Abstract
In this study, a classical model describing a food web in a chemostat involving three species competing for non-reproducing, growth rate-limiting nutrient in which one of the competitors predates on one of the other competitors is considered. Quantitative analyses of non-negativity and boundedness of solution trajectories, dissipativity, behavior around equilibria, global stability and persistence of the model equations are analyzed. We present the global stability of equilibria by constructing a Lyapunov function. Hopf bifurcation theory is applied.
Highlights
In microbiology and population biology, the laboratory device chemostat extensively uses as research technique to culture microorganisms continuously under nutrient limitation in a controlled environment in order to study the general properties of its growth and interaction
We showed a detailed mathematical analysis of a classical food web model describing three species interaction competing for a single nutrient in which one of the competitors predates on one of the other competitors
We showed that the equilibria for system (2) are Locally Asymptotically Stable (LAS) if exist by using Routh-Hurwitz criterion
Summary
In microbiology and population biology, the laboratory device chemostat extensively uses as research technique to culture microorganisms continuously under nutrient limitation in a controlled environment in order to study the general properties of its growth and interaction. The authors (Li and Kuang, 2000) relaxed the assumptions that the functional responses are general monotone functions and discharge rates for the both population are distinct They studied a predator-prey food chain model in a chemostat, where. A tri-tropic level food-chain model is considered in (Chiu and Hsu, 1998) They restricted their attentions to study when the prey grows logistically in the absence of predator and the functional responses of predators are of Holling’s type II and established the criterion where the top-predator goes extinct globally. In this study we confine our interest to explore the criterion for global stability and persistence of the model for which all the individuals coexists in a steady state for all future time This model can be viewed as a system experimentally in lab by considering two bacteria as prey, a protozoan as predator and glucose as the limiting nutrient in the chemostat.
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