# American Institute of Mathematical Sciences

September  2020, 25(9): 3523-3551. doi: 10.3934/dcdsb.2020071

## Dynamics of a stage-structured population model with a state-dependent delay

 School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  August 2019 Revised  November 2019 Published  April 2020

Fund Project: The second author is supported by the NNSF of P.R. China (Grant No. 11671123)

This paper is devoted to the dynamics of a predator-prey model with stage structure for prey and state-dependent maturation delay. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. Secondly, the existence, uniqueness, and local asymptotical stability of (boundary and coexisting) equilibria are investigated by means of degree theory and Routh-Hurwitz criteria. Thirdly, the explicit bounds for the eventual behaviors of the mature population are obtained. Finally, by means of comparison principle and two auxiliary systems, it observed that the local asymptotical stability of either of the positive interior equilibrium and the positive boundary equilibrium implies that it is also globally asymptotical stable if the derivative of the delay function around this equilibrium is small enough.

Citation: Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071
##### References:

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##### References:
Simulations of system (41) with $\tau(x) = 4-2e^{-0.1x}$ and $(r, c) = (0.1, 1)$ illustrate that the positive equilibrium is globally asymptotically stable
Simulations of system (41) with $\tau(x) = 4-2e^{-0.1x}$ and $(r, c) = (0.6, 0.1)$ illustrate that the positive equilibrium is globally asymptotically stable
Simulations of system (41) with $\tau(x)\equiv 4$ and $(r, c) = (0.1, 1)$ illustrate that the positive equilibrium is globally asymptotically stable
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