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Profitless delays for a nonautonomous Lotka-Volterra predator-prey almost periodic system with dispersion. (English) Zbl 1113.92070

Summary: 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 theorems and delay differential equation basic theory, we prove that the system is uniformly persistent under some appropriate conditions. Further, by constructing suitable Lyapunov functionals, 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 the system and existence and uniqueness of positive almost periodic solutions 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.

MSC:

92D40 Ecology
34K20 Stability theory of functional-differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
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