Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The population of a town in the south of Bangladesh has been growing exponentially. However, recent flooding has alarmed residents and people are leaving the town at a rate of \(N\) thousand people per year, where \(N\) is a constant. The rate of change of the population of the town can be modeled by the differential equation $$ \frac{d P}{d t}=0.02 P-N $$ where \(P=P(t)\) is the number of people in the town in thousands. (a) If \(P(0)=100\), what is the largest yearly exodus rate the town can support in the long run? (b) How big must the population of the town be in order to support the loss of 1000 people per year?

Step-by-Step Solution

Answer

The largest yearly exodus rate the town can support in the long run is 2 thousand people per year. The minimum population required to support the loss of 1000 people per year is 50 thousand.

Step 1: Identify the steady state condition

To find the maximum exodus rate , you need to understand when the population remains steady \(i.e., \frac{d P}{d t}=0\). The equation for this scenario would be \(0.02 P=N\). Here, \(P\) denotes population and \(N\) denotes the exodus rate.

Step 2: Solve for the exodus rate

For the town's population to remain steady, we substitute the given initial population \(P=100\) into the equation \(0.02 P=N\) to find \(N\). So, we have \(N = 0.02*100 = 2\). This tells us that the maximum exodus rate the town can support in the long run is 2 thousand people per year, given the initial population was 100 thousand.

Step 3: Calculate minimum population

To find the minimum population required to support a loss of 1000 people per year, you substitute \(N = 1000/1000 = 1\) (since \(N\) is in thousands) into the equation \(0.02 P=N\). Solving for \(P\), you get \(P = \frac{1}{0.02} = 50\). This means that in order to support a loss of 1000 people per year, the population must be at least 50 thousand.

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept often observed in populations, where the number of individuals changes at a rate proportional to its current size.
It is like a snowball effect: as more individuals are present, the rate at which they can multiply also increases. This creates a rapid expansion over time, which can be modeled using differential equations.
In mathematical terms, the differential equation representing exponential growth is usually given by \( \frac{dP}{dt} = rP \), where \( P \) is the current population, and \( r \) is the growth rate.

• The larger the growth rate \( r \), the faster the population will increase.
• Exponential growth cannot continue indefinitely, as resources such as food or space become limited.

When external factors like emigration are involved, as with the population leaving the town in the exercise, the differential equation becomes \( \frac{dP}{dt} = rP - N \). Here, \( N \) accounts for the rate at which people are leaving, which tempers the growth.
Understanding how these components interact is crucial in modeling real-world population dynamics.

Population Dynamics

Population dynamics are driven by various factors that affect the size and growth of biological populations.
These can include birth rates, death rates, immigration, and emigration, all of which contribute to changes in population size over time.
In the context of the exercise, the key factor is emigration, quantified by the constant \( N \), representing the number of people leaving the town each year.
This affects the overall growth rate of the population and can lead to interesting dynamics where, despite a natural tendency to grow, the population may remain stable or even decrease.

• If \( N \) is too high, it counteracts growth completely, resulting in a net decrease.
• Consideration of external environmental factors, like natural disasters, can further complicate dynamics.

The interplay between natural growth and external influences is captured by the differential equation \( \frac{dP}{dt} = 0.02P - N \).
It highlights how constant emigration can impact a population's ability to maintain its numbers.
Understanding these concepts helps us to foresee potential issues in real-life situations, such as managing urban populations or wildlife conservation.

Steady-State Condition

The steady-state condition is an important concept in understanding population dynamics, especially when considering whether a population will grow, shrink, or stabilize over time.
A steady state occurs when the rate of growth is exactly balanced by the exodus rate, leading to no net change in the population size over time (\( \frac{dP}{dt} = 0 \)).
To achieve this, the differential equation \(0.02 P = N\) must hold true. This equation describes a situation where every individual leaving is exactly replaced by one born, keeping the population stable.

• A higher initial population means a higher exodus rate \( N \) can be tolerated without decrease.
• Achieving steady-state can be critical for sustainability, especially in small communities.

In the exercise, calculating the steady-state helps determine how many people the town can afford to lose annually, without affecting its long-term population size.
Similarly, it determines the minimum population required to withstand a certain level of emigration. It's a valuable tool for policymakers in creating sustainable population management strategies.