Abstract
By using the direct Lyapunov method and constructing appropriate Lyapunov functionals, we investigate the global stability for the following scalar delay differential equation with nonlinear term \(y'(t)=f(1-y(t), y(t-\tau ))-cy(t)\), where c is a positive constant and \(f: {\mathbb {R}}^2 \rightarrow {\mathbb {R}}\) is of class \({\mathbf {C}}^1\) and satisfies some additional requirements. This equation is a generalization of the SIS model proposed by Cooke (Rocky Mt J Math 7: 253–263, 1979). Criterions of global stability for the trivial and the positive equilibria of this delay equation are given. A special case of the function f depending only on the variable \(y(t-\tau )\) is also considered. Both general and special cases of this equation are often used in biomathematical modelling.
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Acknowledgments
G.H. was supported by the Fundamental Research Funds for the Central Universities (CUG130415) and National Natural Science Foundation of China (11201435). U.F. was supported by the Polish Ministry of Science and Higher Education, within the Iuventus Plus Grant: “Mathematical modelling of neoplastic processes” Grant No. IP2011 041971. U.F. would like to thank Dr. Marek Bodnar for very useful comments allowing to improve the manuscript.
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Communicated by Paul Newton.
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Huang, G., Liu, A. & Foryś, U. Global Stability Analysis of Some Nonlinear Delay Differential Equations in Population Dynamics. J Nonlinear Sci 26, 27–41 (2016). https://doi.org/10.1007/s00332-015-9267-4
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DOI: https://doi.org/10.1007/s00332-015-9267-4