Anal. Theory Appl., 33 (2017), pp. 206-218.
Published online: 2017-08
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This article is concerned with a system of semilinear parabolic equations with no-flux boundary condition in a mutualistic ecological model. Stability result of the equilibrium about relevant ODE problem is proved by discussing its Jacobian matrix, we give two priori estimates and prove that the model is permanent when $\varepsilon_1+\varepsilon_2\neq 0$. Moreover, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the model are obtained. Nonexistence of nonconstant positive steady states of the model is also given. When $\varepsilon_1+\varepsilon_2 = 0,$ grow up property is derived if the geometric mean of the interaction coefficients is greater than 1 ($\alpha_1\alpha_2 >1$), while if the geometric mean of the interaction coefficients is less than 1 ($\alpha_1\alpha_2<1$), there exists a global solution. Finally, numerical simulations are given.
}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2017.v33.n3.2}, url = {http://global-sci.org/intro/article_detail/ata/10512.html} }This article is concerned with a system of semilinear parabolic equations with no-flux boundary condition in a mutualistic ecological model. Stability result of the equilibrium about relevant ODE problem is proved by discussing its Jacobian matrix, we give two priori estimates and prove that the model is permanent when $\varepsilon_1+\varepsilon_2\neq 0$. Moreover, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the model are obtained. Nonexistence of nonconstant positive steady states of the model is also given. When $\varepsilon_1+\varepsilon_2 = 0,$ grow up property is derived if the geometric mean of the interaction coefficients is greater than 1 ($\alpha_1\alpha_2 >1$), while if the geometric mean of the interaction coefficients is less than 1 ($\alpha_1\alpha_2<1$), there exists a global solution. Finally, numerical simulations are given.