This paper proposes an extension of spiking neural P systems with rules on synapses (SNP-RS systems) working in competitive strategy, called competitive SNP-RS (CSNP-RS systems). In CSNP-RS systems, the spikes are viewed as a kind of competitive resources, and the rules on different synapses will compete the spikes (resources) in neurons. A new strategy is considered: the total amount of spikes consumed by these rules should be greater than or equal to the generated amount of spikes. There are two cases to chosen one rule non-deterministically: 1) two or more rules on the same synapse are enabled; and 2) two or more rules on the different synapses are enabled, and the number of spikes contained in neuron is smaller than the number of spikes consumed by these rules. CSNP-RS systems are a kind of distributed parallel computing models. The computational power of CSNP-RS systems is investigated. Specifically, we prove that CSNP-RS systems are turing universal as number generating/accepting devices and function computing device.