# American Institute of Mathematical Sciences

## Positive solution branches of two-species competition model in open advective environments

 1 School of Computer Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China 2 School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

* Corresponding author

Received  August 2020 Revised  December 2020 Published  December 2020

Fund Project: The work is supported by the National Natural Science Foundation of China (12071270, 61907030), the Fundamental Research Funds for the Central Universities (GK201903088)

The effect of competition is an important topic in spatial ecology. This paper deals with a general two-species competition system in open advective and inhomogeneous environments. At first, the critical values on the interspecific competition coefficients are established, which determine the stability of semi-trivial steady states. Secondly, by analyzing the nonexistence of coexistence steady states and using the theory of monotone dynamical system, we find that the competitive exclusion principle holds if one of the interspecific competition coefficients is large and the other is in a certain range. Thirdly, in terms of these critical values, the structure and direction of bifurcating branches of positive equilibria arising from two semi-trivial steady states are given by means of the bifurcation theory and stability analysis. Finally, we show that multiple coexistence occurs under certain regimes.

Citation: Yan'e Wang, Nana Tian, Hua Nie. Positive solution branches of two-species competition model in open advective environments. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021006
##### References:

show all references

##### References:
The schematic diagrams of the curves $\mathcal{S}^+ = \{(\|u\|_\infty, b): (u,v,b)\in \mathcal{B}^+\}$ with $\mathcal{B}^+$ defined in Section 4.3. Here $b_0<b^*$ in (ⅰ)-(ⅱ), and $b_0>b^*$ in (ⅲ)-(ⅳ)
The schematic diagrams of the curves $\mathcal{S}^+ = \{(\|u\|_\infty, b): (u,v,b)\in \mathcal{B}^+\}$ with $\mathcal{B}^+$ is defined in Section 4.3
The schematic diagrams of the curves $\mathcal{S}^+ = \{(\|u\|_\infty, b): (u,v,b)\in \mathcal{B}^+\}$ with $\mathcal{B}^+$ defined in Section 4.3
The graphs of $b_0-b^*$ vs. $d_1$ in (ⅰ) and vs. $d_2$ in (ⅱ) with other parameters fixed as (5.1)
The graphs of $b_0-b^*$ vs. $d_1$ in (ⅰ) and vs. $d_2$ in (ⅱ) with $q_1 = q_2 = 0$ and other parameters fixed as (5.1)
Schematic diagram of the global dynamics on system (1.3) in $b-c$ plane
 [1] Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013 [2] Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228 [3] Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156 [4] Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294 [5] Lin Niu, Yi Wang, Xizhuang Xie. Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021014 [6] Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283 [7] Qing Li, Yaping Wu. Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3657-3682. doi: 10.3934/dcds.2020051 [8] Sebastian J. Schreiber. The $P^*$ rule in the stochastic Holt-Lawton model of apparent competition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 633-644. doi: 10.3934/dcdsb.2020374 [9] Yu Jin, Xiang-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020362 [10] Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140 [11] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [12] Linfeng Mei, Feng-Bin Wang. Dynamics of phytoplankton species competition for light and nutrient with recycling in a water column. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020359 [13] Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315 [14] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [15] Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227 [16] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [17] Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035 [18] Hongxia Sun, Yao Wan, Yu Li, Linlin Zhang, Zhen Zhou. Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours. Journal of Industrial & Management Optimization, 2021, 17 (2) : 601-631. doi: 10.3934/jimo.2019125 [19] Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial & Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133 [20] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

2019 Impact Factor: 1.27