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

Shangzhi Li; Shangjiang Guo

doi:10.3934/dcdsb.2020071

Article Contents

2020, Volume 25, Issue 9: 3523-3551. Doi: 10.3934/dcdsb.2020071

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

Shangzhi Li and
Shangjiang Guo^,

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

^* Corresponding author: Shangjiang Guo
^* Corresponding author: Shangjiang Guo

Received: July 31, 2019

Revised: October 31, 2019

Early access: April 2020

Published: September 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 34D23, 34K20; Secondary: 92D25.

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Figure 1. 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

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Figure 2. 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

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Figure 3. 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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