scholarly journals Riccati’s differential equation in birth-death processes

1985 ◽  
Vol 36 (3) ◽  
pp. 145-152
Author(s):  
J. Gani
Keyword(s):  
Differential Equation ◽  
Birth Death

scholarly journals Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves

2016 ◽  
Author(s):  
Stuart T. Johnston ◽  
Ruth E. Baker ◽  
D.L. Sean McElwain ◽  
Matthew J. Simpson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Diffusion ◽  
Cell Biology ◽  
Travelling Waves ◽  
Field Approximation ◽  
Travelling Wave ◽  
Mean Field Approximation ◽  
Partial Differential ◽  
Birth Death

AbstractInvasion processes are ubiquitous throughout cell biology and ecology. During invasion, individuals can become isolated from the bulk population and behave differently. We present a discrete, exclusion-based description of the birth, death and movement of individuals. The model distinguishes between individuals that are part of, or are isolated from, the bulk population by imposing different rates of birth, death and movement. This enables the simulation of various co-operative or competitive mechanisms, where there is either a positive or negative benefit associated with being part of the bulk population, respectively. The mean-field approximation of the discrete process gives rise to 22 different classes of partial differential equation, which can include Allee kinetics and nonlinear diffusion. Here we examine the ability of each class of partial differential equation to support travelling wave solutions and interpret the long time behaviour in terms of the individual-level parameters. For the first time we show that the strong Allee effect and nonlinear diffusion can result in shock-fronted travelling waves. We also demonstrate how differences in group and individual motility rates can influence the persistence of a population and provide conditions for the successful invasion of a population.

scholarly journals Measurement and improvement of efficiency of Library by mathematical modelling and finding reliability

2021 ◽  
Vol 7 (2) ◽  
pp. 1-5
Author(s):  
Yakshi Bahl ◽  
Tarun Kumar Garg
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Mathematical Modelling ◽  
Exponential Distribution ◽  
Death Process ◽  
Library System ◽  
The Mathematical Model ◽  
Server System ◽  
Birth Death

Here in this paper we have developed  the mathematical model of the library server using  Markov birth – death process assuming that library system server system is based on exponential distribution. The model so developed by victimisation Chapman Kolmogorov differential equation and is solved by using Mathematica. The solution so obtained is analysed for various rates of failures and repair. The finding so obtained are discussed with the concerned authorities of the library to boost the efficiency of the library.

scholarly journals Gradient flow line near birth–death critical points

2018 ◽  
Vol 12 (02) ◽  
pp. 419-463
Author(s):  
Charel Antony
Keyword(s):  
Differential Equation ◽  
Normal Form ◽  
Critical Points ◽  
Limit Analysis ◽  
Flow Line ◽  
Gradient Flow ◽  
Time Shift ◽  
Gradient Flows ◽  
Index Construction ◽  
Birth Death

Near a birth–death critical point in a one-parameter family of gradient flows, there are precisely two Morse critical points of index difference one on the birth side. This paper gives a self-contained proof of the folklore theorem that these two critical points are joined by a unique gradient trajectory up to time-shift. The proof is based on the Whitney normal form, a Conley index construction, and an adiabatic limit analysis for an associated fast–slow differential equation.

scholarly journals Fractional Linear Birth-Death Stochastic Process—An Application of Heun's Differential Equation

2018 ◽  
Vol 82 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Hidetoshi Konno ◽  
Imre Pázsit
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Birth Death

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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