Stability and Hopf bifurcation for a delay competition diffusion system

Zhou, Li; Tang, Yanbin; Hussein, Shawgy

doi:10.1016/s0960-0779(02)00068-1

Cited by 86 publications

(53 citation statements)

References 8 publications

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“…There has been some papers on spatial patterns in predator-prey systems with time delay [49][50][51][52][53]. However, it has been observed in the literature that the effect of delay on the synchronization, especially combined with spatial diffusion, had been generally overlooked despite their potential ecological reality and intrinsic theoretical interest.…”

Section: Discussionmentioning

confidence: 99%

Analysis of a spatial predator-prey model with delay

Wang

Liu

et al. 2010

Nonlinear Dyn

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In this paper, we present a Holling-Tanner model combined with diffusion and time delay. We found that, when time delay is small, there is no the synchronization of prey and the predator. However, when it is larger, there is antiphase synchronization. Furthermore, a transition from anti-phase synchronization to in-phase synchronization emerges as time delay further increases. Since synchronization of populations could lead them to be extinct, the results obtained may indicate that time delay plays an important role on the persistence of the populations.

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Section: Discussionmentioning

confidence: 99%

Analysis of a spatial predator-prey model with delay

Wang

Liu

et al. 2010

Nonlinear Dyn

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“…We omit the detailed proof for the stability of the bifurcation periodic solutions and their asymptotic expressions as the calculation is tedious. Readers interested in the details of these arguments are referred to [8,9,13,18], where a detailed discussion of these stability results is given.…”

Section: Stability Of the Hopf Bifurcation Solutionsmentioning

confidence: 99%

“…The occurrence of spatial form and pattern evolving from a spatially homogeneous initial state is a fundamental problem in developmental biology. For the memory function f (t) = δ(t) with δ being the Dirac delta function, one gets the case of the so-called zero delay, (u * , v * ) is globally asymptotically stable for all positive initial values [16], for f (t) = δ(t − r), the case of discrete delay, there exists a positive critical value r 0 such that (u * , v * ) is locally asymptotically stable for 0 ≤ r < r 0 and the Hopf bifurcation periodic solutions bifurcate from (u * , v * ) as r passing through r 0 [15,17,18].…”

Section: §1 Introductionmentioning

confidence: 99%

Hopf Bifurcation and Stability of a Competition Diﬀusion System with Distributed Delay

Tang¹,

Zhou²

2005

Publ. Res. Inst. Math. Sci.

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In this paper we investigate the effects of time delay and diffusion rate on the stability of the positive steady state for a competition diffusion system with distributed delay. We obtain the condition of instability of the positive uniform steady state. Regarding the time delay as bifurcation parameter, we reduce the original system on the center manifold and get the stability of the Hopf bifurcation periodic solutions. Finally, regarding the diffusion rate as a bifurcation parameter, we show that the onedimensional competition diffusion system with infinite delay and Dirichlet boundary condition exhibits the spatiotemporal structures near the steady state of the system. §1. Introduction Yamada and Niikura [14] considered the following semilinear parabolic systemwhere u ∈ R n , x ∈ Ω, a bounded domain in R N with smooth boundary, α is a∂u ∂n is the outward normal derivative on ∂Ω. They established the existence conditions of

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“…Some models have been incorporated time delays due to maturation time, capturing time, or other reasons. Many researchers have studied delayed predator-prey models and conclude that time delay in predator-prey system may have significant impact on the underlying dynamics [30][31][32][33][34][35][36]. Especially, Time delays contribute critically to the stable or unstable outcome of prey densities due to predation.…”

Section: Introductionmentioning

confidence: 99%

The effect of prey refuge and time delay on a diffusive predator‐prey system with hyperbolic mortality

Yang

Zhang

2016

Complexity

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In this article, we investigate the effect of prey refuge and time delay on a diffusive predator-prey system with Holling II functional response and hyperbolic mortality subject to Neumann boundary condition. More precisely, we study Turing instability of positive equilibrium by using refuge as parameter, instability and Hopf bifurcation induced by time delay. In addition, by the theory of normal form and center manifold, we derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.

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Stability and Hopf bifurcation for a delay competition diffusion system

Cited by 86 publications

References 8 publications

Analysis of a spatial predator-prey model with delay

Analysis of a spatial predator-prey model with delay

Hopf Bifurcation and Stability of a Competition Diﬀusion System with Distributed Delay

The effect of prey refuge and time delay on a diffusive predator‐prey system with hyperbolic mortality

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