Asymptotic analysis of a size-structured population model with infinite states-at-birth

ABSTRACT

In this paper, we study a size-structured population model with infinite states-at-birth and distributed delay in birth process. We are interested in the asymptotic behavior of the considered system and, particularly, in the effect of time lag on the long-term dynamics. To this end, we formally linearize the system around a steady state and study the linearized system using $C_0$-semigroup framework and spectral analysis methods. In this manner, we achieve some results on linearized stability, instability, and asynchronous exponential growth for the system under some conditions. The obtained results are also illustrated by some examples and simulations.

KEYWORDS:

• Size-structured population model
• delayed birth process
• $C_0$-semigroup
• linearized stability
• asynchronous exponential growth

AMS SUBJECT CLASSIFICATIONS:

• 35F30
• 92D25
• 34D20

Acknowledgements

We would like to thank the referees very much for the careful review and the valuable comments to this manuscript which improve it greatly.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSF of China [grant number 11671142] and [grant number 11771075], Science and Technology Commission of Shanghai Municipality (STCSM) [grant number 13dz2260400] and Shanghai Leading Academic Discipline [project number B407].