Ramsey was appointed as the QPR boss earlier this campaign following the departure of Harry Redknapp.
It was always going to be tough for the 53-year-old to keep Rangers in the Premier League, and on Sunday relegation was confirmed.
QPR needed to pick up all three points from their league game against Manchester City away from home on Sunday afternoon to maintain any chance of remaining in the English top flight for the 2015-16 campaign.
However, the London outfit were hammered 6-0 by City at the Etihad Stadium, with Argentina international striker Sergio Aguero scoring a hat-trick.
Former QPR striker Bothroyd, who is currently on the books of Japanese club Jubilo Iwata, believes that Ramsey should remain the team’s manager next season.
Chris Ramsey should be manager of QPR and in charge of rebuilding the club he took over after Harry gave up on the team and it.....— Jay bothroyd (@jaybothroyd) May 11, 2015
... was always going to be a struggle. With Chris and Les at the club I'm sure Qpr will be back soon... Go on you R'ssss— Jay bothroyd (@jaybothroyd) May 11, 2015