Problem 12 - Calculus: Early Transcendentals

Question

Biologist stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 10,000. The number of fish tripled on the first year. (a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after \( t \) years. (b) How long will it take for the population to increase to 5000?

Step-by-Step Solution

Answer

The population expression is \( P(t) = \frac{10000}{1 + 25e^{-rt}} \), and it takes approximately 3.98 years to reach 5000 fish.

Step 1: Understand the logistic equation

The logistic equation models population growth: \[ P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right)e^{-rt}} \] where \( K \) is the carrying capacity, \( P_0 \) is the initial population, and \( r \) is the growth rate. For this problem, \( K = 10000 \) and \( P_0 = 400 \).

Step 2: Determine the growth rate after the first year

The population triples in the first year, going from 400 to 1200 fish. Substitute into the logistic equation: \[ 1200 = \frac{10000}{1 + \left(\frac{10000 - 400}{400}\right)e^{-r}} \] Simplify to find the growth rate \( r \).

Step 3: Solve for the growth rate

Solve the equation \[ 1200 = \frac{10000}{26e^{-r} + 1} \] to find \( r \). Rearranging: \[ 26e^{-r} = \frac{10000}{1200} - 1 \] Calculate and get \[ e^{-r} \] and use it to find \( r \).

Step 4: Write the expression for the population size

With the value of \( r \), substitute back into the logistic formula: \[ P(t) = \frac{10000}{1 + 25e^{-rt}} \] This expression gives the population at any time \( t \).

Step 5: Find when population reaches 5000

Set \( P(t) = 5000 \) in the equation: \[ 5000 = \frac{10000}{1 + 25e^{-rt}} \] Solve for \( t \) by isolating \( e^{-rt} \) and taking the natural logarithm to find \( t \). Calculate the time required.

Key Concepts

Population Growth

Population growth is a central concept in understanding how populations change over time. It refers to the increase in the number of individuals in a population. In ecology, models like the logistic equation describe how populations grow, considering factors that limit their growth.

Populations usually don't grow indefinitely; they encounter limitations like resources, space, and predation. This is where the logistic model comes into play.

• The growth starts slowly (lag phase), because there are not many individuals to reproduce.
• It speeds up as more individuals reproduce (exponential growth phase).
• Eventually, growth slows and stabilizes due to limiting factors (stationary phase).

In the given problem, the number of fish in the lake increases, but eventually, it will stabilize as it approaches a certain limit, known as the carrying capacity.

Carrying Capacity

Carrying capacity is the maximum population size that an environment can sustain indefinitely. It is a critical concept in the logistic equation as it represents the cap on population growth imposed by limited resources.

In the lake example, the carrying capacity is 10,000 fish. This means that the lake can support up to 10,000 fish with available resources like food, space, and oxygen without degrading the environment. The carrying capacity influences the rate of population growth:

• As the population nears the carrying capacity, growth slows because resources become scarce.
• The carrying capacity depends on environmental factors and can change over time with changes in these factors.

Understanding carrying capacity helps biologists manage wildlife and conserve ecosystems effectively.

Growth Rate

The growth rate in the logistic model determines how quickly a population increases. It is a crucial parameter in the logistic equation because it impacts how soon a population will reach its carrying capacity.

In our problem, the biologist observes the fish population tripling from 400 to 1200 in the first year. This observation provides a basis to calculate the growth rate, denoted as \( r \).

• Higher growth rates lead to faster population growth.
• Lower growth rates mean the population will take longer to approach the carrying capacity.

To find the growth rate, we use the logistic equation with the known values: initial population, observation after the first year, and carrying capacity. Once \( r \) is determined, it allows prediction of future population sizes.

Logarithm in Equations

The logarithm plays a vital role in solving logistic equations for population growth problems. Natural logarithms are used for their properties, which help in isolating variables, particularly the growth rate \( r \).

When solving for \( r \) or \( t \) in logistic models, rearranging the equation often involves the exponential function, which pairs with natural logarithms.

• Using a logarithm helps linearize the exponential components, making them manageable.
• Logarithms translate multiplicative relationships into additive ones, simplifying equation manipulation.

For example, in our problem, after isolating \( e^{-rt} \), we apply natural logarithms to solve for \( t \). This approach is handy in interpreting complex models and forecasting when a population will reach a certain size.