Suppose that a population grows according to a logistic model with carrying capacity $8,000$ and $k = 0.01$ per year. Choose the logistic differential equation for these data.
A) $\frac { d P ( t ) } { d t } = 0.01 P \left( 1 + \frac { p } { 1 } \right)$
B) $\frac { d P ( t ) } { d t } = 8,000 P \left( 1 + \frac { p } { 0.01 } \right)$
C) $\frac { d P ( t ) } { d t } = 0.01 P \left( 1 - \frac { p } { 8,000 } \right)$
D) $\frac { d P ( t ) } { d t } = 0.01 P \left( 1 + \frac { p } { 8,000 } \right)$
E) $\frac { d P ( t ) } { d t } = P \left( 1 + \frac { p } { 8,000 } \right)$

Question

Quizplus · Accepted Answer

The logistic differential equation is given by:
$$\frac { d P ( t ) } { d t } = k P(t)\left(1-\frac{P(t)}{L}\right)$$
where $P(t)$ is the population at time $t$, $k$ is the growth rate and $L$ is the carrying capacity.
Substituting $k=0.01$ and $L=8,000$, we get
$$\frac { d P ( t ) } { d t } = 0.01 P(t)\left(1-\frac{P(t)}{8000}\right)$$
which matches with option C.

Suppose That a Population Grows According to a Logistic Model 8,0008,0008,000

Suppose That a Population Grows According to a Logistic Model $8,000$