An analysis approach to permanence of a delay differential equations model of microorganism flocculation

Abstract

<p style='text-indent:20px;'>In this paper, we develop a delay differential equations model of microorganism flocculation with general monotonic functional responses, and then study the permanence of this model, which can ensure the sustainability of the collection of microorganisms. For a general differential system, the existence of a positive equilibrium can be obtained with the help of the persistence theory, whereas we give the existence conditions of a positive equilibrium by using the implicit function theorem. Then to obtain an explicit formula for the ultimate lower bound of microorganism concentration, we propose a general analysis method, which is different from the traditional approaches in persistence theory and also extends the analysis techniques of existing related works.</p>

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“…Now, we are in a position to discuss the persistence of system (2) in Ω. Following the definition in [2,10], system (2) is said to be uniformly persistent if there exists a ρ > 0 independent of the initial data such that ρ ≤ lim inf t→∞ ψ(t), where ψ = S, E, I, A, Q, R. Based on some analysis methods in [3,13,14,18,29], we will give an explicit eventual lower bound of COVID-19. Now let R c > 1, η ∈ (0, 1) and…”

Dynamical analysis for the sustained harvesting of microorganisms using flocculants in a random environment

“…Recently, there are some relevant studies on the use of mathematical models to describe the continuous culture of microorganisms and the flocculation process of microorganisms. [5][6][7][8][9] Zhang et al [7] investigated a stochastic microorganism flocculation model and obtained the existence of a stationary distribution when the noise intensities are small. In Ref.…”

“…In Ref. [9], Guo et al obtained complete conclusions on the permanence of a time-delayed microorganism flocculation model with general monotonic response functions.…”

Stationary Distribution and Extinction of a Stochastic Microbial Flocculation Model with Regime Switching

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