Profitless delays for a nonautonomous Lotka-Volterra predator-prey almost periodic system with dispersion

This paper studies a nonautonomous Lotka-Volterra almost periodic predator-prey dispersal system with discrete and continuous time delays which consists of two-patches, the prey species can disperse among two-patches, but the predator species is confined to one patch and cannot disperse. By using comparison theorem and delay differential equation basic theory, we prove the system is uniformly persistent under some appropriate conditions. Further, by constructing suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. By using a new method and almost periodic functional hull theory, we show that the almost periodic system has a unique globally asymptotical stable strictly positive almost periodic solution. The conditions for the permanence, global stability of system and the existence, uniqueness of positive almost periodic solution depend on delays, so, time delays are “profitless”. Finally, ecological conclusions and a particular case are given. These results are basically an extension of the known results for nonautonomous Lotka-Volterra systems.