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 \[ \begin{cases} \dot N(t) =r(t)N(t)\big( 1-{1\over K(t)}\int_ 0^ \infty H(s)N(t-s)\,\text{d}s -c(t)u(t)\big),\cr \dot u(t)=-a(t)u(t)+b(t)\int_ 0^ \infty H(s)N^ 2(t-s)\,\text{d}s,\end{cases} \] 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 \[ \int_ 0^ \infty H(s)s\,\text{ d}s<\infty \quad\text{and}\quad \int_ 0^ \infty H(s)\,\text{ d}s=1. \] 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 \((-\infty,0]\).


34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
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