Problem 21 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A drosophila colony (a colony of fruit ies) is being kept in a laboratory for study. It is being provided with essentially unlimited resources, so if left to grow, the colony will grow at a rate proportional to its size. If we let \(N(t)\) be the number of drosophila in the colony at time \(t, t\) given in weeks, then the proportionality constant is \(k\). (a) Write a differential equation re ecting the situation. (b) Solve the differential equation using \(N_{0}\) to represent \(N(0)\). (c) Suppose the drosophila are being cultivated to provide a source for genetic study, and therefore drosophila are being siphoned off at a rate of \(S\) drosophila per week. Modify the differential equation given in part (a) to re ect the siphoning off. (d) One of your classmates is convinced that the solution to the differential equation in part (c) is given by $$ N(t)=N_{0} e^{k t}-S t $$ Show him that this is not a solution to the differential equation. (e) Your classmate is having a hard time giving up the solution he brought up in part (d). He sees that it does not satisfy the differential equation, but he still has a strong gut feeling that it ought to be right. Convince him that it is wrong by using a more intuitive argument. Use words and talk about fruit ies.

Step-by-Step Solution

Answer

The differential equation representing the drosophila colony growth is \(\frac{dN}{dt} = kN\), which integrates into \(N = N_{0} e^{kt}\). When siphoning at a rate of S per week is introduced, the differential equation changes to \(\frac{dN}{dt} = kN - S\). The given function \(N(t)=N_{0} e^{k t}-S t\) is not a solution to the differential equation as it incorrectly models decreasing drosophila at an increasingly negative rate over time.

Step 1: Formulate the Differential Equation

Given that the growth of the colony is proportional to its size, the differential equation that represents that situation would be: \(\frac{dN}{dt} = kN\). This states that the rate of change of the population over time (dN/dt) is proportional (k) to the size of the drosophila population (N).

Step 2: Solve the Differential Equation

To solve the differential equation \(\frac{dN}{dt} = kN\), we separate variables and integrate. On integrating, the two sides respectively, we get: \(\int \frac{1}{N} dN = \int k dt\) leading to \(ln|N| = kt + C\). On exponentiating both sides, and letting \(N_{0}\) represent \(N(0)\), we get : \( N = N_{0} e^{kt}\).

Step 3: Modify the Differential Equation & Solve

The drosophila are constantly being reduced at a rate S per week, the new differential equation becomes \( \frac{dN}{dt} = kN - S\). If we apply the integrating factor method to solve, it yields a more complicated equation, which is not the focus of this exercise.

Step 4: Validity of the Proposed Solution

To test the validity of the equation \(N(t)=N_{0} e^{k t}-S t\), substitute it back into our adjusted differential equation. If rewriting the adjusted differential equation with this function and simplifying does not result in an equality, then it is not a solution.

Step 5: Explanation

An intuitive explanation for why the proposed solution is incorrect is as follows: As time goes on, the term -St will become large and negative, which (if our drosophila could become negative) would imply that the fruit flies are disappearing rapidly, far faster than the siphoning rate S. In reality, the population should be heading towards a steady state where the rate of production (kN) equals the rate of siphoning (S).

Key Concepts

Mathematical Modeling

Mathematical modeling is a powerful tool used to simulate real-world systems and predict outcomes. In the case of the drosophila colony, mathematical modeling helps us understand how the population changes over time. By using differential equations, the rate of change of a system can be captured.
For our given problem, the differential equation captures how the rate of growth of the drosophila population is proportional to its current size. This is an example of exponential growth, a common phenomenon in natural systems. The differential equation formulated in this scenario is given as \( \frac{dN}{dt} = kN \), where \(N\) represents the population size and \(k\) is a positive constant proportionality indicating growth rate.
This means that as the population increases, the rate at which it grows also increases, leading to potentially rapid growth if left unchecked. By modeling populations in this way, scientists can make predictions about future population sizes and manage resources accordingly.

Population Dynamics

Population dynamics is the study of how and why populations change over time. This covers natural growth, declines, and other changes in number due to interactions with the environment or human actions, such as in our drosophila colony.
In the original exercise, the drosophila population is affected by two main factors: exponential growth and siphoning off of the population. In a simplified world with unlimited resources, the population grows exponentially according to the model \(N = N_0 e^{kt}\). However, accounting for siphoning, the differential equation is modified to \( \frac{dN}{dt} = kN - S \).
Here, \(S\) is the constant rate at which drosophila are being removed. The effect of \(S\) changes the nature of the population's dynamics completely, as it introduces a negative feedback loop. This often brings a population to a stable equilibrium, where births minus deaths balance out the siphoning rate. This model showcases how understanding population dynamics helps predict changes and implement effective strategies for managing biological resources.

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. In biological contexts, it describes populations when resources are plentiful and constraints minimal.
The drosophila population is expected to follow an exponential growth model \(N = N_0 e^{kt}\), under ideal conditions. Here, each additional fruit fly leads to more potential births, accelerating overall growth. This makes exponential growth distinctively rapid compared to arithmetic growth.
However, unchecked exponential growth isn't sustainable forever. Eventually, factors such as resources, space, and predation limit growth, causing populations to stabilize or decline. That's why in our problem, the strategy of siphoning helps simulate a more realistic environment. It's essential to recognize that exponential growth is an idealization, offering insights into early-stage population behavior before other effects dominate.