Separated solutions of logistic equation with nonperiodic harvesting (Q323854)

The paper is devoted to the study of a logistic equation of the form NEWLINE\[NEWLINE\dot y=a(t)y-y^2-b(t) \eqno{(1)} NEWLINE\]NEWLINE with continuous non-periodic \(a(t)\) and \(b(t)\). It is also assumed that \(a(t)\) and \(b(t)\) are bounded and positive for \(t\in[0,\infty)\). The concept of two separated solutions (lower and upper) is introduced for the description of the qualitative behavior of the trajectories to equation (1). The main results of the paper concern the properties of these solutions. In particular, it is shown that the considered logistic equation cannot have three mutually separated solutions; the lower solution is unique; the upper solution is not unique. Furthermore, the paper provides a description of the behavior of solutions of the considered logistic equation depending on initial conditions. It is proved that any solution with initial value larger than the initial value of the lower solution tends to the upper solution, and any solution with initial value smaller than the initial value of the lower solution blows up in finite time. The authors also consider logistic equations containing a parameter. For such equations, the conditions for the existence of two separated solutions are investigated.