the derivation of spatial moment equations, in connection with point process theory. The spatial moment equations show a good match between theory and simulations. We parameterized the model based on phytoplankters' ecological and physical characteristics, for both large and small phytoplankton. Defining a zone of potential interactions as the overlap between nutrient depletion volumes, we show that local species composition-within the range of possible interactions-depends on the size class of phytoplankton. In small phytoplankton, individuals remain in mostly monospecific clusters. Spatial structure therefore favours intra- over inter-specific interactions for small phytoplankton, contributing to coexistence. Large phytoplankton cell neighbourhoods appear more mixed. Although some small-scale self-organizing spatial structure remains and could influence coexistence mechanisms, other factors may need to be explored to explain diversity maintenance in large phytoplankton.
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Affiliation(s)
- Coralie Picoche
- Institute of Mathematics of Bordeaux, University of Bordeaux and CNRS, Talence, France
| | | | - Frédéric Barraquand
- Institute of Mathematics of Bordeaux, University of Bordeaux and CNRS, Talence, France.
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Rincón A, Hoyos FE, Restrepo G. Global stability of a continuous bioreactor model under persistent variation of the dilution rate. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:3396-3424. [PMID: 36899587 DOI: 10.3934/mbe.2023160] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
In this work, the global stability of a continuous bioreactor model is studied, with the concentrations of biomass and substrate as state variables, a general non-monotonic function of substrate concentration for the specific growth rate, and constant inlet substrate concentration. Also, the dilution rate is time varying but bounded, thus leading to state convergence to a compact set instead of an equilibrium point. Based on the Lyapunov function theory with dead-zone modification, the convergence of the substrate and biomass concentrations is studied. The main contributions with respect to closely related studies are: i) The convergence regions of the substrate and biomass concentrations are determined as function of the variation region of the dilution rate (D) and the global convergence to these compact sets is proved, considering monotonic and non-monotonic growth functions separately; ii) several improvements are proposed in the stability analysis, including the definition of a new dead zone Lyapunov function and the properties of its gradient. These improvements allow proving convergence of substrate and biomass concentrations to their compact sets, while tackling the interwoven and nonlinear nature of the dynamics of biomass and substrate concentrations, the non-monotonic nature of the specific growth rate, and the time-varying nature of the dilution rate. The proposed modifications are a basis for further global stability analysis of bioreactor models exhibiting convergence to a compact set instead of an equilibrium point. Finally, the theoretical results are illustrated through numerical simulation, showing the convergence of the states under varying dilution rate.
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Affiliation(s)
- Alejandro Rincón
- Grupo de Investigación en Desarrollos Tecnológicos y Ambientales (GIDTA), Facultad de Ingeniería y Arquitectura, Universidad Católica de Manizales, Carrera 23 N. 60-63, Manizales 170002, Colombia
- Grupo de Investigación en Microbiología y Biotecnología Agroindustrial (GIMIBAG), Instituto de Investigación en Microbiología y Biotecnología Agroindustrial, Universidad Católica de Manizales, Carrera 23 N. 60-63, Manizales 170002, Colombia
| | - Fredy E Hoyos
- Departamento de Energía Eléctrica y Automática, Facultad de Minas, Universidad Nacional de Colombia, Sede Medellín, Carrera 80 No. 65-223, Campus Robledo, Medellín 050041, Colombia
| | - Gloria Restrepo
- Grupo de Investigación en Microbiología y Biotecnología Agroindustrial (GIMIBAG), Instituto de Investigación en Microbiología y Biotecnología Agroindustrial, Universidad Católica de Manizales, Carrera 23 N. 60-63, Manizales 170002, Colombia
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Mtar T, Fekih-Salem R, Sari T. Interspecific density-dependent model of predator–prey relationship in the chemostat. INT J BIOMATH 2020. [DOI: 10.1142/s1793524520500862] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The objective of this study is to analyze a model of competition for one resource in the chemostat with general interspecific density-dependent growth rates, taking into account the predator–prey relationship. This relationship is characterized by the fact that the prey species promotes the growth of the predator species which in turn inhibits the growth of the first species. The model is a three-dimensional system of ordinary differential equations. With the same dilution rates, the model can be reduced to a planar system where the two models have the same local and even global behavior. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. Using the nullcline method, we present a geometric characterization of the existence and stability of all equilibria showing the multiplicity of coexistence steady states. The bifurcation diagrams illustrate that the steady states can appear or disappear only through saddle-node or transcritical bifurcations. Moreover, the operating diagrams describe the asymptotic behavior of this system by varying the control parameters and show the effect of the inhibition of predation on the emergence of the bistability region and the reduction until the disappearance of the coexistence region by increasing this inhibition parameter.
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Affiliation(s)
- Tahani Mtar
- University of Tunis El Manar, National Engineering School of Tunis, LAMSIN 1002, Tunis, Tunisia
| | - Radhouane Fekih-Salem
- University of Tunis El Manar, National Engineering School of Tunis, LAMSIN 1002, Tunis, Tunisia
- University of Monastir, Higher Institute of Computer Science of Mahdia, 5111 Mahdia, Tunisia
| | - Tewfik Sari
- ITAP, Univ Montpellier, INRAE, Institut Agro, Montpellier, France
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Fekih-Salem R, Lobry C, Sari T. A density-dependent model of competition for one resource in the chemostat. Math Biosci 2017; 286:104-122. [PMID: 28212840 DOI: 10.1016/j.mbs.2017.02.007] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2016] [Revised: 02/08/2017] [Accepted: 02/13/2017] [Indexed: 11/28/2022]
Abstract
This paper deals with a two-microbial species model in competition for a single-resource in the chemostat including general intra- and interspecific density-dependent growth rates with distinct removal rates for each species. In order to understand the effects of intra- and interspecific interference, this general model is first studied by determining the conditions of existence and local stability of steady states. With the same removal rate, the model can be reduced to a planar system and then the global stability results for each steady state are derived. The bifurcations of steady states according to interspecific interference parameters are analyzed in a particular case of density-dependent growth rates which are usually used in the literature. The operating diagrams show how the model behaves by varying the operating parameters and illustrate the effect of the intra- and interspecific interference on the disappearance of coexistence region and the occurrence of bi-stability region. Concerning the small enough interspecific interference terms, we would shed light on the global convergence towards the coexistence steady state for any positive initial condition. When the interspecific interference pressure is large enough this system exhibits bi-stability where the issue of the competition depends on the initial condition.
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Affiliation(s)
- Radhouane Fekih-Salem
- Université de Tunis El Manar, ENIT, LAMSIN, BP 37, Le Belvédère, 1002 Tunis, Tunisie; Université de Monastir, ISIMa, BP 49, Av Habib Bourguiba, 5111 Mahdia, Tunisie.
| | - Claude Lobry
- Université de Nice et MODEMIC, Le Gd Palais Bt6, 2 BD de Cimiez, 06000 Nice, France.
| | - Tewfik Sari
- IRSTEA, UMR Itap, 361 rue Jean-François Breton, 34196 Montpellier, France; Université de Haute Alsace, LMIA, 4 rue des frères Lumière, 68093 Mulhouse, France.
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Abdellatif N, Fekih-Salem R, Sari T. Competition for a single resource and coexistence of several species in the chemostat. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2016; 13:631-652. [PMID: 27775379 DOI: 10.3934/mbe.2016012] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We study a model of the chemostat with several species in competition for a single resource. We take into account the intra-specific interactions between individuals of the same population of micro-organisms and we assume that the growth rates are increasing and the dilution rates are distinct. Using the concept of steady-state characteristics, we present a geometric characterization of the existence and stability of all equilibria. Moreover, we provide necessary and sufficient conditions on the control parameters of the system to have a positive equilibrium. Using a Lyapunov function, we give a global asymptotic stability result for the competition model of several species. The operating diagram describes the asymptotic behavior of this model with respect to control parameters and illustrates the effect of the intra-specific competition on the coexistence region of the species.
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Affiliation(s)
- Nahla Abdellatif
- Universite de Tunis El Manar, ENIT, LAMSIN, BP 37, Le Belvedere, 1002 Tunis, Tunisia.
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Wade M, Harmand J, Benyahia B, Bouchez T, Chaillou S, Cloez B, Godon JJ, Moussa Boudjemaa B, Rapaport A, Sari T, Arditi R, Lobry C. Perspectives in mathematical modelling for microbial ecology. Ecol Modell 2016. [DOI: 10.1016/j.ecolmodel.2015.11.002] [Citation(s) in RCA: 41] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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El Hajji M, Rapaport A. Practical coexistence of two species in the chemostat - a slow-fast characterization. Math Biosci 2008; 218:33-9. [PMID: 19141299 DOI: 10.1016/j.mbs.2008.12.003] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2008] [Revised: 11/24/2008] [Accepted: 12/05/2008] [Indexed: 11/29/2022]
Abstract
We show that the chemostat model with two species having different but close break-even concentrations exhibits a slow-fast dynamics. Considering small perturbations about the dilution rate for which break-even concentrations are identical, we use the Fenichel theory to show the coexistence of species for large times. Then we determine the reduced dynamics, which is non-trivial and characterized by the slopes of the growth functions about their break-even concentrations.
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Affiliation(s)
- Miled El Hajji
- UMR Analyse des Systèmes et Biométrie, INRA, 2 Place Viala, 34060 Montpellier Cedex, France
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