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  • Differential-difference equations  (2)
  • Springer  (2)
  • Blackwell Publishing Ltd
  • Springer Nature
  • 1
    Electronic Resource
    Electronic Resource
    Averaging methods in the delayed-logistic equation (1980)
    Springer
    Journal of mathematical biology 10 (1980), S. 89-96 
    Details
    ISSN: 1432-1416
    Keywords: Differential-difference equations ; Periodic solutions ; Averaging method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The delayed logistic equation is analyzed using the averaging method. Using the transformation of coordinates v=ln N/K it is shown that the first order term in perturbation theory yields N=K exp(r * cos πt/2) when the delay time T exceeds some critical value T c. The amplitude r* is equal to (40μ/3π − 2)1/2 and μ is an expansion parameter that is proportional to (T − Tc). Comparison of the exponential solution of N and numerical results for the ratio N maximum/N minimum provides a good fit for values of μ larger than the results using the N coordinate as the perturbed coordinate.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276398
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  • 2
    Electronic Resource
    Electronic Resource
    Random effects in population models with hereditary effects (1980)
    Springer
    Journal of mathematical biology 10 (1980), S. 101-112 
    Details
    ISSN: 1432-1416
    Keywords: Differential-difference equations ; Hopf bifurcation ; Fokker-Planck equation ; Averaging method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract This paper discusses the influence of environmental noise on the dynamics of single species population models with hereditary effects. A detailed analysis is carried out for the logistic equation with discrete delay in the resource limitation term (Hutchinson's equation). When the system undergoes Hopf bifurcation, we find the stationary probability density distribution for the amplitude of the periodic solution by means of an averaged Fokker-Planck equation. Finally, we estimate the persistence time of the species when the population density has a lower bound beyond which it goes extinct.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275836
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