Let's analyze and solve the given logistic differential equation:

$\frac{dP}{dt} = P \left(2 - \frac{P}{5000} \right)$

with the initial condition $P(0) = 3000$.

(a) Carrying Capacity

The carrying capacity $K$ is the population level where growth stops, which occurs when $\frac{dP}{dt} = 0$. Setting the growth term to zero:

$2 - \frac{P}{5000} = 0$

Solving for $P$:

$P = 5000$

Thus, the carrying capacity is 5000.

(b) Rate of Maximum Growth

The logistic growth model grows fastest at half of the carrying capacity, i.e., at:

$P = \frac{5000}{2} = 2500$

To find the maximum growth rate, substitute $P = 2500$ into the differential equation:

$\frac{dP}{dt} = 2500 \left(2 - \frac{2500}{5000} \right)$

$= 2500 \left(2 - 0.5\right) = 2500 \times 1.5 = 3750$

Thus, the population grows the fastest at a rate of 3750 individuals per year.

(c) Time of Fastest Growth

For a logistic equation, the fastest growth occurs when $P = \frac{K}{2}$. The general solution to a logistic differential equation is:

$P(t) = \frac{K}{1 + Ce^{-rt}}$

where $r$ is the intrinsic growth rate. To determine the exact time, we need to solve for $t$ when $P = 2500$, given $P(0) = 3000$. If needed, we can proceed with solving this equation explicitly.

(d) Interval Where $P(t)$ is Increasing

The population is increasing when $\frac{dP}{dt} > 0$, which happens when:

$P \left(2 - \frac{P}{5000} \right) > 0$

Since $P$ is always positive, the critical point is:

$2 - \frac{P}{5000} > 0$

$P < 5000$

So, the population is increasing in the interval $0 < P < 5000$. Given $P(0) = 3000$, the population will continue to increase until it reaches 5000.

Follow-up Questions

1. What is the general solution to this logistic equation?
2. How do we determine the value of the intrinsic growth rate?
3. What happens to the population as $t \to \infty$?
4. How does the logistic model compare to an exponential growth model?
5. What real-world examples fit this logistic growth scenario?

Tip: In logistic growth, the fastest population growth happens at half the carrying capacity because of the balance between available resources and reproduction rate.