Global attractivity in delay ``food-limited'' models with exponential impulses.
The authors obtain several sufficient conditions for the global attractivity of the positive equilibrium of the following impulsive system \[ \begin{aligned} N'(t)&= p(t)N(t)\frac{1-N(t-\tau)}{1+\lambda N(t-\tau)},\quad t\geq 0,\;t\neq t_k,\\ N(t_k^{+})&= N(t_k)^{1+b_k},\quad N(t)=\varphi(t),\quad t\in[-\tau,0], \end{aligned} \] with \(p(t)>0,~\lambda>0,~\tau>0,~b_k>-1.\) In particular, they prove the following theorem: Suppose that for any \(\varepsilon>0\) there exists an integer \(N\) such that \[ \prod_{k=n}^{n+m}(1+b_k)<1+\varepsilon,\quad \text{for}\quad n>N,\;m\geq 0. \] If, in addition, \[ \int_0^{\infty} p(s)\prod_{0\leq t_k<s}(1+b_k)^{-1}\,ds =+\infty, \] \[ \limsup_{t\rightarrow\infty} \int_{t-\tau}^t p(s)\prod_{t-\tau\leq t_k<s} (1+b_k)^{-1}ds\leq \tfrac{3}{2}(1+\lambda), \] then every positive solution of (1) tends to the equilibrium \(N\equiv1\) as \(t\) tends to infinity.
scientific article; zbMATH DE number 2006019
scientific article