The given problem is a logistic differential equation:

$\frac{dP}{dt} = 0.1 P \left( 1 - \frac{P}{2000} \right)$

with the initial condition $P(0) = 100$.

a) Solving the Differential Equation

This is a standard logistic growth model. The general solution of a logistic equation of the form:

$\frac{dP}{dt} = r P \left(1 - \frac{P}{K}\right)$

where $r = 0.1$ and $K = 2000$, is:

$P(t) = \frac{K}{1 + \left( \frac{K - P_0}{P_0} \right) e^{-rt}}$

Substituting the known values:

• $K = 2000$ (carrying capacity),
• $P_0 = 100$ (initial population),
• $r = 0.1$ (growth rate),

we get the specific solution:

$P(t) = \frac{2000}{1 + \left( \frac{2000 - 100}{100} \right) e^{-0.1t}} = \frac{2000}{1 + 19 e^{-0.1t}}$

Finding Population at $t = 20$

To find the population when $t = 20$, we substitute $t = 20$ into the solution:

$P(20) = \frac{2000}{1 + 19 e^{-0.1 \times 20}} = \frac{2000}{1 + 19 e^{-2}}$

Using $e^{-2} \approx 0.1353$:

$P(20) = \frac{2000}{1 + 19(0.1353)} = \frac{2000}{1 + 2.5707} = \frac{2000}{3.5707} \approx 560.29$

So, the population after 20 units of time is approximately 560.

b) Finding When the Population Reaches 1200

To find when the population reaches 1200, we solve the equation:

$1200 = \frac{2000}{1 + 19 e^{-0.1t}}$

First, solve for $e^{-0.1t}$:

$1 + 19 e^{-0.1t} = \frac{2000}{1200} = 1.6667$

$19 e^{-0.1t} = 1.6667 - 1 = 0.6667$

$e^{-0.1t} = \frac{0.6667}{19} \approx 0.03509$

Now, take the natural logarithm of both sides:

$-0.1t = \ln(0.03509) \approx -3.350$

$t = \frac{-3.350}{-0.1} \approx 33.5$

So, the population reaches 1200 at approximately $t = 33.5$ units of time.

Summary of Solutions:

• The population at $t = 20$ is approximately 560.
• The population reaches 1200 at $t \approx 33.5$.

Would you like more details on any of the steps? Here are some related questions to explore:

1. How would the solution change if the growth rate $r$ were different?
2. Can we find the limiting behavior of $P(t)$ as $t \to \infty$?
3. How does the carrying capacity $K$ affect the solution?
4. What is the significance of the exponential decay term $e^{-rt}$?
5. Can we derive the general logistic equation from basic principles of population dynamics?

Tip: For logistic models, the population always approaches the carrying capacity $K$ over time, even if the initial population is far from it.