Asymptotic analysis of an age-structured predator-prey model with ratio-dependent Holling Ⅲ functional response and delays

Dongxue Yan; Yuan Yuan; Xianlong Fu

doi:10.3934/eect.2022034

Article Contents

2023, Volume 12, Issue 1: 391-414. Doi: 10.3934/eect.2022034

This issue Previous Article A sharp Gagliardo-Nirenberg inequality and its application to fractional problems with inhomogeneous nonlinearity Next Article

Asymptotic analysis of an age-structured predator-prey model with ratio-dependent Holling Ⅲ functional response and delays

1.
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
2.
School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

^*Corresponding author: Yuan Yuan
^*Corresponding author: Yuan Yuan

Received: March 2022

Revised: June 2022

Early access: July 2022

Published: February 2023

This work is supported by NNSF of China (grant No. 12101323), Science and Technology Commission of Shanghai Municipality (STCSM) (grant No. 18dz2271000), Natural Science Foundation of Jiangsu Province of China (grant No. BK20200749), Nanjing University of Posts and Telecommunications Science Foundation (grant No. NY220093)

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Abstract

This paper studies the dynamical behavior of a radio-dependent predator-prey model with age structure and two delays. The model is first formulated as an abstract non-densely defined Cauchy problem and the conditions for existence of the positive equilibrium point are derived. Then, through determining the distribution of eigenvalues, the globally asymptotic stability of the boundary equilibrium and the locally asymptotic stability for the positive equilibrium are obtained, respectively. In addition, it is also shown that a non-trivial periodic oscillation phenomenon through Hopf bifurcation appears under some conditions. Finally, some numerical examples are provided to illustrate the obtained results.

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Mathematics Subject Classification: Primary: 92D25, 34D20; Secondary: 47E05.

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Figure 1. (a) Time series of $ V(t) $. (b) Time series of $ P(t) $. (c) Solutions of the system (1.1) go to the boundary equilibrium. (d) Time series of $ p(t,a) $. In the four pictures, the lines $ a,\; b,\; c $ correspond to three different sets of initial values

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Figure 2. (a) Time series of $ V(t) $. (b) Time series of $ P(t) $. (c) Solutions of the system (1.1) go to the positive steady state when $ \tau_1 = \tau_2 = 0.3 $. (d) Time series of $ p(t,a) $

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Figure 3. (a) Time series of $ V(t) $. (b) Time series of $ P(t) $. (c) Solutions of the system (1.1) oscillates periodically when $ \tau_1 = \tau_2 = 1.2 $. (d) Time series of $ p(t,a) $

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