Dynamics of an almost periodic logistic integrodifferential equation (Q1898062)

The paper deals with the logistic integrodifferential equation \(dN(t)/dt = N(t) [a(t) - b(t) \int^\infty_0\) \(K_\alpha(s)\) \(N(t - s) ds]\) on \((0, \infty)\), where \(a(t)\) and \(b(t)\) are scalar almost periodic functions, while \(K_\alpha (t)\) is piecewise continuous and integrable on \((0, \infty)\), \(\alpha\) is a nonnegative number. The first problem considered in the paper is the derivation of certain a priori bounds for the solutions. Then the existence of an almost periodic globally attracting solution is proven. The functions \(a(t)\) and \(b(t)\) are assumed to be positive (besides their almost periodicity), and further conditions are imposed on \(K_\alpha\). The analysis is carefully conducted. Solutions with or without level- crossing are then investigated and conditions are provided for the validity of such properties. Two examples and numerical simulation illustrate the theory.