Department of Mathematics & Statistics, McMaster University, Hamilton, ON, Canada.
Department of Mathematics & Statistics, University of New Brunswick, Fredericton, NB, Canada.
Bull Math Biol. 2018 Jul;80(7):1713-1735. doi: 10.1007/s11538-018-0432-4. Epub 2018 Apr 19.
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.
我们研究了一个具有衰减一致性延迟的替代单物种 logistic 分布时滞微分方程（DDE）在生长中。与经典 logistic DDE 的结果相反，种群振荡在自然界中很少观察到。在 Arino 等人提出的替代离散时滞模型中（J Theor Biol 241(1):109-119, 2006），排除了振荡行为。本研究采用了他们关于衰减一致性延迟的思想，并推广了他们的模型。我们建立了一个生存和灭绝的阈值：在前一种情况下，通过 Lyapunov 泛函证实了种群趋近于延迟修正的承载能力；在后一种情况下，通过波动引理证明了灭绝。我们进一步使用自适应动力学得出结论，进化趋势是使生长中的平均延迟尽可能短。这证实了 Hutchinson 的猜想（Hutchinson in Ann N Y Acad Sci 50(4):221-246, 1948），并符合生物学证据。
