97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>


{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.


a. Graph P using a graphing utility. Experiment with different windows until you produce an S-shaped curve characteristic of the logistic model. What window works well for this function? 

Verified step by step guidance

1. Understand the logistic growth function: P(t) = \frac{P_0 K}{P_0 + (K - P_0) e^{-r_0 t}}, where P_0 is the initial population, K is the carrying capacity, and r_0 is the growth rate.
2. Identify the parameters from the problem: P_0 = 50 (initial fish population), K = 8000 (carrying capacity), and r_0 = 0.5 (growth rate). The logistic model given is P(t) = \frac{400,000}{50 + 7950 e^{-0.5t}}.
3. To graph the function, use a graphing utility. Input the function P(t) = \frac{400,000}{50 + 7950 e^{-0.5t}} into the graphing tool.
4. Experiment with different window settings to capture the S-shaped curve. A good starting point for the window might be: x-axis (time, t) from 0 to 20 years, and y-axis (population, P(t)) from 0 to 9000 fish.
5. Adjust the window settings as needed to ensure the entire S-shaped curve is visible, showing the initial growth, the inflection point, and the leveling off as the population approaches the carrying capacity.

Key Concepts

Logistic Growth Function

The logistic growth function models population growth that is initially exponential but slows as the population approaches a maximum limit, known as the carrying capacity (K). The function is represented as P(t) = P₀K / (P₀ + (K - P₀)e^(-r₀t)), where P₀ is the initial population, r₀ is the growth rate, and t is time. This model is crucial for understanding how populations grow in constrained environments.

Carrying Capacity

Carrying capacity (K) refers to the maximum population size that an environment can sustain indefinitely without being degraded. It is a critical concept in ecology and population dynamics, as it influences the growth rate and stability of populations. In the context of the logistic model, as the population approaches K, the growth rate decreases, leading to a leveling off of the population size.

Graphing Logistic Functions

Graphing logistic functions involves plotting the population over time to visualize its growth pattern, which typically forms an S-shaped curve. This curve illustrates the initial rapid growth phase, followed by a slowdown as the population nears the carrying capacity. Understanding how to adjust the graphing window is essential for accurately displaying the behavior of the logistic model and identifying key features such as inflection points.