Persistence and global stability in a delayed predator–prey system with Michaelis–Menten type functional response

Xu, Rui; Chaplain, M. A. J.

doi:10.1016/s0096-3003(01)00111-4

Cited by 18 publications

(11 citation statements)

References 20 publications

(15 reference statements)

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“…From Proposition 6 we can see that the solution of system (1) and (3) is bounded, and so are the derivatives of ( − * ) ( = 1, 2, 3) by the equations in (1). Applying Lemma 1, we obtain that…”

Section: Theorem 8 Assume Thatmentioning

confidence: 88%

“…For example, the predator-prey system for three species with Michaelis-Menten type functional response was studied by many authors [1][2][3][4]. However, the systems in [1][2][3][4] are either with discrete delay or without delay or without diffusion. In view of individuals taking time to move, spatial dispersal was dealt with by introducing diffusion term to corresponding delayed ODE model in previous literatures, namely, adding a Laplacian term to the ODE model.…”

Section: Introductionmentioning

confidence: 99%

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Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

Gan

Tian

Zhang

et al. 2013

Abstract and Applied Analysis

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This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.

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Section: Theorem 8 Assume Thatmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

Gan

Tian

Zhang

et al. 2013

Abstract and Applied Analysis

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“…The notion of global stability is studied by many authors in the predator-prey systems with delay [Beretta & Kuang, 1998;Xu & Chaplain, 2002;Aziz Alaoui & Daher Okiye, 2003].…”

Section: Introduction and Mathematical Modelmentioning

confidence: 99%

Bifurcation and Stability in a Delayed Predator–Prey Model with Mixed Functional Responses

Yafia¹,

Aziz-Alaoui

Merdan

et al. 2015

Int. J. Bifurcation Chaos

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The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003;Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie-Gower functional response as well as that of Beddington-DeAngelis. In this paper, we consider the model with one delay consisting of a unique nontrivial equilibrium E * and three others which are trivial. Their dynamics are studied in terms of local and global stabilities and of the description of Hopf bifurcation at E * . At the third trivial equilibrium, the existence of the Hopf bifurcation is proven as the delay (taken as a parameter of bifurcation) that crosses some critical values.

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“…In (1.1), it has been assumed that the prey grows logistically with growth rate r 1 (t) and carrying capacity r 1 (t)/a 11 (t) in the absence of predation. The predator consumes the prey according to the functional response f (t, x 1 (t)) and grows logistically with growth rate r 2 (t) and carrying capacity x 1 (t)/a 21 (t) proportional to the population size of the prey (or prey abundance).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, many authors have explored the dynamics of predator-prey systems with Holling type functional responses [1,3,4,7,[9][10][11][12][13][14]. Furthermore, some authors [15,16] have also described a type IV functional response that is humped and that declines at high prey densities.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a delayed discrete semi-ratio-dependent predator-prey system with Holling type IV functional response

Wang

2011

Adv Differ Equ

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A discrete semi-ratio-dependent predator-prey system with Holling type IV functional response and time delay is investigated. It is proved the general nonautonomous system is permanent and globally attractive under some appropriate conditions. Furthermore, if the system is periodic one, some sufficient conditions are established, which guarantee the existence and global attractivity of positive periodic solutions. We show that the conditions for the permanence of the system and the global attractivity of positive periodic solutions depend on the delay, so, we call it profitless.

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Persistence and global stability in a delayed predator–prey system with Michaelis–Menten type functional response

Cited by 18 publications

References 20 publications

Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

Bifurcation and Stability in a Delayed Predator–Prey Model with Mixed Functional Responses

Dynamics of a delayed discrete semi-ratio-dependent predator-prey system with Holling type IV functional response

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