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.


d. Graph P' and use the graph to estimate the year in which the population is growing fastest. 

Verified step by step guidance

1. Understand the logistic growth function: P(t) = \(\frac{P_0 K}{P_0 + (K - P_0)e^{-r_0 t}\)}. This function models population growth with an initial population P₀, carrying capacity K, and growth rate r₀.
2. Identify the given parameters for the fish population: P₀ = 50, K = 8000, and r₀ = 0.5. The logistic model provided is P(t) = \(\frac{400,000}{50 + 7950e^{-0.5t}\)}.
3. To find when the population is growing fastest, we need to analyze the derivative of the logistic function, P'(t). The derivative will give us the rate of change of the population over time.
4. Graph the derivative P'(t) using a graphing tool or software. Look for the point where P'(t) reaches its maximum value, as this indicates the fastest growth rate.
5. Estimate the year at which the population growth is fastest by identifying the t-value corresponding to the peak of the graph of P'(t). This t-value represents the year when the population growth rate is at its highest.

Key Concepts

Logistic Growth Model

The logistic growth model describes how a population grows in an environment with limited resources. It is characterized by an initial exponential growth phase, followed by a slowdown as the population approaches a maximum capacity, known as the carrying capacity (K). The model is represented by the equation P(t) = P₀K / (P₀ + (K - P₀)e^(-r₀t)), where P₀ is the initial population, r₀ is the growth rate, and t is time.

Carrying Capacity

Carrying capacity (K) is the maximum population size that an environment can sustain indefinitely without being degraded. It is a crucial concept in ecology and population dynamics, as it determines the upper limit of population growth. In the context of the logistic model, as the population approaches K, the growth rate decreases, leading to a stabilization of the population size.

Derivative and Population Growth Rate

The derivative of the population function, P'(t), represents the rate of change of the population over time, indicating how fast the population is growing or declining. To find when the population is growing fastest, one must analyze the graph of P'(t) to identify its maximum value. This involves determining critical points where the derivative equals zero and assessing the behavior of the function around these points.