Oscillation and attractivity in a differential equation with piecewise constant arguments (Q1335220)

The equation with piecewise constant arguments (1) \(N'(t) = rN(t) [1 - \sum^ m_{j = 0} (a_ j N([t - k_ j]) + b_ j N^ 2([t-k_ j]))]\), in which \([\cdot]\) denotes the greatest integer function, \(r>0\), \(k_ j\) nonnegative integers, \(a_ j\), \(b_ j \geq 0\), \(a_ j + b_ j > 0\), is considered. Necessary and sufficient conditions for the oscillation about the positive equilibrium of (1) of all positive solutions are demonstrated. All the positive solutions are bounded from above and from below by positive constants. When the positive equilibrium \(N^*\) of (1) satisfies \(N^* \exp (r(k + 1)) \sum^ m_{j = 0}\) \((b_ jN^* \exp (r(k + 1)) + a_ j) < 2\), then \(N^*\) is a global attractor of all positive solutions.