Problem 21 - Applied Calculus

Question

A population, \(P\), growing logistically is given by $$ P=\frac{L}{1+C e^{-k t}} $$ (a) Show that $$ \frac{L-P}{P}=C e^{-k t} $$ (b) Explain why part (a) shows that the ratio of the additional population the environment can support to the existing population decays exponentially.

Step-by-Step Solution

Answer

(a) \( \frac{L-P}{P} = C e^{-k t} \). (b) The ratio reflects exponential decay, indicating the population nears carrying capacity over time.

Step 1: Understand the Logistic Growth Formula

The logistic growth model is given by the function \( P = \frac{L}{1 + C e^{-k t}} \), where \( L \) is the carrying capacity of the population, \( C \) is a constant related to the initial population, \( k \) is a growth constant, and \( t \) represents time. The task is to manipulate this formula to express \( \frac{L-P}{P} \).

Step 2: Isolate L - P

Start from the logistic growth formula \( P = \frac{L}{1 + C e^{-k t}} \). Rearrange this equation to solve for \( L - P \). First, multiply both sides by \( 1 + C e^{-k t} \) resulting in \( P(1 + C e^{-k t}) = L \). Then, subtract \( P \) from \( L \): \( L - P = PC e^{-k t} \).

Step 3: Solve for \( \frac{L - P}{P} \)

Using the expression for \( L - P = PC e^{-k t} \) obtained in Step 2, divide both sides by \( P \) to find the ratio: \( \frac{L - P}{P} = C e^{-k t} \). This shows that this ratio is equal to \( C e^{-k t} \), which is exactly what the exercise asks to demonstrate.

Step 4: Interpret the Exponential Decay

The expression \( \frac{L - P}{P} = C e^{-k t} \) shows that the ratio of the remaining capacity (\( L - P \)) to the current population (\( P \)) decreases exponentially over time. This means that as time progresses, the additional population the environment can support reduces exponentially compared to the existing population. This reflects how the population approaches its carrying capacity \( L \).

Key Concepts

Population Dynamics

Population dynamics is the study of how populations change over time. In the context of logistic growth, it's essential to understand how populations start off growing rapidly. This initial fast growth typically happens when resources are abundant. However, as the population increases, resources become scarcer, leading to a slow down in growth rates. With logistic growth, the mathematical representation of this change is used to model real-world scenarios like animal populations.

• Populations start with exponential growth when they're small.
• The growth rate decreases as the population nears its environmental limits.

This change in growth can be observed using the logistic growth equation: \[P = \frac{L}{1 + Ce^{-kt}} \]where \(P\) represents the population size, \(L\) the carrying capacity, and \(C\) a constant. Understanding this equation helps us predict how a population will evolve over time.

Carrying Capacity

Carrying capacity, often represented as \(L\) in the logistic growth model, is the maximum population size that an environment can sustain indefinitely. It's crucial because it marks the upper limit of population growth in a given environment.

• Once the population reaches the carrying capacity, growth will slow to a stop.
• The environment's resources—such as food, space, and water—dictate this maximum.

In our logistic growth equation, carrying capacity is the value that the population approaches as time progresses. As populations near their carrying capacity, the growth rate will decline, reflecting that resources are nearing depletion. Thus, the concept of carrying capacity helps to understand not just the limits but also the sustainable growth rate of a population.

Exponential Decay

Exponential decay in the context of logistic growth describes how the difference between the carrying capacity and current population, divided by the current population, decreases over time. The formula derived in the exercise \[\frac{L-P}{P} = Ce^{-kt} \]presents this phenomenon clearly. Here, \(L-P\) is the unutilized potential of the environment or simply the additional population the environment can sustain.

• As \(t\) (time) increases, \(Ce^{-kt}\) decreases exponentially, reflecting exponential decay.
• This means the population incrementally fills the available space until it stabilizes.

This decay is vital because it demonstrates the limiting nature of an environment's resources. The more the population increases, the slower this growth becomes, eventually reaching a stable point at the carrying capacity. This is how exponential decay links intimately with the balance of population dynamics.