Question content area top

Part $1$

Consider a population$$ P$($t$)$ satisfying the extinction explosion equation

StartFraction dP Over dt EndFraction equals aP squared minus bPdPdt$=$aP$2$bP$,$

where

Upper B equals aP squaredB$=$aP$2$

is the time rate at which births occur and

Upper D equals bPD$=$bP

is the rate at which deaths occur. If the initial population is

Upper P left parenthesis $0$ right parenthesis equals Upper P $0$P$(0)=$P$0$

and

Upper B $0$B$0$

births per month and

Upper D $0$D$0$

deaths per month are occurring at time

t equals $0$t$=0,$

show that the threshold population is

Upper M equals StartFraction Upper D $0$ Upper P $0$ Over Upper B $0$ EndFractionM$=$D$0$P$0$B$0.$

Question content area bottom

Part $1$

StartFraction dP Over dt EndFraction equals aP squared minus bPdPdt$=$aP$2$bP

can be rewritten in the form

StartFraction dP Over dt EndFraction equals kP left parenthesis Upper M minus Upper P right parenthesisdPdt$=$kP$($M$$P$)$

where

kequals$=$enter your response here

and

Mequals$=$enter your response here.