An Alternative Formulation for a Distributed Delayed Logistic Equation

• Published in: Bulletin of mathematical biology Vol. 80; no. 7; pp. 1713 - 1735
• Main Authors: Lin, Chiu-Ju, Wang, Lin, Wolkowicz, Gail S. K.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2018; Springer Nature B.V
• Subjects: Animals; Biological evolution; Carrying capacity; Cell Biology; Conservation of Natural Resources - statistics & numerical data; DDE; Decay; Delay; Differential equations; Ecosystem; Functionals; Humans; Introduced Species - statistics & numerical data; Life Sciences; Logistic Models; Mathematical and Computational Biology; Mathematical Concepts; Mathematics; Mathematics and Statistics; Models, Biological; Mutation; Nitrous oxide; Original Article; Population Dynamics - statistics & numerical data; Population Growth; Species extinction; Species Specificity; Variation; Lyapunov functionals; Dirac delta, gamma, uniform, and tent distributions; Adaptive dynamics; Single species delayed growth models; Integro-differential equations; Hutchinson’s conjecture; Decay-consistent delay
• ISSN: 0092-8240; 1522-9602
• DOI: 10.1007/s11538-018-0432-4

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.