How do consolidate a ranking?

I've been working to outrank a competitor site for months. And for some weeks now, we simply alternate at number 1 and 2. The competitor site is as old as the internet in my country Ghana. Their site is one of our first, so you can imagine the backlinks profile, my site is just over a year old, but we've been working really hard with content creation, traffic and site improvement. We now even have 4 horizontal sitelinks for this keyword... We used to rank nowhere... but now my site is number 1 for a couple of days, and it falls to to number 2, and then they'll be up for about 2 or 3 days, and thats been the trend for almost 2 months now Does anyone know why this is happening... Pls how do I stop this and consolidate the number 1 position? Thanks