Problem 37 - Advanced Engineering Mathematics

Question

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 \(\mathrm{DE}\) this first time using the fact that it is a Bernoulli equation.

Step-by-Step Solution

Answer

The logistic equation is solved as a Bernoulli equation with solution: \( P(t) = \frac{a}{b + Cae^{-at}} \).

Step 1: Recognize the Bernoulli Equation

The given differential equation is \( \frac{dP}{dt} = P(a - bP) \). This can be rewritten as \( \frac{dP}{dt} = aP - bP^2 \), which shows that the equation is in the form of a Bernoulli equation \( \frac{dP}{dt} + f(t)P = g(t)P^n \), where \( n = 2 \), \( f(t) = -a \), and \( g(t) = b \).

Step 2: Modify the Bernoulli Equation

Divide the entire Bernoulli equation by \( P^2 \) to get: \( P^{-2} \frac{dP}{dt} = aP^{-1} - b \). This modification allows us to attempt solving by separating variables.

Step 3: Change in Variable

Introduce a substitution to simplify the equation. Let \( Q = P^{-1} \), thus \( \frac{dQ}{dt} = -P^{-2} \frac{dP}{dt} \). Substitute \( Q \) into the modified equation to get \( \frac{dQ}{dt} = -aQ + b \).

Step 4: Solve the Linear Equation for Q

The equation \( \frac{dQ}{dt} + aQ = b \) is linear. Use the integrating factor method to solve it. The integrating factor is \( e^{\int a \, dt} = e^{at} \). Multiply through by this integrating factor:

Step 5: Integrate Both Sides

The equation becomes \( \frac{d}{dt}(e^{at}Q) = be^{at} \). Integrate both sides with respect to \( t \):\[ \int \frac{d}{dt}(e^{at}Q) \, dt = \int b e^{at} \, dt \]This yields \( e^{at}Q = \frac{b}{a}e^{at} + C \) where \( C \) is a constant of integration. Solve for \( Q \): \( Q = \frac{b}{a} + Ce^{-at} \).

Step 6: Convert Back to Original Variable

Recall the substitution \( Q = P^{-1} \). Thus, \( P^{-1} = \frac{b}{a} + Ce^{-at} \), hence:\[ P(t) = \frac{1}{\frac{b}{a} + Ce^{-at}} \]

Step 7: Simplify and Analyze Solution

The general solution of the logistic equation is:\[ P(t) = \frac{a}{b + Cae^{-at}} \]This describes a population \( P(t) \) that grows over time, initially growing rapidly, then slowing as it approaches a stable carrying capacity of \( \frac{a}{b} \).

Key Concepts

Population Dynamics

Population dynamics is a fascinating field that examines how populations change over time. These changes are influenced by various factors.

• Birth and Death Rates: The natural increase or decrease in the population.
• Immigration and Emigration: Movements of individuals into and out of a population can drastically alter its size.
• Environmental Conditions: Availability of resources, climate, and presence of predators can impact population growth.

Mathematically, population dynamics is often described using differential equations. It's a powerful tool for projecting future changes. Understanding these principles helps in areas such as conservation biology and resource management. By studying how current populations grow, shrink, or remain stable, we can predict future trends and plan accordingly. In our exercise, we're exploring how population growth can be modeled using the logistic equation, which is fundamental to understanding bounded growth.

Logistic Equation

The logistic equation is a pivotal part of modeling population dynamics. Unlike linear growth models, the logistic equation accounts for the limitations of the environment.Mathematically, it is expressed as:\[\frac{dP}{dt} = P(a - bP)\]In this equation, **\(dP/dt\)** represents the rate of change of the population.

• **\(P\)** is the population size at time \(t\).
• **\(a\)** is the growth rate when the population is small.
• **\(b\)** relates to the environmental carrying capacity.

The carrying capacity is the maximum population that the environment can sustain indefinitely. Initially, when the population size is much smaller than the carrying capacity, the growth rate is approximately exponential. As the population nears the carrying capacity, growth slows down, stabilizing the population.The logistic equation highlights that populations cannot grow infinitely. External constraints like resources or territory define real-world limits. This equation is crucial for understanding how populations stabilize over time.

Separable Differential Equations

Separable differential equations are a category of differential equations that can be simplified by separating the variables into individual sides of the equation. The logistic equation is an example used in the original exercise.The general form for separable differential equations is:\[\frac{dy}{dx} = g(x)h(y)\]The goal is to separate the variables \(x\) and \(y\) such that all terms involving \(y\) are on one side and all terms involving \(x\) are on the other:\[h(y) \, dy = g(x) \, dx\]Integration is then used to solve each side, yielding the function \(y(x)\). This method aids in solving problems where traditional techniques may not work.In the logistic equation, substituting terms and rearranging was a crucial step. Recognizing the equation as separable allows for a straightforward integration process, a powerful technique in solving differential equations.