This is primarily a user experience question, IMO.
Are those 5K users a month going to expect to land on your site when they click a link, enter a URL or select a bookmark pointing to your former competitor's site? Or will suddenly finding themselves somewhere they didn't expect to be prove confusing and potentially worrisome? In other words, if you just 301 without warning, will they possibly think the competitor's site has been hacked or that you're some kind of spammers?
I'm guessing the answer is, "probably, at least for a good portion of them."
If it were me, I'd be hitting the back button to get out of there as fast as I could. Probably not the reaction you'd ideally like to see from those 5K visitors.
On the other hand, it's also probably silly to turn your back on 5K visitors per month (which is essentially what you'd be doing if you just close down the other site).
So, how about if you 301 the pages you can to the corresponding pages on your site, but also serve a banner across the top of the page or pop-up window to those visitors letting them know why they got to that page and reminding them to update their bookmarks? That way, they know what's happened, understand it's legit, and they still get to the content they were looking for.
For those where there isn't a corresponding page on your site, send them to a custom landing page that alerts them to the transition, lets them know there's no exact corresponding page, and offers them links to other pages that you think might fill the bill.
If for some reason you can't implement a banner or pop-up, I would probably send all the visitors from the old site to the custom landing page. At least that way they don't think the old site has been hacked, and with any luck at least some of them will stick around to check out your site.