Does local stability imply global attraction for Nicholson’s blowflies equation?
For $p > 1$ and $\tau > 0$ consider Nicholson's blowflies equation in the normalised form $$x'(t) = -x(t) + p\,x(t - \tau)e^{-x(t - \tau)},$$ with an arbitrary continuous initial history $x(s) > 0$ on $[-\tau, 0]$. Its positive equilibrium is $x_* = \log p$, and linearisation at $x_*$ gives the characteristic equation $$z + 1 - (1 - \log p)e^{-z\tau} = 0.$$ The equation $N'(t) = -\delta N(t) + pN(t - h)e^{-\gamma N(t - h)}$ was used by Gurney, Blythe and Nisbet for the dynamics of Nicholson's blowflies; a rescaling of time and population reduces it to the form above. Liz, Tkachenko and Trofimchuk proved that $x_*$ is globally asymptotically stable whenever $1 < p \le e$, and whenever $p > e$ and $$e^{-\tau} > (\log p - 1)\log\frac{(\log p - 1)\log p}{(\log p - 1)^2 + 1},$$ and their computations show the boundary of this region close to that of local stability.
Problem. Suppose that every root $z$ of the characteristic equation has strictly negative real part, so that $x_*$ is locally exponentially stable; parameter values with a purely imaginary characteristic root are excluded. Must every solution with positive initial history satisfy $x(t) \to \log p$ as $t \to \infty$?
Source: Smith (1995), p. 116, as cited by Liz, Tkachenko and Trofimchuk (2003) and Röst and Wu (2007): local asymptotic stability of the positive equilibrium of Nicholson's blowflies equation implies its global asymptotic stability. Stated by AIM — Open Applied Problems in normalised form, with strict local stability as hypothesis.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, delay differential equations, global attractivity, Nicholson's blowflies equation
References
@book{smith1995,
author = {Smith, Hal L.},
title = {Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems},
series = {Mathematical Surveys and Monographs},
volume = {41},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {1995},
doi = {10.1090/surv/041},
}
@article{liz2003,
author = {Liz, Eduardo and Tkachenko, Victor and Trofimchuk, Sergei},
title = {A global stability criterion for scalar functional differential equations},
journal = {SIAM J. Math. Anal.},
volume = {35},
number = {3},
year = {2003},
pages = {596--622},
doi = {10.1137/S0036141001399222},
eprint = {math/0112047},
archivePrefix = {arXiv},
}
@article{rost2007,
author = {R{\"o}st, Gergely and Wu, Jianhong},
title = {Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback},
journal = {Proc. R. Soc. A},
volume = {463},
number = {2086},
year = {2007},
pages = {2655--2669},
doi = {10.1098/rspa.2007.1890},
}
@article{gurney1980,
author = {Gurney, W. S. C. and Blythe, S. P. and Nisbet, R. M.},
title = {Nicholson's blowflies revisited},
journal = {Nature},
volume = {287},
number = {5777},
year = {1980},
pages = {17--21},
doi = {10.1038/287017a0},
}
@article{berezansky2010,
author = {Berezansky, Leonid and Braverman, Elena and Idels, Lev},
title = {Nicholson's blowflies differential equations revisited: Main results and open problems},
journal = {Appl. Math. Model.},
volume = {34},
number = {6},
year = {2010},
pages = {1405--1417},
doi = {10.1016/j.apm.2009.08.027},
}
@misc{aim-nicholson-local-global-stability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does local stability imply global attraction for Nicholson’s blowflies equation?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/285-nicholson-local-global-stability.md},
note = {Entry 285 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00454, "Does local stability imply global attraction for Nicholson’s blowflies equation?", clopen.solutions.