•We build new double-rumors concurrently spreading models.•Without rumors priorities, we introduce a selection parameter for spreaders to express the attractions of different rumors.•We introduce states-vector to describe rumors states of each node, provide the double-rumors dissemination mechanism by states-vectors expressions.•Simulation results indicate that the best start time of new rumor exists explicitly for two rumors spread process.•Large coreness nodes are shown to be the relative better promulgators for the new rumor. For there are always several kinds of rumors spreading simultaneously in real networks, the research about multiple rumors propagation process is necessary and significant. In this paper, considering the spread of two rumors with different launch time, and assuming the content of each rumor may have nothing to do with the others, we study the double-rumors concurrently spreading dynamics in complex networks, and introduce two kinds of double-rumors spreading models: the DSIR model and the C-DSIR model. We then provide the double-rumors dissemination mechanism by states-vectors expressions and derive the mean-field equations of models to describe their dynamics. Particularly, without rumors priorities, we introduce a selection parameter θ for spreaders to express the attractions of different rumors, and study the influence of this parameter on double-rumors spreading. Numerical simulations are performed to explore the interaction between two rumors, and we investigate the spreading peak and the final size of the rumors with various parameters. Simulation results indicate that, the best launch time of new rumor exists explicitly for the DSIR model, the selection parameter θ and the delay time Tin are interdependent quantities, and the closer the start time of new rumor is to the best time, the more obvious the interdependence would be. Meanwhile, Tin is also a network-dependent parameter for our models in a series of BA networks. Furthermore, under the same conditions, the influential nodes identified by large coreness are the relative better promulgators for new rumor, so they are what the strategy should give priorities to. Our experiment reveals some interesting patterns of double-rumors spreading and suggest a possible avenue for further study of interplays of multiple pieces of information in complex network.