Problem 24 - Biocalculus Calculus for the Life Sciences

Question

$$\begin{array}{l}{\text { Sterile insect technique One method of slowing the }} \\ {\text { growth of an insect population without using pesticides is }} \\\ {\text { to introduce into the population a number of sterile males }} \\\ {\text { that mate with fertile females but produce no offspring. }}\end{array}$$. $$ \begin{array}{l}{\text { (The photo shows a screw-worm fly, the first pest effec- }} \\ {\text { tively eliminated from a region by this method.) Let } P} \\ {\text { represent the number of female insects in a population and }} \\\ {S \text { the number of sterile males introduced each generation. }}\end{array}$$. $$\begin{array}{c}{\text { Let } r \text { be the per capita rate of production of females by }} \\ {\text { females, provided their chosen mate is not sterile. Then the }} \\ {\text { female population is related to time } t \text { by }} \\ {t=\int \frac{P+S}{P[(r-1) P-S]} d P}\end{array}$$ $$\begin{array}{l}{\text { Suppose an insect population with } 10,000 \text { females grows at }} \\ {\text { a rate of } r=1.1 \text { and } 900 \text { sterile males are added. Evaluate }} \\ {\text { the integral to give an equation relating the female popula- }} \\ {\text { tion to time. (Note that the resulting equation can't be }} \\ {\text { solved explicitly for } P .}\end{array}$$

Step-by-Step Solution

Answer

The equation is \( t = -\ln |P| + 100 \ln |0.1P - 900| + C \).

Step 1: Identify the Given Function

The problem provides us with the function to integrate: \( t = \int \frac{P+S}{P[(r-1)P-S]} \, dP \). This function describes the relation between the number of females, added sterile males, and time.

Step 2: Substitute Known Values

The problem states that \( r = 1.1 \), \( P = 10000 \), and \( S = 900 \). Substitute these values into the expression to get: \( t = \int \frac{P + 900}{P[(0.1)P - 900]} \, dP \).

Step 3: Simplify the Integrand

Simplify the expression inside the integral. The expression becomes: \( t = \int \frac{P + 900}{0.1P^2 - 900P} \, dP \). This can be further simplified to \( t = \int \frac{P + 900}{P(0.1P - 900)} \, dP \).

Step 4: Partial Fraction Decomposition

Use partial fraction decomposition to separate the integrand: \( \frac{P + 900}{P(0.1P - 900)} = \frac{A}{P} + \frac{B}{0.1P - 900} \). Solve for \( A \) and \( B \).

Step 5: Solve the System of Equations

Equate coefficients to solve for \( A \) and \( B \). You get the equations: \( A + 0.1B = 1 \) and \( -900A = 900 \). Solving these, we find \( A = -1 \) and \( B = 10 \).

Step 6: Integrate Each Term

Integrate \( t = \int \frac{-1}{P} \, dP + \int \frac{10}{0.1P - 900} \, dP \). The first term integrates to \(-\ln |P| \) and the second term to \(100 \ln |0.1P - 900| \).

Step 7: Construct the General Solution

Combine the integrated terms to form the equation: \( t = -\ln |P| + 100 \ln |0.1P - 900| + C \), where \( C \) is the integration constant.

Step 8: Relate the Female Population to Time

The final equation relating the female population \( P \) to time \( t \) is \( t = -\ln |P| + 100 \ln |0.1P - 900| + C \). This equation can't be explicitly solved for \( P \).

Key Concepts

Population Dynamics

Population dynamics is the study of how and why populations of organisms change over time. In the context of the Sterile Insect Technique, understanding population dynamics helps us track how populations grow or shrink when sterile males are introduced. The aim is to slow down the growth of an insect population, which can be particularly useful for controlling pests.

• Growth Rate: A key factor in population dynamics is the growth rate, which describes how quickly a population can increase.
• Sterile Insect Technique: By introducing sterile males, we disrupt the reproduction cycle, which can slow down the growth rate.
• Modeling: We use mathematical models to predict population changes and the effect of interventions like sterlization.

Understanding population dynamics allows us to better plan and manage pest control strategies.

Partial Fraction Decomposition

Partial fraction decomposition is a mathematical technique used to simplify complex rational expressions. It's particularly important when integrating rational functions, as seen in the provided problem.

• Purpose: The method breaks down a complicated fraction into simpler parts that are easier to integrate.
• Method: Involves expressing the original fraction as a sum of fractions with simpler denominators.
• Steps: Identify distinct parts of the denominator, such as linear or quadratic expressions, and solve for constants that make the equation true.

In the solution to the original problem, we used partial fraction decomposition to separate the integrand of the equation into simpler components, making it possible to integrate.

Integral Calculus

Integral calculus is a critical branch of mathematics focusing on integrals and their properties. In the given problem, integral calculus helps relate population changes over time.

• Integration: The process of finding the integral of a function, giving us a family of solutions or an area under a curve.
• Definite vs. Indefinite: Definite integrals give a number, representing an area, while indefinite integrals can contain an extra constant (C).
• Applications: Used to find out how quantities accumulate, such as population growth over a period.

The integral in the problem reflects changes in the female population over time, an example of using calculus to solve real-world issues.

Per Capita Rate

The per capita rate refers to the average rate of production or change per individual within a population. It's a core concept in understanding how populations, like insects, grow or decline.

• Definition: Represents the average change contributed by a single individual.
• Importance: Helps determine how effective the introduction of sterile males might be in reducing reproduction.
• Calculation: Often expressed in terms of births or deaths per individual, providing insight into potential growth rates.

In the Sterile Insect Technique scenario, the per capita rate ( ")) provides insight into how quickly the insect population could grow without intervention, thus guiding the number of sterile males needed.