 A population of Christmas beetles in Brisbane grows at a rate proportional to their
current population. In the absence of external factors, the population will double in one
week’s time. On any given day, there is a net migration into the area of 10 beetles, 11
beetles are eaten by a local Magpie population, and 2 die of natural causes.
a) Write a initial value problem to describe the change in population at time t, given
that the initial population of Christmas beetles is P(0) = 100. You must write all
the parameters in your model explicitly.

To describe the change in population of Christmas beetles at time \( t \), we need to formulate a differential equation based on the given information. The rate of change in the population will depend on both the growth rate of the beetles (which is proportional to their current population) and external factors like migration, predation, and natural deaths.

### Key information:
1. **Population growth**: The population grows at a rate proportional to the current population, i.e., \( \frac{dP}{dt} = kP \), where \( k \) is the proportionality constant (growth rate).
   
2. **Population doubling time**: The population doubles in 1 week. This gives a way to determine the value of \( k \).

3. **Migration**: There is a net migration of 10 beetles per day into the area, which adds to the population.

4. **Magpie predation**: 11 beetles are eaten by Magpies per day, reducing the population.

5. **Natural deaths**: 2 beetles die of natural causes per day, also reducing the population.

6. **Initial condition**: The initial population is \( P(0) = 100 \).

---

### Step 1: Write the differential equation for population change.

We can combine all these factors into the following initial value problem:

\[
\frac{dP}{dt} = kP + (\text{migration}) - (\text{predation}) - (\text{natural deaths})
\]

Substituting the values for migration, predation, and natural deaths:

\[
\frac{dP}{dt} = kP + 10 - 11 - 2
\]

This simplifies to:

\[
\frac{dP}{dt} = kP - 3
\]

---

### Step 2: Determine the value of \( k \) (the proportional growth rate).

We are told that the population doubles in 1 week. In other words, if \( P(0) = P_0 \), then \( P(7) = 2P_0 \). Using the solution to the differential equation for exponential growth, we know that:

\[
P(t) = P_0 e^{kt}
\]

Thus, at \( t = 7 \) days, we have:

\[
2P_0 = P_0 e^{7k}
\]

Dividing both sides by \( P_0 \):

\[
2 = e^{7k}
\]

Taking the natural logarithm of both sides:

\[
\ln(2) = 7k
\]

Thus, the growth rate \( k \) is:

\[
k = \frac{\ln(2)}{7} \approx 0.099
\]

---

### Step 3: Initial value problem.

Now that we have \( k \approx 0.099 \), the initial value problem that describes the population at any time \( t \) is:

\[
\frac{dP}{dt} = 0.099P - 3
\]

with the initial condition:

\[
P(0) = 100
\]

---

This is the initial value problem that models the change in the population of Christmas beetles over time.

---

Do you want more details on how to solve this equation or have any other questions?

Here are some related questions:
1. How would the model change if the beetles' population tripled in one week instead of doubling?
2. What would happen to the population if the Magpies started eating 20 beetles per day instead of 11?
3. How does the population behave in the long run if migration and deaths remain constant?
4. What would the population be after two weeks if no beetles were eaten by Magpies?
5. How would the dynamics change if there was a beetle migration out of the area as well?

**Tip**: When solving growth-related differential equations, always focus on identifying key external factors that could significantly affect the population's long-term behavior.

A population of Christmas beetles in Brisbane grows at a rate proportional to their current population. In the absence of external factors, the population will double in one week’s time. On any given day, there is a net migration into the area of 10 beetles, 11 beetles are eaten by a local Magpie population, and 2 die of natural causes. a) Write a initial value problem to describe the change in population at time t, given that the initial population of Christmas beetles is P(0) = 100. You must write all the parameters in your model explicitly.

Explore the initial value problem describing the population growth of Christmas beetles in Brisbane, incorporating factors such as migration, predation, and natural deaths. Learn how to solve the differential equation using exponential growth principles. Suitable for advanced high school students (Grades 11-12).

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