# American Institute of Mathematical Sciences

## On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect

 School of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

* Corresponding author. G. Shang's current address is Advanced Microscopy and Instrumentation Research Center, Harbin Institute of Technology, Harbin, China

Dedicated to Professor Sze-Bi Hsu on the occasion of his 70th birthday and retirement

Received  April 2020 Revised  November 2020 Published  December 2020

For a stage-structured population model in periodic discrete habitat, with periodic initial values it reduces to a system of two differential equations with time delay. Assuming the birth rate is of unimodal type, we obtain the influence of time delay on the local and global dynamics. In particular, large delay leads to population vanishing. As delay decreases, we found three critical values of delay for the emergence of different dynamics, by appealing to a combination of monotone dynamical system theory, Hopf bifurcation theory and the fluctuation method. Numerical simulations are also performed to illustrate the results.

Citation: Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021005
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Parameter regions for different dynamics in $(\tau,p)$ plane
Plot of $S_0(\tau)$ and $S_1(\tau)$
(a) $\tau = 15\le \tau_1$ and the solution converges to the positive equilibrium; (b) $\tau = 20>\tau_1$ but close to $\tau_1$, and the solution converges to a periodic solution; (c) $\tau = 80<\tau_2$ but close to $\tau_2$, and the solution still converges to periodic solution which has a larger period than in (b); (d) $\tau = 90\in (\tau_2,\tau^*)$, the solution converges to the positive equilibrium
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