For the exponential population growth with a constant growth rate of $R$, $$ n(t+1) = R*n(t) $$ where $n$ is the number of individuals in the population. Therefore, we can sample from a Poisson distribution since the sum of $n$ numbers drawn from a Poisson distribution with mean $R$ is known to follow a Poisson distribution with mean $Rn$.
If $n=K$ remains constant and $R$ is a function of $t$, what distribution would, $$ R(t+1) = R(t)*K $$ would follow? Note that $R$ need not be an integer and so Poisson may not apply.