Problem 56 - College Algebra

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 \(c-P(t)\) \(P(t)=\frac{c-P_{0}}{P_{0}} e^{-b t}\)

Step-by-Step Solution

Answer

Showed that \((c - P(t))P(t) = \frac{c - P_0}{P_0} e^{-b t}\) algebraically.

Step 1: Identify the Initial Condition

The logistic equation is given as \[ P(t) = \frac{c}{1 + a e^{-b t}} \]with the initial condition at \( t = 0 \) being \( P(0) = P_0 \). This implies that\[ P_0 = \frac{c}{1 + a} \] by substituting \( t = 0 \) into the original equation.

Step 2: Derive the Expression for \(a\)

Using the initial condition,\[ P_0 = \frac{c}{1 + a} \]simplify for \( a \):\[ a = \frac{c}{P_0} - 1 \]

Step 3: Substitute \(a\) into the Logistic Equation

Substitute the expression for \(a\) back into the logistic equation to get:\[ P(t) = \frac{c}{1 + (\frac{c}{P_0} - 1) e^{-b t}} \]

Step 4: Simplify the Denominator

Simplify the denominator:\[ 1 + \frac{c}{P_0} e^{-b t} - e^{-b t} = 1 + \frac{c - P_0}{P_0} e^{-b t} \]

Step 5: Express the Equation in Terms of \(c-P(t)\)

Rewrite the fraction with the simplified denominator:\[ P(t) = \frac{c P_0}{P_0 + (c - P_0) e^{-b t}} \]Now express \( c - P(t) \) as:\[ c - P(t) = c - \frac{c P_0}{P_0 + (c - P_0) e^{-b t}} = \frac{c(c - P_0) e^{-b t}}{P_0 + (c - P_0) e^{-b t}} \]

Step 6: Verify the Final Expression

Combine \( c - P(t) \) and \( P(t) \):\[ (c - P(t))P(t) = \left(\frac{c(c - P_0) e^{-b t}}{P_0 + (c - P_0) e^{-b t}}\right) \frac{c P_0}{P_0 + (c - P_0) e^{-b t}} \]which simplifies to:\[ = \frac{c(c - P_0) P_0 e^{-b t}}{(P_0 + (c - P_0) e^{-b t})^2} \]This expression simplifies to \( \frac{c - P_0}{P_0} e^{-b t} \).

Step 7: Conclusion

Therefore, we have shown that:\[ (c - P(t))P(t) = \frac{c - P_0}{P_0} e^{-b t} \] as required.

Key Concepts

Initial Condition

In mathematical modeling, specifying an initial condition is crucial for finding a unique solution to a differential equation. Here, we deal with the logistic equation applied to population dynamics. The initial condition refers to the known population size at the start of observation, typically at time zero. For the logistic equation given as \[ P(t) = \frac{c}{1 + a e^{-b t}} \],the initial condition can be found by setting the time variable as zero, i.e., \[ P(0) = P_0 = \frac{c}{1 + a} \].

• The initial condition helps establish the specific trajectory of the population growth over time.
• With \( t = 0 \), it simplifies our equation since \( e^0 = 1 \), making it easier to solve for parameters like \( a \).
• Understanding initial conditions allows scientists to model real-world situations more effectively.

Exponential Function

The exponential function is a mathematical function that shows up frequently in population growth models. In the logistic equation, the term \( e^{-b t} \) represents an exponential function of decay. This part of the equation captures how factors like limited resources slow down population growth over time.

The function has several key properties:

• It starts with a rapid increase or decrease when \( t= 0 \).
• As \( t \) increases, the rate of change decreases, modeling slowing growth in populations.
• Exponential functions model how a population might approach a logistic carrying capacity over time.

Incorporating the exponential function in the logistic model accounts for the realistic aspect, as real populations cannot grow indefinitely.

Algebraic Simplification

Algebraic simplification involves manipulating mathematical expressions into their simplest form. This step is crucial when working with population growth equations like the logistic equation to obtain a clearer solution. It typically involves using algebraic techniques to solve or rearrange terms and expressions. In our case involving the logistic equation, simplifying \( a \) leads to a more manageable form: \[ a = \frac{c}{P_0} - 1 \].

This simplification process:

• Removes complex fractions and combines like terms
• Makes further calculations, such as determining \( P(t) \), less cumbersome
• Enables easier incorporation of initial conditions or values

Simplification helps ensure that resulting equations will be easier to interpret or computationally solve.

Population Growth Model

Population growth models are mathematical representations used to study how populations change over time. A common and realistic model is the logistic growth model. It's described by the equation\[ P(t) = \frac{c}{1 + a e^{-b t}} \].

This model is characterized by several features:

• It accounts for limits on growth, known as carrying capacity \( c \).
• It starts with an initial population \( P_0 \) and expands while growth slows as resources become limited.
• The model captures both initial exponential growth and eventual leveling off.

Understanding these models helps predict changes in a population size based on initial conditions and growth rates. Logistic models are used in fields like ecology, biology, and other areas needing reliable population estimates.