Oscillations in a delay-logistic equation

Abstract

Sufficient conditions are derived for all nonconstant nonnegative solutions of the equations of the form \[ d x ( t ) d t = x ( t ) { a − ∑ j = 1 n b j x ( t − τ j ) } \frac {{dx\left ( t \right )}}{{dt}} = x\left ( t \right )\left \{ {a - \sum \limits _{j = 1}^n {{b_j}x\left ( {t - {\tau _j}} \right )} } \right \} \] and \[ d x ( t ) d t = x ( t ) { a − b ∫ − ∞ t k ( t − s ) x ( s ) d s } \frac {{dx\left ( t \right )}}{{dt}} = x\left ( t \right )\left \{ {a - b\int _{ - \infty }^t {k\left ( {t - s} \right )x\left ( s \right )ds} } \right \} \] to be oscillatory about their respective positive steady states. The results are complementary to those in [15].