In a newly made preserve in a zoo, 400 rabbits were released. The monthly maximum population growth rate is 8%. Population ecologists have determined that the forest preserve can sustain a population of 6,000 rabbits, and the population will follow a logistic growth model. Which of the following is a logistic differential equation that models the rabbit population, R(t)?

A) $\frac { d R } { d t } = 8 R \left( 1 - \frac { R } { 400 } \right) , R ( 0 ) = 6,000$
B) $\frac { d R } { d t } = 0.08 R \left( 1 - \frac { R } { 6,000 } \right) , R ( 0 ) = 400$
C) $\frac { d R } { d t } = 0.08 R \left( 1 - \frac { R } { 400 } \right) , R ( 0 ) = 6,000$
D) $\frac { d R } { d t } = 0.08 \left( 1 - \frac { R } { 6,000 } \right) , R ( 0 ) = 400$
E) $\frac { d R } { d t } = 0.08 \left( 1 - \frac { R } { 6,000 } \right) , R ( 0 ) = 400$.

Question

In a newly made preserve in a zoo, 400 rabbits were released. The monthly maximum population growth rate is 8%. Population ecologists have determined that the forest preserve can sustain a population of 6,000 rabbits, and the population will follow a logistic growth model. Which of the following is a logistic differential equation that models the rabbit population, R(t) ?

A) $\frac { d R } { d t } = 8 R \left( 1 - \frac { R } { 400 } \right) , R ( 0 ) = 6,000$
B) $\frac { d R } { d t } = 0.08 R \left( 1 - \frac { R } { 6,000 } \right) , R ( 0 ) = 400$
C) $\frac { d R } { d t } = 0.08 R \left( 1 - \frac { R } { 400 } \right) , R ( 0 ) = 6,000$
D) $\frac { d R } { d t } = 0.08 \left( 1 - \frac { R } { 6,000 } \right) , R ( 0 ) = 400$
E) $\frac { d R } { d t } = 0.08 \left( 1 - \frac { R } { 6,000 } \right) , R ( 0 ) = 400$.

Quizplus · Accepted Answer

The logistic differential equation that models population growth with a carrying capacity is given by $\frac{dR}{dt} = rR\left(1 - \frac{R}{K}\right)$, where $r$ is the growth rate, $R$ is the population at time $t$, and $K$ is the carrying capacity. In this case, $r = 0.08$ (or 8%), $K = 6,000$ (the carrying capacity of the preserve), and the initial population $R(0) = 400$. Therefore, option B correctly represents the logistic growth model for the rabbit population.

In a Newly Made Preserve in a Zoo, 400 Rabbits dRdt=8R(1−R400),R(0)=6,000\frac { d R } { d t } = 8 R \left( 1 - \frac { R } { 400 } \right) , R ( 0 ) = 6,000dtdR​=8R(1−400R​),R(0)=6,000

In a Newly Made Preserve in a Zoo, 400 Rabbits $\frac { d R } { d t } = 8 R \left( 1 - \frac { R } { 400 } \right) , R ( 0 ) = 6,000$