$(10$ points$)$ Suppose the population $P(t)$ of fish in a lake after $t$ months is modeled by the logistic equation $d\frac{P}{d}t=kP(M-P)$ when no fishing occurs. Now suppose that, because of fishing, fish are harvested from the lake at a rate proportional to the size of the existing fish population. Therefore, the population is modeled by

$\frac{dP}{dt}=kP(M-P)-hP$

where $k,M,$ and $h$ are all positive constants.

$($a$)$ If $h,kMhkMP(0)>00$ here?$0,$ show that the population $is$ still obeys a logistic model. Find the new carrying capacity $in$ terms $ofh,k$ and $M.(Whydowe$ require $0$ here?$)$

$(b)IfhkM$ and $P(0)>0,$ show that this population will eventually $go$ extinct due $to$ over$-$fishing.