We spend a lot of effort worrying about the competition and what they are up to. We often view them with suspicion and try to "outwork" or outplay them. This is a healthy thing to do in business (but don't let it rule you!)
However, if there's some synergy: working in different areas or services perhaps - you can often benefit from sharing resources and working together to both parties success.
We often work with other agents - helping with viewings, supplying cleaning and maintenance staff and being "on hand" during holiday periods. Though we obviously charge for this help, we are able to give our professional, personable touch in a timely manner.
So when you are met with a new competitor, see if there is a way of collaborating- and maybe you will both find a profitable solution to an age-old problem.