Problem 37 - Advanced Engineering Mathematics

Question

Population Growth In the study of population dynamics one of the most famous models for a growing but bounded population is the logistic equation $$ \frac{d P}{d t}=P(a-b P), $$ where \(a\) and \(b\) are positive constants. Although we will come back to this equation and solve it by an alternative method in Section 2.8, solve the DE this first time using the fact that it is a Bernoulli equation.

Step-by-Step Solution

Answer

The solution to the logistic equation is \( P(t) = \frac{a}{b + Ce^{at}} \).

Step 1: Recognize the Bernoulli Equation

A Bernoulli equation has the form \( \frac{dP}{dt} + Q(t) P = R(t) P^n \). Our given equation is \( \frac{dP}{dt} = P(a-bP) \). We can rewrite this as \( \frac{dP}{dt} - aP = -bP^2 \), which matches the Bernoulli form with \( Q(t) = -a \), \( R(t) = -b \), and \( n = 2 \).

Step 2: Transform the Equation

For Bernoulli equations, we substitute \( y = P^{1-n} = P^{-1} \), so \( \frac{dy}{dt} = -P^{-2} \frac{dP}{dt} \). Substituting \( \frac{dP}{dt} = P(a - bP) \) gives \( \frac{dy}{dt} = -P^{-2}(P(a - bP)) = -aP^{-1} + b \).

Step 3: Simplify Using Substitution

Replace \( P^{-1} \) with \( y \) to obtain \( \frac{dy}{dt} = ay - b \), which is a linear first-order differential equation.

Step 4: Solve the Linear Differential Equation

The equation \( \frac{dy}{dt} = ay - b \) is separable. Rearrange it as \( \frac{dy}{dt} - ay = -b \). Integrate the left side with respect to \( y \) and the right side with respect to \( t \) to find the general solution. Its integrating factor is \( e^{-at} \), thus \( y(t) = Ce^{at} + \frac{b}{a} \).

Step 5: Back-Substitute for Original Variable

Since we initially set \( y = P^{-1} \), revert to the original variable: \( P(t) = \frac{1}{y(t)} = \frac{1}{Ce^{at} + \frac{b}{a}} \).

Step 6: Simplify the Expression

Simplify the expression for \( P(t) \) to have the final form: \( P(t) = \frac{a}{b + Ce^{at}} \), incorporating the constant \( C \) which can be determined by initial conditions if available.

Key Concepts

Understanding the Bernoulli Equation

The Bernoulli equation is a type of differential equation that has a distinct form. Specifically, it is written as \( \frac{dP}{dt} + Q(t)P = R(t)P^n \). This equation is nonlinear due to the presence of the term \( P^n \), distinguishing it from linear differential equations.

To solve a Bernoulli equation, a clever substitution is often used. By transforming the equation, we effectively convert it into a linear equation, which is easier to manage. In our earlier example, we took \( y = P^{1-n} \). Solving this transformed equation allows us to eventually find the solution to the original differential equation.

Bernoulli equations appear frequently in mathematical modeling, especially when dealing with multiplicative relationships like population dynamics or fluid flow. Recognizing and solving them is an essential skill in calculus and applied mathematics.

Exploring Population Dynamics

Population dynamics is a fascinating field where mathematics, ecology, and biology intersect. It involves studying how populations change over time and what factors influence their growth or decline. The logistic differential equation, like the one presented, is a key model in understanding these changes.

The equation \( \frac{dP}{dt} = P(a-bP) \) suggests that a population will grow proportionally to its current size \( P \), but this growth is limited by a carrying capacity due to the \( -bP^2 \) term.

This type of equation is highly relevant in resource management, conservation efforts, and ecosystem studies. By predicting future population sizes, scientists and policymakers can make informed decisions about sustainable practices and environmental protections. Understanding the underlying mathematics is crucial for these applications.

Basics of First-Order Differential Equations

First-order differential equations are the simplest type of differential equations and involve only the first derivative of an unknown function with respect to a given variable. They are essential for modeling growth, decay, and other natural processes.

A key feature of first-order equations is that they can often be solved by separable techniques or integrating factors, as seen in our example where we solved the transformed linear equation.

These equations are foundational in both pure mathematics and applied contexts such as physics, engineering, and economics. They allow us to describe how systems evolve over time, from cooling coffee to radioactive decay, making them an indispensable tool for understanding dynamic systems.