## 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 $$N^*=(a^ 0/b_ 0)e^{a^ 0T}$$, $$N_ *=(a_ 0/b^ 0)exp[a_ 0T(1-(a^ 0b^ 0/a_ 0b_ 0)e^{a^ 0T})]$$.

### MSC:

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