Spatio-temporal delays in a nutrient-plankton model on a finite domain: linear stability and bifurcations

Gourley, Stephen A.; Ruan, Shigui

doi:10.1016/s0096-3003(02)00494-0

Cited by 19 publications

(8 citation statements)

References 16 publications

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“…These studies show that a Turing instability is possible in such systems (Ruan, 1998;Boushaba and Ruan, 2001), but taking into account the diffusive transport of individuals during the time delay results in no Turing instability on a bounded domain (Gourley and Ruan, 2002). The Turing bifurcation has also been observed to fail for the modified Lengyel-Epstein twovariable model with time delays, used to simulate a delayed illumination feedback for the photosensitive chlorine dioxide-iodine-malonic acid (CDIMA) reaction (Li and Ji, 2004).…”

Section: The Reaction Diffusion Mechanismmentioning

confidence: 97%

“…There have been a number of studies of reaction diffusion systems in the presence of time delays (Ruan, 1998;Boushaba and Ruan, 2001;Gourley and Ruan, 2002) in the context of nutrient recycling for plankton populations, with an overall emphasis on the Turing instability. These studies show that a Turing instability is possible in such systems (Ruan, 1998;Boushaba and Ruan, 2001), but taking into account the diffusive transport of individuals during the time delay results in no Turing instability on a bounded domain (Gourley and Ruan, 2002).…”

Section: The Reaction Diffusion Mechanismmentioning

confidence: 99%

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Gene Expression Time Delays and Turing Pattern Formation Systems

Gaffney

Monk

2006

Bltn. Mathcal. Biology

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The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this paper, we investigate the effects of incorporating gene expression time delays into a one-dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. While oscillatory behaviour is rare, we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning.

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Section: The Reaction Diffusion Mechanismmentioning

confidence: 97%

Section: The Reaction Diffusion Mechanismmentioning

confidence: 99%

Gene Expression Time Delays and Turing Pattern Formation Systems

Gaffney

Monk

2006

Bltn. Mathcal. Biology

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“…He has found diffusion-driven instability of the reaction-diffusion system and a family of traveling waves in the presence of delay. Further, Gourley and Ruan [34] examined the role of time delay with a weighted spatial averaging to account for the movement of individuals during the time delay period on a diffusive NP model and found a homogenizing role of the averaging. In the present study, we have also investigated a NP system with diffusion, but our aim was completely different; we have examined the role of toxic effect on the spatial distribution of phytoplankton.…”

Section: Discussionmentioning

confidence: 99%

Spatial dynamics of a nutrient–phytoplankton system with toxic effect on phytoplankton

Chakraborty

Tiwari

Misra

et al. 2015

Mathematical Biosciences

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Chattopadhyay, J. (2015). Spatial dynamics of a nutrientphytoplankton system with toxic effect on phytoplankton. Mathematical Biosciences, 264, 94-100. https://doi.org/10.1016Biosciences, 264, 94-100. https://doi.org/10. /j.mbs.2015 Spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton AbstractThe production of toxins by some species of phytoplankton is known to have several economic, ecological, and human health impacts. However, the role of toxins on the spatial distribution of phytoplankton is not well understood. In the present study, the spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton is investigated. We analyze the linear stability of the system and obtain the condition for Turing instability. In the presence of toxic effect, we find that the distribution of nutrient and phytoplankton becomes inhomogeneous in space and results in different patterns, like stripes, spots, and the mixture of them depending on the toxicity level. We also observe that the distribution of nutrient and phytoplankton shows spatiotemporal oscillation for certain toxicity level.

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“…Since the boundary conditions are homogeneously Neumann, according to [Xu & Yuan, 2015;Pang & Wang, 2004;Gourley & Ruan, 2003] the characteristic equation of system (4) takes the following form…”

Section: Now We Letmentioning

confidence: 99%

Dynamic Analysis of a Reaction–Diffusion Rumor Propagation Model

Zhao

Zhu

2016

Int. J. Bifurcation Chaos

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The rapid development of the Internet, especially the emergence of the social networks, leads rumor propagation into a new media era. Rumor propagation in social networks has brought new challenges to network security and social stability. This paper, based on partial differential equations (PDEs), proposes a new SIS rumor propagation model by considering the effect of the communication between the different rumor infected users on rumor propagation. The stabilities of a nonrumor equilibrium point and a rumor-spreading equilibrium point are discussed by linearization technique and the upper and lower solutions method, and the existence of a traveling wave solution is established by the cross-iteration scheme accompanied by the technique of upper and lower solutions and Schauder's fixed point theorem. Furthermore, we add the time delay to rumor propagation and deduce the conditions of Hopf bifurcation and stability switches for the rumor-spreading equilibrium point by taking the time delay as the bifurcation parameter. Finally, numerical simulations are performed to illustrate the theoretical results.

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Spatio-temporal delays in a nutrient-plankton model on a finite domain: linear stability and bifurcations

Cited by 19 publications

References 16 publications

Gene Expression Time Delays and Turing Pattern Formation Systems

Gene Expression Time Delays and Turing Pattern Formation Systems

Spatial dynamics of a nutrient–phytoplankton system with toxic effect on phytoplankton

Dynamic Analysis of a Reaction–Diffusion Rumor Propagation Model

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