I have data:
dat <- tibble( day = 200:210, x = sample(-10:10, size = 11,replace = T))
I have a variable y
y
df %>% mutate(y = 5 + cumsum(x))
or, with your extra conditions
df %>% mutate(y = (5 + cumsum(x)) %>% pmin(10) %>% pmax(0))