• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1269-1297
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• 2020

In this paper, we consider a two-species parabolic-parabolic-elliptic chemotaxis system with weak competition and a fractional diffusion of order s ∈ (0, 2). It is proved that for s > 2p₀, where p₀ is a nonnegative constant depending on the system's parameters, there admits a global classical solution. Apart from this, under the circumstance of small chemotactic strengths, we arrive at the global asymptotic stability of the coexistence steady state.

• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.593-611
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• 2008

Global attractivity and oscillatory behavior of the following nonlinear impulsive parabolic differential equation which is a general form of many population models $$\array{\{{{\frac {{\partial}u(t,x)}{{\partial}t}=\Delta}u(t,x)-{\delta}u(t,x)+f(u(t-\tau,x)),\;t{\neq}t_k,\\u(t^+_k,x)-u(t_k,x)=g_k(u(t_k,x)),\;k{\in}I_\infty,}\;\;\;\;\;\;\;\;(*)$$ are considered. Some new sufficient conditions for global attractivity and oscillation of the solutions of (*) with Neumann boundary condition are established. These results no only are true but also improve and complement existing results for (*) without diffusion or impulses. Moreover, when these results are applied to the Nicholson's blowflies model and the model of Hematopoiesis, some new results are obtained.

• The Pure and Applied Mathematics
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• v.17 no.1
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• pp.93-106
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• 2010

We employ the methods of Lattice Dynamical System to establish a global model extending the Walrasian evolutionary cobweb model in an independent single local market to the global market evolution over an infinite chain of many local markets with interaction of each other through a diffusion of prices between them. For brevity of the model, we assume linear decreasing demands and logistic supplies with naive predictors, and investigate the traveling wave behaviors of global price dynamics and show that their dynamics are conjugate to those of H$\acute{e}$non maps and hence can exhibit complicated behaviors such as period-doubling bifurcations, chaos, and homoclic orbits etc.

• The Pure and Applied Mathematics
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• v.16 no.2
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• pp.243-254
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• 2009

The main purpose of this paper is to use the methods of Lattice Dynamical System to establish a global model, which extends the Walrasian evolutionary cobweb model in an independent single local market to the global market evolution over an infinite chain of many local markets interacting each other through a diffusion of prices between them. For brevity of the model, we assume linear decreasing demands and quadratic supplies with naive predictors, and investigate the spatially homogeneous global price dynamics and show that the dynamics is topologically conjugate to that of well-known logistic map and hence undergoes a period-doubling bifurcation route to chaos as a parameter varies through a critical value.

• 한국연소학회:학술대회논문집
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• 2002.06a
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• pp.68-75
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• 2002

The dynamic behaviors of the single vortex and flame-vortex interaction in a $CH_4/Air$ diffusion flame with addition of $CO_2$ were investigated numerically. The numerical method was based on a predictor-corrector for low Mach number flow. A two-step global reaction mechanism was adopted as a combustion model. Through comparison of results by effect of $CO_2$ added either on the fuel or oxidizer side, it was found that the growth of single vortex and entrainment of surrounding fluid by $CO_2$addition on the fuel side are larger than those by $CO_2$ addition on oxidizer side. Also, when $CO_2$ is added on fuel side, flame-vortex interaction becomes more significant than on air side.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.402-418
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• 2022

Noise is a fundamental factor to increased validity and regularity of spike propagation and neuronal firing in the nervous system. In this paper, we examine the stochastic version of the Izhikevich-FitzHugh neuron dynamical model. This approach is based on techniques presented by Luo and Guo, which provide a general framework for the bifurcation and stability analysis of two dimensional stochastic dynamical system as an Itô averaging diffusion system. By using largest lyapunov exponent, local and global stability of the stochastic system at the equilibrium point are investigated. We focus on the two kinds of stochastic bifurcations: the P-bifurcation and the D-bifurcations. By use of polar coordinate, Taylor expansion and stochastic averaging method, it is shown that there exists choices of diffusion and drift parameters such that these bifurcations occurs. Finally, numerical simulations in various viewpoints, including phase portrait, evolution in time and probability density, are presented to show the effects of the diffusion and drift coefficients that illustrate our theoretical results.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.35 no.6
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• pp.625-632
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• 2011

The ignition characteristics of a $CH_4$/hot air counterflow diffusion flame were investigated numerically using a flame-controlling continuation method. For the chemical reactions, the GRI-v1.2 reaction mechanism was used in the simulation. The maximum flame temperature was presented in the space of the inverse global strain rate, and showed S-curve-type behavior. The flame temperatures and velocities of the upper and middle branches were compared for different global strain rates. In addition, the global strain rate was compared with the local strain rates defined at the flame surface and the boundaries of the fuel and oxidizer sides of the fuel/air mixing layer. These local strain rates correlated well with the global strain rate.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.30 no.10 s.253
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• pp.996-1002
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• 2006

Experiments in methane-air low strain rate counterflow diffusion flames diluted with nitrogen have been conducted to study the behavior of flame extinction and edge flame oscillation in which flame length is less than the burner diameter and thus lateral conduction heat loss in addition to radiative heat loss could be remarkable at low global strain rates. Critical mole fraction at flame extinction is examined with velocity ratio and global strain rate. Onset conditions of edge flame oscillation and flame oscillation modes are also provided with global strain rate and added nitrogen mole fraction to fuel stream (fuel Lewis number). It is seen that flame length is closely relevant to lateral heat loss, and this affects flame extinction and edge flame oscillation considerably. Edge flame oscillations in low strain rate flames are experimentally described well and are categorized into three: a growing oscillation mode, a decaying oscillation mode, and a harmonic oscillation mode. The regime of flame oscillation is also provided at low strain rate flames. Important contribution of lateral heat loss even to edge flame oscillation is clarified

• Bulletin of the Korean Chemical Society
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• v.23 no.11
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• pp.1595-1603
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• 2002

In this paper we have presented the results of diffusion behavior of model systems for eight liquid n-alkanes ($C_{12}$-$C_{44}$) in a canonical (NVT) ensemble at several temperatures using molecular dynamics simulations. For these n-alkanes of small chain length n, the chains are clearly <$R_{ee}^2$>/6<$R_g^2$>>1 and non-Gaussian. This result implies that the liquid n-alkanes over the whole temperatures considered are far away from the Rouse regime, though the ratio becomes close to the unity as n increases. Calculated self-diffusion constants $D_{self}$ are comparable with experimental results and the Arrhenius plot of self-diffusion constants versus inverse temperature shows a different temperature dependence of diffusion on the chain length. The global rotational motion of n-alkanes is examined by characterizing the orientation relaxation of the end-to-end vector and it is found that the ratio ${\tau}1/{\tau}2$ is less than 3, the value expected for a isotropically diffusive rotational process. The friction constants ${\xi}$of the whole molecules of n-alkanes are calculated directly from the force auto-correlation (FAC) functions and compared with the monomeric friction constants ${\xi}_D$ extracted from $D_{self}$. Both the friction constants give a correct qualitative trends: decrease with increasing temperature and increase with increasing chain length. The friction constant calculated from the FAC's decreases very slowly with increasing temperature, while the monomeric friction constant varies rapidly with temperature. By considering the orientation relaxation of local vectors and diffusion of each site, it is found that rotational and translational diffusions of the ends are faster than those of the center.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.29-44
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• 2008

By employing the continuation theorem of coincidence degree theory, we derive a sufficient condition for the existence and attractricity of a positive periodic solution for a generalized predator-prey model with diffussion feedback controls.