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Positive periodic solutions of a single species model with feedback regulation and distributed time delay. (English) Zbl 1087.34051

The authors consider the following nonautonomous system modeling the growth of a single species with feedback regulation and distributed time delay

N ˙(t)=r(t)N(t)1 - 1 K(t) 0 H (s) N (t-s) d s - c (t) u (t),u ˙(t)=-a(t)u(t)+b(t) 0 H(s)N 2 (t-s)ds,

where r,a,b,cC[0,) are nonnegative ω-periodic functions, KC[0,) is a positive ω-periodic function (the capacity of the environment), ω>0 is a constant, N(t) denotes the density of the species at time t, u(t) is the regulator, and the kernel H(t)>0 satisfies the conditions

0 H(s)sds<and 0 H(s)ds=1·

For any ϕ,ψC((-,0],(0,)), the authors prove the existence of an ω-periodic solution (N,u) satisfying the initial value conditions N=ϕ and u=ψ on (-,0].

MSC:
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)