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.
['Gang Huang', 'Anping Liu', 'Urszula Foryś']
Global Stability Analysis of Some Nonlinear Delay Differential Equations in Population Dynamics