# American Institute of Mathematical Sciences

June  2018, 7(2): 293-316. doi: 10.3934/eect.2018015

## Asymptotic behavior of a hierarchical size-structured population model

 School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

* Corresponding author: Xianlong Fu

Received  October 2017 Revised  December 2017 Published  May 2018

Fund Project: This work is supported by NSF of China (Nos. 11671142 and 11771075), STCSM (No. 18dz2271000) and Shanghai Leading Academic Discipline Project (No. B407).

We study in this paper a hierarchical size-structured population dynamics model with environment feedback and delayed birth process. We are concerned with the asymptotic behavior, particularly on the effects of hierarchical structure and time lag on the long-time dynamics of the considered system. We formally linearize the system around a steady state and study the linearized system by $C_0-{\rm{semigroup}}$ framework and spectral analysis methods. Then we use the analytical results to establish the linearized stability, instability and asynchronous exponential growth conclusions under some conditions. Finally, some examples are presented and simulated to illustrate the obtained results.

Citation: Dongxue Yan, Xianlong Fu. Asymptotic behavior of a hierarchical size-structured population model. Evolution Equations & Control Theory, 2018, 7 (2) : 293-316. doi: 10.3934/eect.2018015
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##### References:
Choosing the parameters with $m = 10, ~~T = 120$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.18}{1+s^3}$; (c) $u_0(s) = \frac{0.09}{1+s^3}+e^{-3s}$
Choosing the parameters with $m = 10, ~~T = 160$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.18}{1+s^3}$; (c) $u_0(s) = \frac{0.09}{1+s^3}+e^{-3s}$
Choosing the parameter $m = 100, ~~T = 1050$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.01}{0.01+s^2}+0.0215$; (c) $u_0(s) = \frac{0.05}{2+s^3}+0.0365$
Choosing the parameter $m = 80, ~~T = 1200$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.01}{0.01+s^2}+0.0215$; (c) $u_0(s) = \frac{0.05}{2+s^3}+0.0365$
Choosing the parameters with $m = 40$, $T = 100$. The initial condition corresponding to the curve is $u_0(s) = s(16.5-s), ~0\leq s \leq 16.5$, $-1\leq t \leq100$; $u_0(s) = (s-16.5)(39-s), ~16.5\leq s \leq 39$, $-1\leq t \leq100$
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