Dynamic transitions for the S-K-T competition system

Ruikuan Liu; Dongpei Zhang

doi:10.3934/dcdsb.2021277

Article Contents

2022, Volume 27, Issue 9: 5343-5365. Doi: 10.3934/dcdsb.2021277

This issue Previous Article Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems Next Article

Dynamic transitions for the S-K-T competition system

Ruikuan Liu^1, and
Dongpei Zhang^2, ,

1.
School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China
2.
College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

^* Corresponding author: Dongpei Zhang
^* Corresponding author: Dongpei Zhang

Received: March 2021

Early access: November 2021

Published: September 2022

The work is supported by the Young Scholars Development Fund of SWPU (Grant No. 201899010079), the scientific research starting project of SWPU (Grant No. 2018QHZ029), Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and Its Applications (No. 18TD0013), Southwest Petroleum Univ. under Grant 2019CXTD08, Youth Science and Technology Innovation Team of Southwest Petroleum Univ. for Nonlinear Systems (No. 2017CXTD02)

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Abstract

This paper is concerned with dynamical transition for biological competition system modeled by the S-K-T equations. We study the dynamical behaviour of the S-K-T equations with two different boundary conditions. For the system under non-homogeneous Dirichlet boundary condition, we show that the system undergoes a mixed dynamic transition from the homogeneous state to steady state solutions when the bifurcation parameter cross the critical surface. For the system with Neumann boundary condition, we prove that the system undergoes a mixed dynamic transition, a jump transition and a continuous transition when the bifurcation parameter cross the critical number. Finally, two examples are provided to validate the effectiveness of the theoretical results.

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Mathematics Subject Classification: Primary: 35B32, 35B35, 35K57; Secondary: 37G35.

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Figure 1. cross-section graph of $ \Lambda_k $

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Figure 2. The dynamical transitions of $ (13) $ when $ \int_{\Omega}\psi_{1}^{3}dx \neq 0 $

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