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Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential systems. (English) Zbl 0883.34074

The authors consider the system \[ A\dot x_i(t)=x_i(t)F_i(t,x_1(t),\dots ,x_n(t),x_1(t-\tau (t)), \dots ,x_n(t-\tau (t)), \tag{1} \] where \(F_i\) are periodic with respect to \(t\) for \(n=1,\dots ,n\). System (1) is a generalization of the nonautonomous Lotka-Volterra system, which plays a very important role in mathematical population biology. Existence and uniqueness theorems for periodic solutions of (1) are established by combining the theory of monotone flow generated by FDEs. Application to a delay nonautonomous predator-prey system is presented.

MSC:

34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
34C25 Periodic solutions to ordinary differential equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
34K13 Periodic solutions to functional-differential equations
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