Dynamics of Notch Activity in a Model of Interacting Signaling Pathways

Bani-Yaghoub, Majid; Amundsen, David E.

doi:10.1007/s11538-009-9469-8

Cited by 6 publications

(6 citation statements)

References 52 publications

(69 reference statements)

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“…Namely the individuals never cross the boundaries, although they can live and freely move on the boundaries. This has been used in several studies (see for example [3,13,15]). Combining the zero-flux and Dirichlet boundary conditions gives rise to mixed boundary conditions, where the flux at each boundary is proportional to the population density.…”

Section: Model Developmentmentioning

confidence: 99%

Modeling and Numerical Simulations of Single Species Dispersal in Symmetrical Domains

Bani-Yaghoub¹,

Yao²,

Reed³

2014

Preprint

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We develop a class of nonlocal delay Reaction-Diffusion (RD) models in a circular domain. Previous modeling efforts include RD population models with respect to onedimensional unbounded domain, unbounded strip and rectangular spatial domain. However, the importance of an RD model in a symmetrical domain lies in the increasing number of empirical studies conducted with respect to symmetrical natural habitats of single species. Assuming that the single species has no directional preference to spread in the symmetrical domain, the RD model is reduced to an equation with no angular dependance. The model can be further reduced by considering the birth function in the form of the Bessel function of the first kind. We numerically simulate the reduced forms of the nonlocal delay RD model to study the dispersal and growth of behaviors of the single species in a circular domain. Although spatial patterns of population densities are gradually developed, it is numerically shown that the single species population goes extinct in the absence of the birth function or it may converge to a positive equilibrium in the presence of the birth function.

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Section: Model Developmentmentioning

confidence: 99%

Modeling and Numerical Simulations of Single Species Dispersal in Symmetrical Domains

Bani-Yaghoub¹,

Yao²,

Reed³

2014

Preprint

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“…Ordinary and partial differential equations (ODEs and PDEs) have proven to be effective instruments in the study of real world problems in biology and medicine [7] [5] [9]. They will unquestionably continue to serve as interdisciplinary tools in future research in mathematical biology.…”

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confidence: 99%

Introduction to Delay Models and Their Wave Solutions

Bani-Yaghoub¹

2017

Preprint

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In this paper, a brief review of delay population models and their applications in ecology is provided. The inclusion of diffusion and nonlocality terms in delay models has given more capabilities to these models enabling them to capture several ecological phenomena such as the Allee effect, waves of invasive species and spatio-temporal competitions of interacting species. Moreover, recent advances in the studies of traveling and stationary wave solutions of delay models are outlined. In particular, the existence of stationary and traveling wave solutions of delay models, stability of wave solutions, formation of wavefronts in the special domain, and possible outcomes of delay models are discussed.

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“…Note that wavefronts and backs are qualitatively similar and general outcomes are satisfied for both of them. In addition to vast applications of traveling wavefronts in biology, various kinds of waves have been observed in chemical reactions ( [81], [179], [17] and references herein) as well as nonlinear optics [5], water waves [38], [39], [79], gas dynamics [160] and solid mechanics [171], [172], [173], [47]. Our primary concern in the present work is traveling and stationary wave solutions of the forms mentioned above.…”

Section: Waveforms In Biology and Ecologymentioning

confidence: 99%

“…Combining the zero-flux and Dirichlet boundary conditions, we get mixed boundary conditions where the flux at each boundary is proportional to the density at the boundary. and separating terms with h from terms with T we find two ordinary differential equations: (5 " 16) h rr + -h r + \h e e = A h, (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) /p…”

Section: A Nonlocal Rd Model In Symmetrical Domainmentioning

confidence: 99%

“…The former case is known as Turing instability or diffusion driven stability where spatial patterns arise from a steady state of an RD system that is stable in absence of diffusion but becomes unstable when diffusion is introduced. In our previous studies[17],[20],[18],[21] we demonstrated the relevance of neurite spontaneous symmetry breaking with Turing instabilities in a proposed RD model of interacting signaling pathways. In what follows we will show that in the population model (2.7), diffusion has stabilizing effects and the model (2.7) does not exhibit any Turing instability.A solution of (2.7) in a neighborhood of the steady state fa has the form It should be noted that the wave number k is not restricted to discrete numbers.…”

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confidence: 91%

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Wave solutions of nonlocal delayed reaction-diffusion equations

Bani-Yaghoub¹

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Dynamics of Notch Activity in a Model of Interacting Signaling Pathways

Cited by 6 publications

References 52 publications

Modeling and Numerical Simulations of Single Species Dispersal in Symmetrical Domains

Modeling and Numerical Simulations of Single Species Dispersal in Symmetrical Domains

Introduction to Delay Models and Their Wave Solutions

Wave solutions of nonlocal delayed reaction-diffusion equations

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