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. {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.b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Ch. 3 - Derivatives

Problem 3.9.97b

Chapter 3, Problem 3.9.97b

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.

b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

Verified step by step guidance

Identify the logistic growth function given in the problem: P(t) = \(\frac{400,000}{50 + 7950e^{-0.5t}\)}.

To find the time it takes for the population to reach 5000 fish, set P(t) = 5000 and solve for t: \(\frac{400,000}{50 + 7950e^{-0.5t}\)} = 5000.

Rearrange the equation to isolate the exponential term: 50 + 7950e^{-0.5t} = \(\frac{400,000}{5000}\).

Calculate \(\frac{400,000}{5000}\) to simplify the equation: 50 + 7950e^{-0.5t} = 80.

Subtract 50 from both sides and solve for e^{-0.5t}: 7950e^{-0.5t} = 30. Then, divide both sides by 7950 and take the natural logarithm to solve for t.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logistic Growth Model

The logistic growth model describes how a population grows in a limited environment, 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. This S-shaped curve illustrates how populations stabilize as resources become scarce.

Carrying Capacity

Carrying capacity (K) refers to the maximum population size that an environment can sustain indefinitely without degrading the habitat. In the context of the logistic growth model, it acts as a threshold that limits population growth as resources become limited. Understanding carrying capacity is crucial for predicting population dynamics and managing ecological systems, as it helps determine when a population will stabilize.

Exponential Growth and Decay

Exponential growth occurs when the growth rate of a population is proportional to its current size, leading to rapid increases when resources are abundant. Conversely, exponential decay describes a decrease in population size when resources are limited or when mortality rates exceed birth rates. In logistic growth, the initial phase often resembles exponential growth until the effects of limited resources begin to slow the growth, transitioning the population towards the carrying capacity.

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