Dynamic behaviors of a nonlinear amensalism model
Abstract A nonlinear amensalism model of the form dN1dt=r1N1(1−(N1P1)α1−u(N2P1)α2),dN2dt=r2N2(1−(N2P2)α3), $$\begin{aligned} &\frac{dN_{1}}{dt}= r_{1}N_{1} \biggl(1- \biggl( \frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl(\frac{N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr), \\ &\frac{dN_{2}}...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-05-01
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| Series: | Advances in Difference Equations |
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| Online Access: | http://link.springer.com/article/10.1186/s13662-018-1624-9 |
| _version_ | 1818059192055889920 |
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| author | Runxin Wu |
| author_facet | Runxin Wu |
| author_sort | Runxin Wu |
| collection | DOAJ |
| description | Abstract A nonlinear amensalism model of the form dN1dt=r1N1(1−(N1P1)α1−u(N2P1)α2),dN2dt=r2N2(1−(N2P2)α3), $$\begin{aligned} &\frac{dN_{1}}{dt}= r_{1}N_{1} \biggl(1- \biggl( \frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl(\frac{N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr), \\ &\frac{dN_{2}}{dt}= r_{2}N_{2} \biggl(1- \biggl( \frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}} \biggr), \end{aligned}$$ where ri,Pi,u,i=1,2,α1,α2,α3 $r_{i}, P_{i}, u, i=1, 2, \alpha_{1}, \alpha_{2}, \alpha_{3}$ are all positive constants, is proposed and studied in this paper. The dynamic behaviors of the system are determined by the sign of the term 1−u(P2P1)α2 $1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} $. If 1−u(P2P1)α2>0 $1-u (\frac {P_{2}}{P_{1}} )^{\alpha_{2}}>0$, then the unique positive equilibrium D(N1∗,N2∗) $D(N_{1}^{*},N_{2}^{*})$ is globally attractive, if 1−u(P2P1)α2<0 $1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0$, then the boundary equilibrium C(0,P2) $C(0, P_{2})$ is globally attractive. Our results supplement and complement the main results of Xiong, Wang, and Zhang (Advances in Applied Mathematics 5(2):255–261, 2016). |
| first_indexed | 2024-12-10T13:12:37Z |
| format | Article |
| id | doaj.art-0848e0106e00440fa89326c1259099df |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-10T13:12:37Z |
| publishDate | 2018-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-0848e0106e00440fa89326c1259099df2022-12-22T01:47:38ZengSpringerOpenAdvances in Difference Equations1687-18472018-05-012018111310.1186/s13662-018-1624-9Dynamic behaviors of a nonlinear amensalism modelRunxin Wu0Mathematics and Physics Institute, Fujian University of TechnologyAbstract A nonlinear amensalism model of the form dN1dt=r1N1(1−(N1P1)α1−u(N2P1)α2),dN2dt=r2N2(1−(N2P2)α3), $$\begin{aligned} &\frac{dN_{1}}{dt}= r_{1}N_{1} \biggl(1- \biggl( \frac{N_{1}}{P_{1}} \biggr)^{\alpha _{1}}-u \biggl(\frac{N_{2}}{P_{1}} \biggr)^{\alpha_{2}} \biggr), \\ &\frac{dN_{2}}{dt}= r_{2}N_{2} \biggl(1- \biggl( \frac{N_{2}}{P_{2}} \biggr)^{\alpha_{3}} \biggr), \end{aligned}$$ where ri,Pi,u,i=1,2,α1,α2,α3 $r_{i}, P_{i}, u, i=1, 2, \alpha_{1}, \alpha_{2}, \alpha_{3}$ are all positive constants, is proposed and studied in this paper. The dynamic behaviors of the system are determined by the sign of the term 1−u(P2P1)α2 $1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}} $. If 1−u(P2P1)α2>0 $1-u (\frac {P_{2}}{P_{1}} )^{\alpha_{2}}>0$, then the unique positive equilibrium D(N1∗,N2∗) $D(N_{1}^{*},N_{2}^{*})$ is globally attractive, if 1−u(P2P1)α2<0 $1-u (\frac{P_{2}}{P_{1}} )^{\alpha_{2}}<0$, then the boundary equilibrium C(0,P2) $C(0, P_{2})$ is globally attractive. Our results supplement and complement the main results of Xiong, Wang, and Zhang (Advances in Applied Mathematics 5(2):255–261, 2016).http://link.springer.com/article/10.1186/s13662-018-1624-9Amensalism modelDifferential inequality theoryGlobal stability |
| spellingShingle | Runxin Wu Dynamic behaviors of a nonlinear amensalism model Advances in Difference Equations Amensalism model Differential inequality theory Global stability |
| title | Dynamic behaviors of a nonlinear amensalism model |
| title_full | Dynamic behaviors of a nonlinear amensalism model |
| title_fullStr | Dynamic behaviors of a nonlinear amensalism model |
| title_full_unstemmed | Dynamic behaviors of a nonlinear amensalism model |
| title_short | Dynamic behaviors of a nonlinear amensalism model |
| title_sort | dynamic behaviors of a nonlinear amensalism model |
| topic | Amensalism model Differential inequality theory Global stability |
| url | http://link.springer.com/article/10.1186/s13662-018-1624-9 |
| work_keys_str_mv | AT runxinwu dynamicbehaviorsofanonlinearamensalismmodel |