Problem 8 - Single Variable Calculus

Question

Use the given logistic equation to find (a) the growth constant, (b) the carrying capacity of the environment, and (c) the initial population. $$ P(t)=\frac{100 e^{0.2 t}}{e^{0.2 t}+19} $$

Step-by-Step Solution

Answer

For the given logistic equation \(P(t)=\frac{100 e^{0.2 t}}{e^{0.2 t}+19},\) (a) the growth constant is \(r = 0.2,\) (b) the carrying capacity of the environment is \(K = 100,\) and (c) the initial population is \(Pe = 5\).

Step 1: Determine the carrying capacity K

To find the value of K, we need to analyze the given equation and find the maximum value of the population. In the logistic equation, the population will reach its carrying capacity when the growth rate (rt) is relatively large compared to the initial population (Pe). As the value of 't' tends to infinity, the exponential term 'e^{0.2t}' becomes very large, and the denominator approaches K. Hence, the carrying capacity K is the limiting value of P(t) as t approaches infinity. In this case, we can find K by looking at the fraction part of the equation: $$ K = \lim_{t\to\infty} P(t) = \lim_{t\to\infty} \frac{100e^{0.2 t}}{e^{0.2 t}+19} $$ As t approaches infinity, the exponential term 'e^{0.2t}' dominates the denominator, and we get: $$ K = \frac{100e^{0.2 t}}{e^{0.2 t}} = 100 $$ So, the carrying capacity of the environment is K = 100.

Step 2: Determine the growth constant r

The growth constant r is the coefficient of t in the exponential term. In the given equation, the exponential term is 'e^{0.2t}', and the coefficient of t is 0.2. Therefore, the growth constant r is 0.2.

Step 3: Determine the initial population Pe

The initial population Pe is the value of the logistic equation P(t) at t=0. To find the initial population, substitute t = 0 in the given equation: $$ P(0)=\frac{100 e^{0.2(0)}}{e^{0.2(0)}+19} = \frac{100}{1+19} $$ $$ P(0) = \frac{100}{20} = 5 $$ Thus, the initial population Pe is 5. In conclusion, for the given logistic equation $$P(t)=\frac{100 e^{0.2 t}}{e^{0.2 t}+19},$$ (a) the growth constant is r = 0.2, (b) the carrying capacity of the environment is K = 100, and (c) the initial population is Pe = 5.

Key Concepts

Growth Constant

The growth constant, often denoted as \( r \), is a crucial element in the study of population dynamics with logistic equations. It represents the rate at which the population grows when conditions for growth are optimal—meaning no limiting factors are immediately impacting the population size.
In the formula we are examining, \( P(t) = \frac{100 e^{0.2 t}}{e^{0.2 t} + 19} \), the growth constant is extracted directly from the exponential term.
The component \( e^{0.2t} \) reveals that \( 0.2 \) is the coefficient aligned with \( t \), making the growth constant \( r = 0.2 \).
This means that, under unrestricted and ideal circumstances, the population would grow at a consistent rate of 20% per time period. The higher \( r \) is, the faster the population can increase under favorable conditions.

This fundamental aspect helps in predicting how rapidly a population can expand initially, given that other factors (like resources and space) don't become limiting.

Carrying Capacity

The carrying capacity, often symbolized by \( K \), describes the maximum population size that an environment can sustainably support. It reflects the limit set by factors like food resources, habitat space, water availability, and other necessities.
In a logistic equation like \( P(t) = \frac{100 e^{0.2 t}}{e^{0.2 t} + 19} \), we determine the carrying capacity by examining the behavior of the function as \( t \) goes towards infinity.
When \( t \) becomes very large, the exponential term \( e^{0.2t} \) in both the numerator and denominator grows very large as well. Eventually, this term dominates the denominator, simplifying it to approximate the form of its coefficient.
This results in the expression \( K = 100 \), indicating that 100 is the largest population the environment can support under the given conditions.

The concept of carrying capacity is essential as it helps to buffer against unchecked population growth, preventing depletion of resources and environmental degradation.

Initial Population

The initial population is denoted as \( Pe \) and represents the size of the population at the starting point of the observation, which is typically taken at \( t = 0 \). Understanding the initial population size gives insight into the early conditions of population growth.
In our logistic function \( P(t) = \frac{100 e^{0.2 t}}{e^{0.2 t} + 19} \), we find the initial population by setting \( t = 0 \).
This results in the equation \( P(0) = \frac{100 e^{0}}{e^{0} + 19} = \frac{100}{20} = 5 \). Therefore, the initial population \( Pe \) is 5.

Starting with the initial population allows us to predict how the population will change over time under the effects of both the growth constant and the carrying capacity. This helps in understanding the early stages of population dynamics under logistic growth.