Short answer: "There is an increase in population during the summer time." is correct. Here increase is used as a noun.
With increase, the verb is written the same as the noun. And, depending on the context, increased is the past participle of the verb increase and also an adjective.
As a noun (increase):
An increase in interest rates is expected as a result of the Reserve Bank's decision.
Tourist season provides a temporary increase in revenue for local business.
Improved facilities will lead to an increase in employee satisfaction.
As a verb (increase):
We increase our competitive advantage by becoming more responsive to customers.
I increase my chances of getting a job by applying for the right job.
The water pressure increases as you dive deeper.
As the past participle (increased):
People's life expectancy has increased compared to previous centuries.
The demand for our products has increased.
Losses for businesses have increased when they have not adapted to changing markets.
As an adjective (increased):
People now have an increased life expectancy compared to previous centuries.
We are experiencing increased demand for our products.
Businesses can expect increased losses if they can't compete in today's market.