Threshold dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey system

Baifeng Zhang; Guohong Zhang; Xiaoli Wang

doi:10.3934/dcdsb.2021260

Article Contents

2022, Volume 27, Issue 9: 4969-4993. Doi: 10.3934/dcdsb.2021260

This issue Previous Article Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation Next Article Attractors for a class of perturbed nonclassical diffusion equations with memory

Threshold dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey system

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

^* Corresponding author: Guohong Zhang
^* Corresponding author: Guohong Zhang

Received: May 2021

Revised: September 2021

Early access: November 2021

Published: September 2022

G.-H. Zhang is partially supported by grants from National Science Foundation of China(11871403); X.-L. Wang is partially supported by grants from Fundamental Research Funds for the Central Universities (XDJK2020B050)

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Abstract

In this paper, we investigate the global dynamics of a Leslie-Gower predator-prey system in advective homogeneous environments. We discuss the existence and uniqueness of positive steady-state solutions. We study the large time behavior of solutions and establish threshold conditions for persistence and extinction of two species when they live in open advective environments. Numerical simulations indicate that the introduction of advection leads to the evolution of spatial distribution patterns of species and specially it may induce spatial separation of the prey and predator under some conditions.

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Mathematics Subject Classification: Primary: 35K57, 35B32; Secondary: 35B35, 35B36, 37L15, 92C15.

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Figure 1. Bifurcation diagrams of steady state solutions to (3) in open advective environments with the bifurcation parameter $ q $. The fixed parameter values $ b = 1 $, $ a = 1, c = 10, L = 10 $. (a) $ d_1 = d_2 = 0.01,r_1 = 5,r_2 = 6 $; (b) $ d_1 = 0.1,d_2 = 0.15,r_1 = r_2 = 5 $; (c) $ d_1 = d_2 = 0.1,r_1 = r_2 = 16 $; (d) $ d_1 = d_2 = 0.15,r_1 = 5,r_2 = 2 $; (e) $ d_1 = 0.15,d_2 = 0.1,r_1 = r_2 = 12 $

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Figure 2. Evolution of spatial distribution patterns on $ [0,L] $ with different advection speeds after a long time (t = 3000). Here $ b = 1 $ and $ q^*(d_1,r_1,b)\le q^*(d_2,r_2,b) $. The fixed parameter values $ a = 1, c = 10, L = 10,d_1 = 0.1,d_2 = 0.15,r_1 = r_2 = 5 $.(a) $ q = 1.7 $, (b) $ q = 1.9 $, (c) $ q = 2 $

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Figure 3. Evolution of spatial distribution patterns on $ [0,L] $ with different advection speeds after a long time (t = 3000). Here $ b = 1 $ and $ q^*(d_1,r_1,b)>q^*(d_2,r_2,b) $. The fixed parameter values $ a = 1, c = 10, L = 10,d_1 = 0.15,d_2 = 0.1,r_1 = r_2 = 12 $. (a) $ q = 1.5 $, (b) $ q = 3.3 $, (c) $ q = 3.38 $, (d) $ q = 3.8 $, (e)$ q = 3.875 $, (f) $ q = 4 $

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Figure 4. Bifurcation diagrams of steady state solutions to (3) with the bifurcation parameter $ q $ when the advective environments is closed. The fixed parameter values $ b = 0, $ $ a = 1, c = 10, L = 10 $, $ d_1 = 0.15,d_2 = 0.1,r_1 = 30,r_2 = 9 $

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Figure 5. Evolution of the stable steady-state spatial distribution patterns of species on $ [0,L] $ with different advection speeds after a long time (t = 3000). Here $ b = 0 $. The fixed parameter values $ a = 1, c = 10, L = 10,d_1 = 0.15,d_2 = 0.1,r_1 = 30,r_2 = 9 $. (a) $ q = 0.1 $, (b) $ q = 2.5 $, (c) $ q = 3 $, (d) $ q = 7.2 $, (e) $ q = 20 $

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