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Persistence and stability in general nonautonomous single-species Kolmogorov systems with delays. (English) Zbl 1119.34056

The author studies a nonautonomous single species Kolmogorov system with delays of the type

dx(t) dt=x(t)f(t,x t ),t[0,),

where x t (s)=x(t+s) s[-τ,0]· This equation includes many special delayed nonautonomous population growth models of a single species. The results obtained in this paper extend the main results given by R. R. Vance and E. A. Coddington [J. Math. Biol. 27, 491–506 (1989; Zbl 0716.92016)], establishing in particular different general criteria on the boundedness, persistence, permanence, global asymptotic stability and the existence of positive periodic solutions for the equation above.

34K25Asymptotic theory of functional-differential equations
92D25Population dynamics (general)
35K20Second order parabolic equations, initial boundary value problems
34K13Periodic solutions of functional differential equations