Problem 41 - Advanced Engineering Mathematics

Question

Population Model 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

(a) \( P(t) = C'e^{(k_1 - k_2)t} \). (b) For \( k_1 > k_2 \), growth; \( k_1 = k_2 \), constant; \( k_1 < k_2 \), decay.

Step 1: Express Differential Equation

The differential equation given is: \( \frac{dP}{dt} = \frac{dB}{dt} - \frac{dD}{dt} \). Substitute \( \frac{dB}{dt} = k_1 P \) and \( \frac{dD}{dt} = k_2 P \). Thus, the differential equation becomes: \( \frac{dP}{dt} = k_1 P - k_2 P \). Simplify it to: \( \frac{dP}{dt} = (k_1 - k_2)P \).

Step 2: Solve Separable Differential Equation

Re-write the equation \( \frac{dP}{dt} = (k_1 - k_2)P \) as \( \frac{1}{P} dP = (k_1 - k_2) dt \). Integrate both sides: \( \int \frac{1}{P} dP = \int (k_1 - k_2) dt \). This results in \( \ln |P| = (k_1 - k_2)t + C \), where \( C \) is the integration constant.

Step 3: Solve for P(t)

Exponentiate both sides of the equation \( \ln |P| = (k_1 - k_2)t + C \) to get \( |P| = e^{(k_1 - k_2)t + C} \). We can express this as \( P(t) = C'e^{(k_1 - k_2)t} \), where \( C' = e^C \) is an arbitrary constant that can be determined by initial conditions.

Step 4: Analyze Case k_1 > k_2

For \( k_1 > k_2 \), the exponent \( (k_1 - k_2) \) is positive, resulting in an exponential growth: \( P(t) = C'e^{(k_1 - k_2)t} \) grows as \( t \) increases.

Step 5: Analyze Case k_1 = k_2

If \( k_1 = k_2 \), the exponent \( (k_1 - k_2) \) is zero, so the population remains constant:\( P(t) = C'e^{0 \cdot t} = C' \).

Step 6: Analyze Case k_1 < k_2

When \( k_1 < k_2 \), the exponent \( (k_1 - k_2) \) is negative, leading to exponential decay: \( P(t) = C'e^{(k_1 - k_2)t} \) decreases as \( t \) increases, eventually approaching zero.

Key Concepts

Differential Equations

Differential equations are formulas that involve variables and their rates of change. In essence, they are used to describe how a particular quantity changes over time and are crucial in understanding population dynamics. In the context of the exercise, the differential equation \( \frac{dP}{dt} = \frac{dB}{dt} - \frac{dD}{dt} \) helps describe the change in population \( P(t) \) of a community over time. Here, \( dB/dt \) represents the birth rate, while \( dD/dt \) is the death rate.

Essentially, by subtracting the death rate from the birth rate, we can determine the net change in the population. This net change can be either positive or negative, indicating whether the population is growing or declining.

Solving these equations usually involves integration, which allows us to track how the population size changes over time. The solution to a differential equation isn’t just a number; it’s a function that forecasts the future of the population.

Birth and Death Rates

Birth and death rates are fundamental to understanding how populations change. They are often expressed in relation to the existing population size, meaning they can be represented proportionally. In mathematical terms, these rates are often proportional to the current population; the exercise defines them as \( dB/dt = k_1 P \) and \( dD/dt = k_2 P \).

Here, \( k_1 \) and \( k_2 \) are constants that represent the birth and death rate constants, respectively. These constants determine the strength or impact these rates have on the population.

• A high birth rate coefficient \( k_1 \) means the potential for faster population growth.
• A high death rate coefficient \( k_2 \) can suggest rapid population decline.

By understanding these rates, we can predict not only how a population might grow or shrink, but also assess the balance between them. When \( k_1 \) is greater than \( k_2 \), the births outpace deaths leading to population growth. Conversely, if \( k_2 \) is greater, deaths overtake births causing the population to shrink.

Exponential Growth and Decay

Populations can experience exponential growth or decay based on the relationship between birth and death rates. This happens when changes occur proportionally to the current amount, often leading to rapid results.

When the birth rate \( k_1 \) exceeds the death rate \( k_2 \), the population grows exponentially, a process which is captured by the equation \( P(t) = C'e^{(k_1 - k_2)t} \) with a positive exponent. This model suggests that as time progresses, the population can grow very quickly, doubling or even tripling over consistent time intervals.

However, if the death rate \( k_2 \) is higher than the birth rate \( k_1 \), the equation manifests in exponential decay. In this scenario, the population decreases over time as fewer individuals are born compared to the number dying, causing the population to decline progressively.

• Exponential growth leads populations to increase rapidly when resources are unlimited.
• Exponential decay occurs when mortality exceeds natality, potentially leading a population towards extinction.

The exponential models are simplified but provide an insightful foundation into how populations might react under various environmental circumstances.