• 한국수학사학회지
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• 제27권3호
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• pp.197-210
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• 2014

Species like insects and fishes have, in many cases, non-overlapping time intervals of one generation and their descendant one. So the population dynamics of such species can be formulated as discrete models. In this paper various discrete population models are introduced in chronological order. The author's investigation starts with the Malthusian model suggested in 1798, and continues through Verhulst model(the discrete logistic model), Ricker model, the Beverton-Holt stock-recruitment model, Shep-herd model, Hassell model and Sigmoid type Beverton-Holt model. We discuss the mathematical and practical significance of each model and analyze its properties. Also the stability properties of stationary solutions of the models are studied analytically and illustratively using GSP, a computer software. The visual outputs generated by GSP are compared with the analytical stability results.

• Journal of applied mathematics & informatics
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• 제25권1_2호
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• pp.363-374
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• 2007

In this paper, we consider the discrete nonlinear delay model which describe the control of a single population of cells. We establish a sufficient condition for oscillation of all positive solutions about the positive equilibrium point and give a sufficient condition for the global attractivity of the equilibrium point. The oscillation condition guarantees the prevalence of the population about the positive steady sate and the global attractivity condition guarantees the nonexistence of dynamical diseases on the population.

• Journal of the Korean Physical Society
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• 제73권11호
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• pp.1774-1786
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• 2018

Phenotypic variation among clones (individuals with identical genes, i.e. isogenic individuals) has been recognized both theoretically and experimentally. We investigate the effects of phenotypic variation on evolutionary dynamics of a population. In a population, the individuals are assumed to be haploid with two genotypes : one genotype shows phenotypic variation and the other does not. We use an individual-based Moran model in which the individuals reproduce according to their fitness values and die at random. The evolutionary dynamics of an individual-based model is formulated in terms of a master equation and is approximated as the Fokker-Planck equation (FPE) and the coupled non-linear stochastic differential equations (SDEs) with multiplicative noise. We first analyze the deterministic part of the SDEs to obtain the fixed points and determine the stability of each fixed point. We find that there is a discrete phase transition in the population distribution when the probability of reproducing the fitter individual is equal to the critical value determined by the stability of the fixed points. Next, we take demographic stochasticity into account and analyze the FPE by eliminating the fast variable to reduce the coupled two-variable FPE to the single-variable FPE. We derive a quasi-stationary distribution of the reduced FPE and predict the fixation probabilities and the mean fixation times to absorbing states. We also carry out numerical simulations in the form of the Gillespie algorithm and find that the results of simulations are consistent with the analytic predictions.

• 한국수학사학회지
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• 제26권2_3호
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• pp.197-210
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• 2013

18세기 후반, 맬더스는 최초로 집단의 개체군 성장에 대해 연구하였고 버룰스트는 맬더스 모델을 수정하여 로지스틱 모델을 창안하였다. 종간의 포식경쟁에 대한 모델로서 록카-볼테라모델이 만들어졌으며 가우스는 박테리아를 이용한 실험을 통해 록카-볼테라 모델을 변형 발전시켰다. 종간의 포식 작용과 경쟁에 대해 연구하는 와중에 불안정 공존 부동점의 존재가 밝혀지면서 솔로몬과 홀링은 피식자에 대한 포식자의 제한된 능력을 고려한 기능 반응과 수반응을 록카-볼테라 모델에 적용하였다. 니콜슨과 베일리는 숙주와 기생포식자 사이의 포식활동을 연구하여 이산 모델을 만들었다. 20세기에 들어와서 질병 역학에 대한 수학적 모델이 연구되었고 실제 자료와의 비교 연구가 진행되었다. 질병 역학 모델은 역학적 현상에 따라 SIS, SIR 또는 SEIR과 같은 다양한 모델로 명명되었는데, 이들 대부분은 SlR모델을 기본으로 하여 발전되었다.

• Bulletin of the Korean Chemical Society
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• 제18권4호
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• pp.415-424
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• 1997

Conformational flexibilities of the GlcNAc(β1,3)Gal(β)OMe are investigated through NMR spectroscopy and molecular modeling. Adiabatic energy map generated with a dielectric constant of 50 contains three local minima. All of the molecular dynamics simulations on three local minimum energy structures show fluctuations between two low energy structures, N2 at φ=80° and ψ=60° and N3 at φ=60° and ψ=-40°. We have presented adequate evidences to state that GlcNAc(β1,3)Gal(β)OMe exists in two conformationally discrete forms. Two state model of N2 and N3 conformers with a population ratio of 40:60 is used to calculate the effective cross relaxation rate and reproduces the experimental NOEs very well. Molecular dynamics simulation in conjunction with two state model proves successfully the dynamic equilibrium existed in GlcNAc(β1,3)Gal(β)OMe and can be considered as a powerful method to analyze the motional properties in the structure of carbohydrate. This observation also cautions against the indiscriminate use of a rigid model to analyze NMR data.