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.