Neurons form networks by growing out neurites that synaptically connect to other neurons. During this process, neurites develop complex branched trees. Interestingly, the outgrowth of neurite branches is often accompanied by the simultaneous withdrawal of other branches belonging to the same tree. This apparent competitive outgrowth between branches of the same neuron is relevant for the formation of synaptic connectivity, but the underlying mechanisms are unknown. An essential component of neurites is the cytoskeleton of microtubules, long polymers of tubulin dimers running throughout the entire neurite. To investigate whether competition between neurites can emerge from the dynamics of a resource such as tubulin, we developed a multi-compartmental model of neurite growth. In the model, tubulin is produced in the soma and transported by diffusion and active transport to the growth cones at the tip of the neurites, where it is assembled into microtubules to elongate the neurite. Just as in experimental studies, we find that the outgrowth of a neurite branch can lead to the simultaneous retraction of its neighboring branches. We show that these competitive interactions occur in simple neurite morphologies as well as in complex neurite arborizations and that in developing neurons competition for a growth resource such as tubulin can account for the differential outgrowth of neurite branches. The model predicts that competition between neurite branches decreases with path distance between growth cones, increases with path distance from growth cone to soma, and decreases with a higher rate of active transport. Together, our results suggest that competition between outgrowing neurites can already emerge from relatively simple and basic dynamics of a growth resource. Our findings point to the need to test the model predictions and to determine, by monitoring tubulin concentrations in outgrowing neurons, whether tubulin is the resource for which neurites compete.