Bulletin of Mathematical Biology

, Volume 80, Issue 7, pp 1713–1735 | Cite as

An Alternative Formulation for a Distributed Delayed Logistic Equation

  • Chiu-Ju Lin
  • Lin Wang
  • Gail S. K. Wolkowicz
Original Article


We study an alternative single species logistic distributed delay differential equation (DDE) with decay-consistent delay in growth. Population oscillation is rarely observed in nature, in contrast to the outcomes of the classical logistic DDE. In the alternative discrete delay model proposed by Arino et al. (J Theor Biol 241(1):109–119, 2006), oscillatory behavior is excluded. This study adapts their idea of the decay-consistent delay and generalizes their model. We establish a threshold for survival and extinction: In the former case, it is confirmed using Lyapunov functionals that the population approaches the delay modified carrying capacity; in the later case the extinction is proved by the fluctuation lemma. We further use adaptive dynamics to conclude that the evolutionary trend is to make the mean delay in growth as short as possible. This confirms Hutchinson’s conjecture (Hutchinson in Ann N Y Acad Sci 50(4):221–246, 1948) and fits biological evidence.


Single species delayed growth models Decay-consistent delay Integro-differential equations Dirac delta, gamma, uniform, and tent distributions Adaptive dynamics Lyapunov functionals Hutchinson’s conjecture 



The research of Chiu-Ju Lin and Gail S. K. Wolkowicz was supported by a Natural Sciences and Engineering Research Council (NSERC) of Canada Discovery Grant and Accelerator supplement. The research of Lin Wang was also supported by NSERC.


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Department of Mathematics & StatisticsUniversity of New BrunswickFrederictonCanada

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