• Journal of the Korean Data and Information Science Society
• /
• 제25권5호
• /
• pp.1137-1149
• /
• 2014

Since the bifurcating autoregressive (BAR) model was developed by Cowan and Staudte (1986) to analyze cell lineage data, a lot of research has been directed to BAR and its generalizations. Based mainly on the author's works, this paper is concerned with a contemporary review on the BAR in terms of an overview and perspectives. Specifically, bifurcating structure is extended to multi-cast tree and to branching tree structure. The AR(1) time series model of Cowan and Staudte (1986) is generalized to tree structured random processes. Branching correlations between individuals sharing the same parent are introduced and discussed. Various methods for estimating parameters and related asymptotics are also reviewed. Consequently, the paper aims to give a contemporary overview on the BAR model, providing some perspectives to the future works in this area.

• Journal of the Korean Statistical Society
• /
• 제35권4호
• /
• pp.409-417
• /
• 2006

This article is concerned with threshold modeling of the bifurcating autoregressive model (BAR) originally suggested by Cowan and Staudte (1986) for tree structured data of cell lineage study where each individual $(X_t)$ gives rise to two off-spring $(X_{2t},\;X_{2t+1})$ in the next generation. The triplet $(X_t,\;X_{2t},\;X_{2t+1})$ refers to mother-daughter relationship. In this paper we propose a threshold model incorporating the difference of 'fertility' of the mother for the first and second off-springs, and thereby extending BAR to threshold-BAR (TBAR, for short). We derive a sufficient condition of stationarity for the suggested TBAR model. Also various inferential methods such as least squares (LS), maximum likelihood (ML) and quasi-likelihood (QL) methods are discussed and relevant limiting distributions are obtained.

• Journal of applied mathematics & informatics
• /
• 제29권1_2호
• /
• pp.1-22
• /
• 2011

In this paper, a Lotka-Volterra model with time delays is considered. A set of sufficient conditions for the existence of Hopf bifurcation are obtained via analyzing the associated characteristic transcendental equation. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form method and center manifold theory. Finally, the main results are illustrated by some numerical simulations.

• 대한수학회논문집
• /
• 제26권2호
• /
• pp.321-338
• /
• 2011

In this paper, a class of delayed epidemic model with diffusion is investigated. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulation are also carried out to support our analytical findings. Finally, biological explanations and main conclusions are given.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제23권2호
• /
• pp.139-155
• /
• 2019

In this paper, we study the asymptotic behavior and Hopf bifurcation of the modified Holling-Tanner models for the predator-prey interactions in the absence of diffusion. Further the direction of Hopf bifurcation and stability of bifurcating periodic solutions are investigated. Diffusion driven instability of the positive equilibrium solutions and Turing instability region regarding the parameters are established. Finally we illustrate the theoretical results with some numerical examples.

• Journal of applied mathematics & informatics
• /
• 제28권1_2호
• /
• pp.193-207
• /
• 2010

In this paper, a predator-prey model with stage structure and distributed delay is investigated. Mathematical analyses of the model equation with regard to boundedness of solutions, nature of equilibria, permanence, extinction and stability are performed. By the comparison theorem, a set of easily verifiable sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model. Taking the product of the per-capita rate of predation and the rate of conversing prey into predator as the bifurcating parameter, we prove that there exists a threshold value beyond which the positive equilibrium bifurcates towards a periodic solution.

• Journal of applied mathematics & informatics
• /
• 제38권5_6호
• /
• pp.375-406
• /
• 2020

The anti-predator factor due to fear of predator in eco- epidemiological models has a great importance and cannot be evaded. The present paper consists of a modified Lesli-Gower predator-prey model with contagious disease in the predator population only and also consider the fear effect in the prey population. Boundedness and positivity have been studied to ensure the eco-epidemiological model is well-behaved. The existence and stability conditions of all possible equilibria of the model have been studied thoroughly. Considering the fear constant as bifurcating parameter, the conditions for the existence of limit cycle under which the system admits a Hopf bifurcation are investigated. The detailed study for direction of Hopf bifurcation have been derived with the use of both the normal form and the central manifold theory. We observe that the increasing fear constant, not only reduce the prey density, but also stabilize the system from unstable to stable focus by excluding the existence of periodic solutions.

• 대한수학회보
• /
• 제50권2호
• /
• pp.353-373
• /
• 2013

In this paper, a class of delayed predator-prey models of prey migration and predator switching is considered. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.

• 대한수학회보
• /
• 제57권3호
• /
• pp.677-690
• /
• 2020

In this article, we study a reaction-diffusion system with homogeneous Dirichlet boundary conditions, which describing a three-species food chain model. Under some conditions, the predator-prey subsystem (u₁ ≡ 0) has a unique positive solution (${\bar{u_2}}$, ${\bar{u_3}}$). By using the birth rate of the prey r1 as a bifurcation parameter, a connected set of positive solutions of our system bifurcating from semi-trivial solution set (r₁, (0, ${\bar{u_2}}$, ${\bar{u_3}}$)) is obtained. Results are obtained by the use of degree theory in cones and sub and super solution techniques.

• 한국화재소방학회논문지
• /
• 제22권2호
• /
• pp.121-128
• /
• 2008

화재에 의한 흡입연기의 중장기 인체 유해성은 흡입연기가 폐에 침착되는 양과 밀접한 관련이 있다. 연기의 폐 내 침착량을 구하기 위해서는 인체 실험이 불가능한 만큼 폐모델을 이용한 실험이 필요하나 실제 폐형태에서 나타나는 연속적으로 감소되는 분지관의 제작상 어려움으로 인하여 하위 세대에서는 모델실험을 통한 침착 연구가 힘들다. 본 문제를 해결하기 위하여 이 연구에서는 아래로 갈수록 직경이 단계적으로 감소하는 구형 충전층을 이용한 폐모델을 개발하고 이를 폐침착 실험에 적용하였다. 실험장치는 각 입자크기별 호흡 패턴 변화에 따른 입자의 침착량을 측정하도록 구성되었으며 표준입자에 대한 실험 값을 실제 폐에 대한 결과와 비교함으로써 개발된 폐모델의 타당성을 검증하였다. 이 폐모델은 화재시 발생하는 여러 가지 연기입자의 흡입에 의한 인체 피해 연구에 도움이 될 것으로 생각된다.