Oscillations and Convergence in an Almost Periodic Competition System

Sufficient conditions are derived for the existence of a globally attractive almost periodic solution of a competition system modelled by the nonautonomous Lotka–Volterra delay differential equations $$\begin{gathered} \frac{{{\text{d}}N_1 (t)}}{{{\text{d}}t}} = N_1 (t)\left {r_1 (t) - a_{11} (t)N_1 (t - \tau (t)) - a_{12} (t)N_2 (t - \tau (t))} \right], \hfill \\ \frac{{{\text{d}}N_2 (t)}}{{{\text{d}}t}} = N_2 (t)\left {r_2 (t) - a_{21} (t)N_1 (t - \tau (t)) - a_{22} (t)N_2 (t - \tau (t))} \right], \hfill \\ \end{gathered} $$ in which $ \tau ,r_i ,a_{ij} (i,j = 1,2) $ are continuous positive almost periodic functions; conditions are also obtained for all positive solutions of the above system to 'oscillate' about the unique almost periodic solution. Some ecobiological consequences of the convergence to almost periodicity and delay induced oscillations are briefly discussed.