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Zbl 1087.34051
Yin, Fuqi; Li, Yongkun
Positive periodic solutions of a single species model with feedback regulation and distributed time delay. (English)
[J] Appl. Math. Comput. 153, No. 2, 475-484 (2004). ISSN 0096-3003

The authors consider the following nonautonomous system modeling the growth of a single species with feedback regulation and distributed time delay $$ \cases \dot N(t) =r(t)N(t)\big( 1-{1\over K(t)}\int\sb 0\sp \infty H(s)N(t-s)\,\text{d}s -c(t)u(t)\big),\cr \dot u(t)=-a(t)u(t)+b(t)\int\sb 0\sp \infty H(s)N\sp 2(t-s)\,\text{d}s,\endcases$$ where $r,a,b,c\in C[0,\infty)$ are nonnegative $\omega$-periodic functions, $K\in C[0,\infty)$ is a positive $\omega$-periodic function (the capacity of the environment), $\omega>0$ 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 $$ \int\sb 0\sp \infty H(s)s\,\text{ d}s<\infty \quad\text {and}\quad \int\sb 0\sp \infty H(s)\,\text{ d}s=1. $$ For any $\phi,\psi\in C((-\infty,0],(0,\infty))$, the authors prove the existence of an $\omega$-periodic solution $(N,u)$ satisfying the initial value conditions $N=\phi$ and $u=\psi$ on $(-\infty,0]$.
[Vigirdas Mackevičius (Vilnius)]

MSC 2000:: *34K13 Periodic solutions of functional differential equations
92D25 Population dynamics

Keywords: feedback regulation; positive periodic solution; distributed time delay; nonautonomous system

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