Consider a population $P=P(t)$ with constant relative birth and death rates $$ and $,$ respectively, and a constant rate of change of the population at time $t$ is modeled by the differential equation

$\frac{dP}{dt}=kP-m,$ where $k=a-.$

$($a$)$ Find the solution of this equation that satisfies the intial condition $P(0)=P_0.($Write an equation involving

$$

$($b$)$ What condition on $m$ will lead to an exponential expansion of the population?

$mk{p}_0$

$m>k{p}_0$

$mk{p}_0$

$m=kpmmk{P}_{0}m>k{P}_{0}mk{P}_{0}m=k{P}_{0}mmk{p}_{0}m>k{p}_{0}mk{P}_{0}mm$

$m=k{P}_0$

What condition $onm$ will result $in$ a population decline?

$mk{p}_0$

$m>k{p}_0$

$mk{P}_0$

$mm$

$m=kp$

$(c)$ What condition $onm$ will result $in$ a constant population?

$mk{P}_0$

$m>k{P}_0$

$mk{P}_0$

$m$

$m=k{P}_0$

What condition $onm$ will result $in$ a population decline?

$mk{p}_0$

$m>k{p}_0$

$mk{P}_0$

$m$