The predatory bacterium Myxococcus xanthus can invade prey bacteria using two distinct motility apparatuses. It is commonly acknowledged that adventurous motility is used for isolated bacteria, while social motility corresponds to bacterial clusters. Inspired by recent biological findings, we propose a simple model of predatory invasion focusing on the co-occurrence of these two mechanisms and their possible synergistic effects. At microscopic scale, cell motion is persistent; therefore, we opt for a transport-reaction model, extending previous reaction-diffusion models. Another specificity is the structuration of the bacterial population into clusters with varying speeds and persistence times. In the linear regime, we find a transition from normal speed to anomalous speed, consistent with reaction-diffusion theory but with specificities due to the hyperbolic nature of the model. For the nonlinear regime, we numerically observe and study the existence of transitions between pulled and pushed fronts. Finally, we reproduced biological experiments with mutants lacking each of the motility apparatuses based on relevant modifications of the model. Moreover, we propose a rational basis for the reported synergistic effects. Our work paves the way for a better understanding of the complex waves of bacterial population advance, which are precursors to biofilm formation.