Problem 49 - Precalculus

Question

In Exercises 49 and 50 , refer to the logistic model \(f(t)=\frac{a}{1+c e^{-k t}},\) where \(a\) is the carrying capacity. As \(c\) increases, does the model reach the carrying capacity in less time or more time?

Step-by-Step Solution

Answer

Increasing \(c\) causes the model to reach the carrying capacity in more time.

Step 1: Understand the Logistic Model

The given logistic model is \(f(t) = \frac{a}{1+c e^{-kt}}\). In this function, \(a\) is the carrying capacity, \(c\) is a positive constant, \(e\) is the base of the natural logarithm, and \(k\) is the growth rate. The equation models how a quantity \(f(t)\) approaches the carrying capacity \(a\) over time \(t\).

Step 2: Analyze Impact of Increasing \(c\)

At the beginning, when \(t=0\), the expression becomes \(f(0) = \frac{a}{1+c}\). As \(t\) increases, the \(e^{-kt}\) term decreases to zero, making the denominator closer to 1, and \(f(t)\) closer to \(a\). Increasing \(c\) makes the initial value \(f(0)\) smaller (as \(c\) appears in the denominator), which means the function \(f(t)\) starts further from the carrying capacity.

Step 3: Determine Time to Reach the Carrying Capacity

The rate at which \(f(t)\) approaches \(a\) is influenced by \(c\). A larger \(c\) means that initially, \(f(t)\) is smaller, so it takes more time for \(f(t)\) to grow and reach close to the carrying capacity. Hence, increasing \(c\) results in \(f(t)\) taking more time to reach the carrying capacity \(a\).

Key Concepts

Carrying Capacity

In the logistic growth model, the carrying capacity is represented by the parameter \( a \). It signifies the maximum population size or quantity that the environment can sustain indefinitely. The carrying capacity is a crucial aspect because it represents a limiting factor in many real-world scenarios. For instance, in a biological context, it may define the maximum number of organisms an ecosystem can support due to resource limitations such as food or space.

Understanding carrying capacity helps in predicting the ultimate stabilization of a system. In the model \( f(t) = \frac{a}{1 + ce^{-kt}} \), this value \( a \) (- indicates the horizontal asymptote of the function,- reflects the equilibrium state when growth levels off, and- shows the saturation point of the concerned quantity.) Eventually, as time progresses, no matter the initial changes in parameters like \( c \) or \( k \), \( f(t) \) tends to converge to this constant \( a \).

Growth Rate

The growth rate in the logistic model is represented by the constant \( k \). This parameter defines how rapidly the population or quantity increases towards the carrying capacity \( a \). A higher growth rate \( k \) implies that the quantity \( f(t) \) grows at a faster pace.

In practical terms, the growth rate can be affected by factors such as abundance of resources, favorable environmental conditions, or other external growth stimuli. Mathematically, a larger \( k \) means the depletion of the \( e^{-kt} \) term to zero happens swiftly, allowing \( f(t) \) to approach the carrying capacity faster.

However, due to the nature of logistical growth,

• if \( k \) is too high, you may encounter overshooting, where the population exceeds the carrying capacity and fluctuates around it before stabilizing.
• appropriate tuning of \( k \) ensures a smooth transition to \( a \) without excessive oscillations.

Thus, understanding the growth rate is key to analyzing how quickly a system responds to its constraints.

Effect of Constants

In the logistic growth model, the constant \( c \) significantly impacts the initial behavior and time to reach the carrying capacity \( a \). This constant \( c \) serves as a scaling factor for the initial denominator term \( 1 + ce^{-kt} \). When analyzing its effect:

• Higher values of \( c \) lead to smaller initial values of \( f(t) \), because they increase the denominator. This means the quantity starts off further away from \( a \).
• Consequently, the time it takes to get close to the carrying capacity extends, as the starting point of \( f(t) \) is lower, requiring a longer growth period.

Additionally, the choice of \( c \) can model different scenarios. For instance, in populations, a larger \( c \) could simulate an initial condition where resources are scarce, or a significant initial resistance to growth exists.

In essence, \( c \) adjusts how quickly \( f(t) \) gets underway in its journey towards \( a \), making it an important factor to consider for accurately depicting real-world dynamics.