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Analysis of the power law logistic population model with slowly varying coefficients
Author(s) -
Shepherd J.J.,
Stacey A.,
Grozdanovski T.
Publication year - 2012
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1561
Subject(s) - mathematics , logistic function , limiting , range (aeronautics) , population , power law , zero (linguistics) , statistical physics , mathematical analysis , statistics , demography , physics , philosophy , sociology , engineering , composite material , mechanical engineering , linguistics , materials science
We apply a multiscale method to construct general analytic approximations for the solution of a power law logistic model, where the model parameters vary slowly in time. Such approximations are a useful alternative to numerical solutions and are applicable to a range of parameter values. We consider two situations—positive growth rates, when the population tends to a slowly varying limiting state; and negative growth rates, where the population tends to zero in infinite time. The behavior of the population when a transition between these situations occurs is also considered. These approximations are shown to give excellent agreement with the numerical solutions of test cases. Copyright © 2012 John Wiley & Sons, Ltd.