Below is a chart which shows traffic from Google for a new website. What could be the reason for those dynamics: fast growth and slow decline?
I see this often with new websites (although less common over the past 2 or 3 years) - at one point, it appeared that when you completed a new site and got it live it would rank fairly well and after a while, it would disappear and then eventually settle out. This is not a million miles away from the Google Sandbox although does differ in some areas.
Further investigation also revealed that a lot of this traffic is bots - trying to get everything entered on their records (whether it's spiders for engines, chaching websites or just people crawling new websites to collect phone numbers/emails etc).
I'm convinced that (and I cannot prove this - it's just what experience suggests) after Google indexes a chain of events occur (such as those already mentioned) and this is why the bot count is so high.
However, there is also no reason why the following hasn't occured
1) Your website has changed with a negative SEO result 2) A competitor(s) website(s) have changed, performing better than yours, moving above your site in the rankings for various keywords, therefore meaning less people click your website.
I mention this for fullness only (I know it's not part of the question) but remember, SEO is not just about how well you've SEO'd your website, it's also about how well the competition are doing!
That's not fast growth, it's a spike. (Though note you're providing no context.)
Chop off those two highest points, which represent all of two weeks(probably two days) over ~5 months, and the graph is much more even.
What is the subject matter of your website?
On the other hand, the decline could simply be seasonal variation. If, for example, it's related to some corporate event which happens around late May, then an annual growth-decline pattern is to be expected. Also, most websites experience declines in August due to northern hemisphere vacations. Everyone's on the beach instead of sat at their computers.