Problem 41 - Advanced Engineering Mathematics

Question

In one model of the changing population \(P(t)\) of a community, it is assumed that $$ \frac{d P}{d t}=\frac{d B}{d t}-\frac{d D}{d t} $$ where \(d B / d t\) and \(d D / d t\) are the birth and death rates, respectively. (a) Solve for \(P(t)\) if \(d B / d t=k_{1} P\) and \(d D / d t=k_{2} P\). (b) Analyze the cases \(k_{1}>k_{2}, k_{1}=k_{2}\), and \(k_{1}

Step-by-Step Solution

Answer

Population grows if \(k_1 > k_2\), stays constant if \(k_1 = k_2\), and decreases if \(k_1 < k_2\).

Step 1: Write Down the Differential Equation

The given differential equation for the population is \( \frac{dP}{dt} = \frac{dB}{dt} - \frac{dD}{dt} \). We know that \( \frac{dB}{dt} = k_1 P \) and \( \frac{dD}{dt} = k_2 P \). Substituting these into the population model gives us \( \frac{dP}{dt} = k_1 P - k_2 P \).

Step 2: Simplify the Equation

Combine the terms to simplify the differential equation: \( \frac{dP}{dt} = (k_1 - k_2) P \). This is a first-order linear ordinary differential equation.

Step 3: Solve the Differential Equation

The solution to this type of differential equation \( \frac{dP}{dt} = aP \) is \( P(t) = P_0 e^{at} \), where \( a = k_1 - k_2 \) and \( P_0 \) is the initial population. Therefore, the solution is \( P(t) = P_0 e^{(k_1 - k_2)t} \).

Step 4: Analyze Case \(k_1 > k_2\)

If \( k_1 > k_2 \), then \( a = k_1 - k_2 > 0 \). This implies \( P(t) = P_0 e^{at} \) will grow exponentially over time, as the birth rate is greater than the death rate.

Step 5: Analyze Case \(k_1 = k_2\)

If \( k_1 = k_2 \), then \( a = k_1 - k_2 = 0 \). Here, \( P(t) = P_0 e^{0} = P_0 \), indicating that the population remains constant since the birth and death rates are equal.

Step 6: Analyze Case \(k_1 < k_2\)

If \( k_1 < k_2 \), then \( a = k_1 - k_2 < 0 \). This results in \( P(t) = P_0 e^{at} \), which decreases exponentially over time since the death rate exceeds the birth rate.

Key Concepts

Differential Equations

In the context of population dynamics, differential equations provide a mathematical framework to understand how a population changes over time. A differential equation expresses the rate of change of a variable, in this case, the population \( P(t) \), concerning time.

In this exercise, the differential equation is given by:

• \( \frac{d P}{d t} = \frac{d B}{d t} - \frac{d D}{d t} \)

This equation suggests that the rate at which the population changes depends on the difference between the birth rate (\( \frac{d B}{d t} \)) and the death rate (\( \frac{d D}{d t} \)).

We can further simplify this equation using given relationships, where the birth rate is \( k_1 P \) and the death rate is \( k_2 P \). Substituting these into the equation provides:

• \( \frac{d P}{d t} = (k_1 - k_2) P \)

This is a first-order linear ordinary differential equation, which is often used to model processes like population growth or decay where the change depends directly on the current state of the system.

Exponential Growth and Decay

Exponential functions describe processes that increase or decrease at rates proportional to their current value. In our context, once we have the differential equation \( \frac{d P}{d t} = (k_1 - k_2) P \), the solution tells us how the population \( P(t) \) evolves over time.

The general solution for such an equation is:

• \( P(t) = P_0 e^{(k_1 - k_2)t} \)

Where \( P_0 \) is the initial population at time \( t = 0 \). The term \( e^{(k_1 - k_2)t} \) is an exponential function.

- **If \( k_1 > k_2 \)**: The population grows exponentially because the birth rate exceeds the death rate. This positive growth factor \( (k_1 - k_2) \) means each successive population count is larger than the last.
- **If \( k_1 = k_2 \)**: The population remains constant over time, as there is no net growth or decay. The exponential factor effectively turns into \( e^0 = 1 \), indicating stability.
- **If \( k_1 < k_2 \)**: The population declines exponentially since the death rate is greater than the birth rate, resulting in a negative growth factor \( (k_1 - k_2) \). Each successive population count is smaller than the previous one.

Birth and Death Rates

Birth and death rates are crucial components in the study of population dynamics as they directly influence the size and growth of a population over time. They can be expressed as functions of the population size \( P \).

- **Birth Rate (\( \frac{d B}{d t} \))**: This is typically modeled as \( k_1 P \), where \( k_1 \) is a constant rate. It signifies that the birth rate proportionally increases with the population size, as more individuals result in more births.
- **Death Rate (\( \frac{d D}{d t} \))**: Similarly, this is modeled as \( k_2 P \), with \( k_2 \) being a constant. The death rate increases with the population size, assuming more individuals result in more deaths.

These rates help determine the overall population change. Understanding the relative values of \( k_1 \) and \( k_2 \) can predict whether a population will increase, decrease, or remain stable. These basic concepts form the core of more complex models that include additional factors like migration, resource limitations, or changes in environmental conditions.