Problem 54 - Precalculus

Question

Suppose that both logistic growth models \(f(t)=\frac{a_{1}}{1+c_{1} e^{-k_{i} t}}\) and \(g(t)=\frac{a_{2}}{1+c_{2} e^{-k_{s} t}}\) have horizontal asymptote \(y=100\) What can you say about the corresponding carrying capacities?

Step-by-Step Solution

Answer

Both logistic models have the same carrying capacity of 100.

Step 1: Understand the Logistic Function

A logistic growth model is represented by the formula \( f(t)=\frac{a}{1+ce^{-kt}} \). The parameter \( a \) represents the horizontal asymptote of the function, which is the carrying capacity of the population model described by the logistic function.

Step 2: Identify Given Information

In the problem, both functions \( f(t) \) and \( g(t) \) have the horizontal asymptote \( y=100 \). This indicates that the value of \( a_1 \) in function \( f(t) \) and the value of \( a_2 \) in function \( g(t) \) are both 100.

Step 3: Conclude the Carrying Capacities

Since the horizontal asymptotes for both functions \( f(t) \) and \( g(t) \) are equal to 100, it means their carrying capacities are also 100.

Key Concepts

Carrying Capacity

In logistic growth models, the **carrying capacity** is a vital concept. It represents the maximum population size that an environment can sustain indefinitely. This threshold is determined by resources availability, habitat size, and environmental conditions. In the context of the logistic growth formula \( f(t) = \frac{a}{1 + ce^{-kt}} \), the parameter \( a \) signifies the carrying capacity. This means it is the upper limit that the population \( f(t) \) approaches as time \( t \) tends to infinity.For instance, if a population is modeled using a logistic function where \( a = 100 \), this implies the environment can support up to 100 individuals in the long run. The population will grow rapidly when it's small, but as it approaches 100, the growth will slow down. This diminishing growth rate is due to limited resources or space, reflecting real-world conditions.

Horizontal Asymptote

A **horizontal asymptote** in a logistic growth function represents the line that the function approaches as the population increases over time. In the formula used for logistic growth models, \( f(t) = \frac{a}{1+ce^{-kt}} \), the variable \( a \) is the horizontal asymptote.When solving problems involving logistic growth, the horizontal asymptote provides essential information about the system's carrying capacity. For example, given the horizontal asymptote \( y = 100 \), this indicates the population will level off at 100, as seen in exercise calculations. It acts as a visual guideline to understand how a population's size stabilizes over time.

Exponential Function

The **exponential function** is a core component of logistic growth models. It is represented in the logistic equation by \( e^{-kt} \), where \( e \) is the base of the natural logarithm, and \( t \) is time.Initially, the exponential function dominates the growth pattern, causing rapid increase when the population size is much less than the carrying capacity. However, as the population approaches this limit, the effect of \( e^{-kt} \) diminishes. This results in a slowing growth rate, avoiding the unrealistic indefinite increase depicted in simple exponential models. This shift underlies why the logistic model is often more accurate for real-world population studies.

Population Model

A **population model** like the logistic growth model is designed to describe how populations change over time. These models incorporate factors such as growth rate and carrying capacity to predict future population sizes. In practice, these models help ecologists and scientists in forecasting population trends, aiding in habitat management and ecological conservation. Logistic models, in particular, account for limits on population growth, unlike simple exponential models, providing a more realistic picture. By understanding these dynamics, stakeholders can make informed decisions about resource management and environmental protection.