  Evolving network models under a dynamic growth rule which comprises the
addition and deletion of nodes are investigated. By adding a node with a
probability $P_a$ or deleting a node with the probability $P_d=1-P_a$ at each
time step, where $P_a$ and $P_d$ are determined by the Logistic population
equation, topological properties of networks are studied. All the fat-tailed
degree distributions observed in real systems are obtained, giving the evidence
that the mechanism of addition and deletion can lead to the diversity of degree
distribution of real systems. Moreover, it is found that the networks exhibit
nonstationary degree distributions, changing from the power-law to the
exponential one or from the exponential to the Gaussian one. These results can
be expected to shed some light on the formation and evolution of real complex
real-world networks.
