Global attractivity and oscillations in a periodic logistic integrodifferential equation. (English) Zbl 0735.45006

The authors consider the integrodifferential equation \[ dN(t)/dt=N(t)[a(t)-b(t)\int_ 0^ T H(s)N(t-s)ds]\tag{1} \] where \(a\), \(b\) are continuous positive periodic functions of period \(T\) and the kernel \(H:[0,T]\to[0,\infty)\) is piecewise continuous and \(N(s)=\varphi(s)\geq0\) for \(s\in[-T,0]\) and \(\varphi(0)>0\). The authors prove that (1) possesses a globally attractive positive periodic solution \(\tilde N(t)\), i.e., one such that all other solutions \(N(t)\to\tilde N(t)\) as \(t\to\infty\), provided that \((b^ 0)^ 2N^*T<b_ 0\). It is established in addition that all positive solutions of (1) oscillate about the periodic solution if \(b_ 0N_ *\int_ 0^ T H(s)e^{\lambda s}ds>\lambda\) for \(\lambda>0\). In these formulas \(a_ 0\), \(a^ 0\) denote the minimum and maximum values of the function \(a(t)\) on the interval \([0,T ]\) and \(b_ 0\), \(b^ 0\) denote the minimum and maximum values of \(b(t)\) on the same interval and \(N^*=(a^ 0/b_ 0)e^{a^ 0T}\), \(N_ *=(a_ 0/b^ 0)exp[a_ 0T(1-(a^ 0b^ 0/a_ 0b_ 0)e^{a^ 0T})]\).


45J05 Integro-ordinary differential equations
45M15 Periodic solutions of integral equations
92D25 Population dynamics (general)
92D40 Ecology
45M20 Positive solutions of integral equations