Problem 481 - Precalculus

Question

Recall that the general form of a logistic equation for a population is given by \(P(t)=\frac{c}{1+a e^{-b t}},\) such that the initial population at time \(t=0\) is \(P(0)=P_{0}\) . Show algebraically that \(\frac{c-P(t)}{P(t)}=\frac{c-P_{0}}{P_{0}} e^{-b t}\).

Step-by-Step Solution

Answer

We verified the expression \( \frac{c-P(t)}{P(t)} = \frac{c-P_{0}}{P_{0}} e^{-b t} \) using algebraic manipulations of the logistic equation.

Step 1: Understand the Logistic Equation

The logistic equation is given by \[ P(t) = \frac{c}{1 + ae^{-bt}} \] Here, \( c \) represents the carrying capacity, \( a \) and \( b \) are positive constants, \( e \) is the base of the natural logarithm, and \( P(t) \) is the population at time \( t \).

Step 2: Substitute Initial Condition

The initial condition is \( P(0) = P_0 \). Substitute \( t = 0 \) into the logistic equation:\[ P(0) = \frac{c}{1 + a e^{0}} = \frac{c}{1 + a} \]So, \( P_0 = \frac{c}{1 + a} \). Rearrange to find \( a \).

Step 3: Express 'a' in terms of known variables

Rearrange the equation \( P_0 = \frac{c}{1 + a} \) to solve for \( a \):\[ a = \frac{c - P_0}{P_0} \]

Step 4: Substitute 'a' back into the Logistic Equation

Use the expression for \( a \) from Step 3 and substitute it into the logistic equation:\[ P(t) = \frac{c}{1 + \frac{c - P_0}{P_0}e^{-bt}} \]Simplify the denominator. It becomes:\[ P(t) = \frac{c}{\frac{P_0 + (c - P_0)e^{-bt}}{P_0}} \]

Step 5: Simplify the Expression

Simplify the expression:\[ P(t) = \frac{c P_0}{P_0 + (c - P_0)e^{-bt}} \]

Step 6: Derive the Required Form

We need to show:\[ \frac{c - P(t)}{P(t)} = \frac{c - P_0}{P_0}e^{-bt} \]Substitute \( P(t) = \frac{c P_0}{P_0 + (c - P_0)e^{-bt}} \) into the left side:\[ \frac{c - \frac{c P_0}{P_0 + (c - P_0)e^{-bt}}}{\frac{c P_0}{P_0 + (c - P_0)e^{-bt}}} \]

Step 7: Simplify Both Numerator and Denominator

The expression becomes:\[ \frac{c(P_0 + (c - P_0)e^{-bt}) - c P_0}{c P_0} \]Which simplifies to:\[ \frac{cP_0 + c(c - P_0)e^{-bt} - cP_0}{cP_0} \]\[ = \frac{c(c - P_0)e^{-bt}}{cP_0} \]

Step 8: Verify the Final Expression

Cancel \( c \) from numerator and denominator:\[ = \frac{c - P_0}{P_0}e^{-bt} \]This confirms the required form.

Key Concepts

Carrying Capacity

The term 'carrying capacity' is pivotal in understanding logistic equations in population dynamics. In this context, carrying capacity, denoted as \( c \), refers to the maximum population size that the environment can sustain indefinitely. It represents a balance where resources are available at a rate that can support this maximum population size without depleting resources.
Here are some key points about carrying capacity:

• It is a limit that a population cannot surpass, defined by the environment's resources.
• An increasing population approaches carrying capacity, which slows growth due to resource limitations.
• Once at carrying capacity, the population stabilizes, barring other events like long-term changes in environment or species.

In a logistic equation, as the population \( P(t) \) approaches \( c \), its growth rate reduces, creating an S-shaped curve over time. This is due to a reduced amount of resources per individual, causing competitive inhibition.

Natural Logarithm

The natural logarithm is an important concept when dealing with exponential functions, such as those found in logistic equations. Its base, \( e \), is approximately equal to 2.718281828, and is a fundamental mathematical constant.
In the logistic equation \( P(t) = \frac{c}{1 + ae^{-bt}} \), the term \( e^{-bt} \) is crucial. Here, \( e^{-bt} \) is an exponential decay factor, which demonstrates how quickly the effect of the initial population decreases over time.
Key aspects of natural logarithms include:

• They are often used to simplify calculations involving exponential growth or decay.
• The function \( e^{x} \) paired with \( \ln(x) \) are inverses, meaning computing the natural log of an exponent retraces steps back to its original number.
• Understanding its properties and manipulation is important to handle expressions such as in the logistic equation’s derivations.

The ability to work with and understand natural logarithms is crucial for dissecting and manipulating equations featuring exponential growth or decay like the logistic model.

Initial Population

The initial population, denoted by \( P_0 \), is a key parameter in logistic equations that reflects the starting point of population studies. In the context of the equation \( P(t) = \frac{c}{1 + ae^{-bt}} \), it specifies the population size at \( t = 0 \).
Let's break it down:

• The expression \( P(0) = P_0 = \frac{c}{1 + a} \) helps establish a baseline, allowing predictions of how populations change over time.
• It directly influences the term \( a \), derived from rearranging \( a = \frac{c - P_0}{P_0} \), where \( a \) affects how quickly the carrying capacity is reached.
• Understanding \( P_0 \) helps inform how initial conditions (like population size, distribution, or environment) affect future dynamics.

Having a firm grasp on the role of the initial population allows for a solid foundation in predicting future population trends within logistic growth models.