The effect of delays on stability and persistence in plankton models

Ruan, Shigui

doi:10.1016/0362-546x(95)93092-i

Cited by 65 publications

(42 citation statements)

References 22 publications

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“…Recently, Beretta & Takeuchi (1994a, b) and studied the global stability of this model by applying the Liapunov functional method. For other related work, we refer to , Ruan (1995) and the references cited therein. Freedman & Xu (1993) extended the single species model in Beretta et al (1990) to a competition model of chemostat-type with delayed nutrient recycling.…”

Section: Introductionmentioning

confidence: 99%

Oscillations in Plankton Models with Nutrient Recycling

Ruan

2001

Journal of Theoretical Biology

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Plankton}nutrient interaction models with both instantaneous and delayed nutrient recycling are considered. The system consists of three components: autotrophic phytoplankton, herbivorous zooplankton and dissolved limiting nutrient. Local stability of the equilibria is analysed. It is shown that the positive equilibrium loses its stability when the nutrient input concentration passes through a critical value and the Hopf bifurcation occurs that induces oscillations of the populations. Numerical simulations are carried out to illustrate the obtained results.

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Section: Introductionmentioning

confidence: 99%

Oscillations in Plankton Models with Nutrient Recycling

Ruan

2001

Journal of Theoretical Biology

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“…His results show that when the system is characterized by oscillating behavior, an increase of the delay can have a stabilizing effect. The results in [26] indicate that if the kernel f (s) is a weak kernel, then the system (2.1) is uniformly persistent, a term used to describe long term survival of the interacting species. Global stability of some models related to system (2.1) has been studied by Beretta and Takeuchi [2][3][4].…”

Section: Continuously Delayed Nutrient Recyclingmentioning

confidence: 99%

“…Following Caperon [6] and Cunningham and Nisbet [8], in a previous paper [26] one of us introduced a discrete delay in the growth response of the species to nutrient uptake in the model of Beretta, Bischi and Solimano [1]. By using the discrete delay as a bifurcation parameter, it was shown that the model undergoes a Hopf bifurcation.…”

Section: Introductionmentioning

confidence: 99%

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Global stability in chemostat-type plankton models with delayed nutrient recycling

Ruan²

1998

Journal of Mathematical Biology

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Abstract. In this paper, we consider chemostat-type plankton models in which plankton feeds on a limiting nutrient and the nutrient is supplied at a constant rate and is partially recycled after the death of plankton by bacterial decomposition. We use a distributed delay to describe nutrient recycling and a discrete delay to model the planktonic growth response to nutrient uptake. When one or both delays occur, by constructing Liapunov functionals, we obtain some sufficient conditions for the global attractivity of the positive equilibrium, which can be regarded as estimates of the delays for persistence of global attractivity.

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“…Chemostats with periodic inputs were studied [1,2], those with periodic washout rate [3,4], and those with periodic input and washout [5]. In recent years, those with nutrient recycling [6][7][8][9][10] have been investigated and some investing results were obtained. Now many scholars pointed out that it was necessary to consider models with periodic perturbations, since those phenomena might be exposed in many real words.…”

Section: Introductionmentioning

confidence: 99%

Global Analysis of Beddington-DeAngelis Type Chemostat Model with Nutrient Recycling and Impulsive Input

Rehim¹,

Sun²,

Muhammadhaji³

2013

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In this paper, a Beddington-DeAngelis type chemostat model with nutrient recycling and impulsive input is considered. Except using Floquet theorem, introducing a new method combining with comparison theorem of impulse differential equation and by using the Liapunov function method, the sufficient and necessary conditions on the permanence and extinction of the microorganism are obtained. Two examples are given in the last section to verify our mathematical results. The numerical analysis shows that if only the system is permanent, then it also is globally attractive.

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The effect of delays on stability and persistence in plankton models

Cited by 65 publications

References 22 publications

Oscillations in Plankton Models with Nutrient Recycling

Oscillations in Plankton Models with Nutrient Recycling

Global stability in chemostat-type plankton models with delayed nutrient recycling

Global Analysis of Beddington-DeAngelis Type Chemostat Model with Nutrient Recycling and Impulsive Input

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