A model for the population P$($t$)$ in a suburb of a large city is given by the initial$-$value problem dPdt $=$ P$(101107$P$),$ P$(0)=4,000,$where t is measured in months.Find the population P of the suburb at time t$.$P$($t$)=$ What is the limiting value of the population? At what time $($in months$)$ will the population be equal to one$-$half of this limiting value? $($Round your answer to one decimal place.$)$ months