Problem 43 - Precalculus with Limits

Question

A conservation organization released 100 animals of an endangered species into a game preserve. The preserve has a carrying capacity of 1000 animals. The growth of the pack is modeled by the logistic curve \(p(t)=\frac{1000}{1+9 e^{-0.1656 t}}\) where \(t\) is measured in months (see figure). (a) Estimate the population after 5 months. (b) After how many months is the population \(500 ?\) (c) Use a graphing utility to graph the function. Use the graph to determine the horizontal asymptotes, and interpret the meaning of the asymptotes in the context of the problem.

Step-by-Step Solution

Answer

(a) We get the population size after 5 months by substituting \(t = 5\) into the equation. (b) The time it takes for the population to reach 500 can be found by solving the equation for \(t\) when \(p(t)=500\). (c) The horizontal asymptotes of the function represent the maximum population size (carrying capacity) and the minimum population size that the species can reduce to over time.

Step 1: Estimate the population after 5 months

Let's substitute \(t = 5\) in the logistic growth model equation: \[ p(5)= \frac{1000}{1+9e^{-0.1656 \cdot 5}} \]

Step 2: Finding the time for the population to reach 500

We need to solve the equation \[ p(t)=500=\frac{1000}{1+9e^{-0.1656 t}} \] for \(t\).

Step 3: Graph the function and find the horizontal asymptotes

To graph the function, plug in a range of \(t\) values into the equation. To find the horizontal asymptotes, remember that these occur where the function approaches but never reaches a certain value. In this logistic growth model, the horizontal asymptotes will be at \(p=0\) and \(p=1000\), which represents the carrying capacity of the preserve.

Key Concepts

Carrying Capacity

The concept of "carrying capacity" is fundamental in understanding population dynamics, especially in ecological contexts. It refers to the maximum number of individuals in a population that an environment can sustain indefinitely. For instance, in the exercise, the game preserve has a carrying capacity of 1000 animals.
Carrying capacity is influenced by several factors:

• Resources: Availability of food, water, and shelter.
• Environmental conditions: Climate and weather patterns.
• Space: The physical area available for the population.

In practice, when a population reaches this capacity, it tends to stabilize. Growth slows down as resources become limited, preventing further increase in population size beyond a certain point.

Population Modeling

Population modeling is a mathematical way of predicting how a population will change over time. One common model used is the logistic growth model, as seen in the exercise. This model considers not just the growth potential of a population, but also limits imposed by environmental factors, like carrying capacity.
The logistic growth equation is given by:\[ p(t) = \frac{K}{1 + Ae^{-rt}} \]where:

• \( p(t) \) is the population at time \( t \).
• \( K \) is the carrying capacity.
• \( A \) is a constant related to the initial population size.
• \( r \) is the rate of growth.

This model initially shows rapid growth, which then slows as the population nears the carrying capacity, creating an S-shaped curve. It's more accurate than linear models when resources are limited.

Differential Equations

Differential equations are a tool used in mathematical modeling to describe the rate at which one quantity changes with respect to another. They are integral to understanding dynamic systems, like population changes over time.
In the context of logistic growth, a differential equation can take the form:\[ \frac{dp}{dt} = rp(1 - \frac{p}{K}) \]Here,

• \( \frac{dp}{dt} \) represents the change in population over time.
• \( r \) is the intrinsic rate of increase.
• \( p \) is the current population size.
• \( K \) is the carrying capacity.

This equation shows how the growth rate changes as the population size approaches the carrying capacity. Understanding these equations helps in predicting future trends and managing populations sustainably.