# American Institute of Mathematical Sciences

April  2021, 26(4): 2273-2297. doi: 10.3934/dcdsb.2021006

## 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  April 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (4) : 2273-2297. doi: 10.3934/dcdsb.2021006
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##### 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
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