Tewfik Sari

The Competitive Exclusion Principle is a cornerstone of ecology, predicting that n species cannot coexist on fewer than n limiting resources. While this principle is robustly validated in constant environments like the chemostat, it fails to account for the vast biodiversity observed in nature. In this study, we challenge this classical paradigm by investigating how periodic fluctuations in a thermal environment -rather than artificial variations in operating parameters -influence the outcome of single-resource competition. Combining mathematical analysis based on Floquet theory with numerical exploration, we demonstrate that periodic temperature variations can trigger robust, stable coexistence over a wide parameter space. We show that the domain of coexistence expands with increased nutrient enrichment (larger Sin) and slower environmental cycles. However, we also identify a critical trade-off: slower fluctuations induce high-amplitude population oscillations, which may increase the risk of extinction during periodic troughs. Our findings provide a biologically realistic mechanism for biodiversity and suggest that environmental periodicity is a fundamental driver of species coexistence in natural systems.

We investigate the classical model of competition of two populations in the chemostat when a linear coupling between the populations is taken into account and the removal rates of the populations are distinct from the dilution rate and their yield coefficients are also distinct. This model extends a model of wall growth and a model of lateral gene transfer, previously studied in the literature. We show the existence and uniqueness of the coexistence equilibrium at which the populations coexist, provided that the input concentration of the chemostat exceeds a critical value, or, equivalently the dilution rate does not exceed a critical value that can be computed explicitly. In contrast with the particular cases of this model, previously studied in the literature, the positive equilibrium can be unstable with the appearance of Hopf bifurcations and sustainable oscillations. We construct the operating diagram of the system, which is the two-parameter bifurcation diagram with respect to the operating parameters, that are the dilution rate of the chemostat and its input nutrient concentration. This study reveals a rich variety of dynamical behaviors, including the emergence and disappearance of stable and unstable limit cycles through Hopf bifurcations and limit point of cycles bifurcations. Furthermore, codimension-two bifurcations such as cusp and generalized Hopf points are identified, highlighting complex transitions in the system’s dynamics.

This paper is a follow-up to a previous work where we considered populations with time-varying growth rates living in patches and irreducible migration matrix between the patches. Each population, when isolated, would become extinct. Dispersal-induced growth (DIG) occurs when the populations are able to persist and grow exponentially when dispersal among the populations is present. In this paper, we consider the situation where the migration matrix is not necessarily irreducible. We provide a mathematical analysis of the DIG phenomenon, in the context of a deterministic model with periodic variation of growth rates and migration. Our results apply in the case, important for applications, where there is migration in one direction in one season and in the other direction in another season. We also consider dispersal-induced decay (DID), where each population, when isolated, grows exponentially, while populations die out when dispersal between populations is present.

In this paper, a flocculation model is considered in a chemostat with a single resource and one species in two forms: planktonic and aggregated bacteria. The removal rates are distinct and include the specific death rates. Considering distinct yield coefficients with a large class of growth rates, we present a mathematical analysis of the model by establishing the necessary and sufficient conditions of existence and local asymptotic stability of all equilibria. From these conditions, we first theoretically determine the operating diagram describing the asymptotic behavior of the system according to two operating parameters. The bifurcation analysis shows a rich behavior: transcritical, saddle-node, Hopf, and homoclinic bifurcations. Using the numerical method with MATCONT software, we detect fold bifurcation points of two stable and unstable limit cycles that emerge through supercritical and subcritical Hopf bifurcations, as well as two-parameter bifurcations such as Bogdanov–Takens, Cusp, and Generalized Hopf bifurcations. Our analysis shows that the flocculation process promotes the coexistence of isolated and attached bacteria around a stable limit cycle or positive equilibrium. These theoretical messages explain and reinforce the experimental observations about the positive effect of flocculation on the maintenance of species coexistence and the protection of the most relevant species among microbial ecosystems.

This paper is a follow-up to a previous work where we considered populations with time-varying growth rates living in patches and irreducible migration matrix between the patches. Each population, when isolated, would become extinct. Dispersal-induced growth (DIG) occurs when the populations are able to persist and grow exponentially when dispersal among the populations is present. In this paper, we consider the situation where the migration matrix is not necessarily irreducible. We provide a mathematical analysis of the DIG phenomenon, in the context of a deterministic model with periodic variation of growth rates and migration. Our results apply in the case, important for applications, where there is migration in one direction in one season and in the other direction in another season. We also consider dispersal-induced decay (DID), where each population, when isolated, grows exponentially, while populations die out when dispersal between populations is present.

This paper investigates the dynamics of a chemostat model incorporating two populations of one bacterial species: susceptible and virus-infected. Through the two operating parameters of the model, represented by the input concentration of the nutrient and the dilution rate of the chemostat, we analyze the existence and stability conditions of all possible equilibria, and then describe the operating diagram of the model, which is the bifurcation diagram giving its behavior with respect to those operating parameters, that visually depicts the various regions of stability of those equilibria. This diagram gives a better understanding of the complex interplay between bacterial populations growth, viral infection and environmental factors, in a controlled environment.