Bifurcation for the Lotka-Volterra competition model
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929321929605120 |
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| author | Li, Zaizheng Terracini, Susanna |
| author_facet | Li, Zaizheng Terracini, Susanna |
| contents | We analyze the bifurcation phenomenon for the following two-component competition system:
\begin{equation*}
\begin{cases}
-Δu_1=μu_1(1-u_1)-βαu_1u_2,& \text{in}\ B_1\subset \mathbb{R}^N,
-Δu_2=σu_2(1-u_2)-βγu_1u_2,& \text{in}\ B_1\subset \mathbb{R}^N,
\frac{\partial u_1}{\partial n}= \frac{\partial u_2}{\partial n} =0,&\text{on}\ \partial B_1,
\end{cases}
\end{equation*}
where $N\ge 2$, $α>γ>0$, $σ\geμ>0$ and $β>\fracσγ$. More precisely, treating $β$ as the bifurcation parameter, we initially perform a local bifurcation analysis around the positive constant solutions, obtaining precise information of where bifurcation could occur, and determine the direction of bifurcation. As a byproduct, the instability of the constant solution is provided. Furthermore, we extend our exploration to the global bifurcation analysis.
Lastly, under the condition $σ=μ$, we demonstrate the limiting configuration on each bifurcation branch as the competition rate $β\rightarrow+\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13410 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bifurcation for the Lotka-Volterra competition model Li, Zaizheng Terracini, Susanna Analysis of PDEs 35B32, 92D25 We analyze the bifurcation phenomenon for the following two-component competition system: \begin{equation*} \begin{cases} -Δu_1=μu_1(1-u_1)-βαu_1u_2,& \text{in}\ B_1\subset \mathbb{R}^N, -Δu_2=σu_2(1-u_2)-βγu_1u_2,& \text{in}\ B_1\subset \mathbb{R}^N, \frac{\partial u_1}{\partial n}= \frac{\partial u_2}{\partial n} =0,&\text{on}\ \partial B_1, \end{cases} \end{equation*} where $N\ge 2$, $α>γ>0$, $σ\geμ>0$ and $β>\fracσγ$. More precisely, treating $β$ as the bifurcation parameter, we initially perform a local bifurcation analysis around the positive constant solutions, obtaining precise information of where bifurcation could occur, and determine the direction of bifurcation. As a byproduct, the instability of the constant solution is provided. Furthermore, we extend our exploration to the global bifurcation analysis. Lastly, under the condition $σ=μ$, we demonstrate the limiting configuration on each bifurcation branch as the competition rate $β\rightarrow+\infty$. |
| title | Bifurcation for the Lotka-Volterra competition model |
| topic | Analysis of PDEs 35B32, 92D25 |
| url | https://arxiv.org/abs/2404.13410 |