Logistic Growth: The growth rate of many different populations depends not only on the number of individuals (leading to exponential growth) but also on a "carrying capacity" of the environment. If x is the population at time t and the growth rate of x is proportional to the product of the population and the carrying capacity M minus the population, then the growth rate is described by the differential equation dx/dt = kx(M - x) where k and M are constants for a given species in a given environment. 15. Let k = 1 and M = 100, and assume the initial population is x(0) = 5 . a. Solve the differential equation dx/dt = x(100 - x) for x . b. Graph the population x(t) for 0 ? t ? 20. c. When will the population be 20? 50? 90? 100? d. What is the population after a "long" time? (Find the limit, as t becomes arbitrarily large, of x)
Added by Debbie B.
Close
Step 1
We first solve for x(100-x) using the differential equation: dx/dt = x(100-x) Show more…
Show all steps
Your feedback will help us improve your experience
Madhur L and 60 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Logistic growth Scientists often use the logistic growth function $P(t)=\frac{P_{0} K}{P_{0}+\left(K-P_{0}\right) e^{-r_{d}}}$ to model population growth, where $P_{0}$ is the initial population at time $t=0, K$ is the carrying capacity, and $r_{0}$ 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. World population (part 1 ) The population of the world reached 6 billion in $1999(t=0)$. Assume Earth's carrying capacity is 15 billion and the base growth rate is $r_{0}=0.025$ per year. a. Write a logistic growth function for the world's population (in billions) and graph your equation on the interval $0 \leq t \leq 200$ using a graphing utility. b. What will the population be in the year 2020? When will it reach 12 billion?
Derivatives
Derivatives of Logarithmic and Exponential Functions
Logistic growth Scientists often use the logistic growth function $P(t)=\frac{P_{0} K}{P_{0}+\left(K-P_{0}\right) e^{-r_{0} t}}$ to model population growth where $P_{0}$ is the initial population at time $t=0, K$ is the carrying capacity, and $r_{0}$ 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. (FIGURE CAN'T COPY) The population of the world reached 6 billion in $1999(t=0) .$ Assume Earth's carrying capacity is 15 billion and the base growth rate is $r_{0}=0.025$ per year. a. Write a logistic growth function for the world's population (in billions), and graph your equation on the interval $0 \leq t \leq 200$ using a graphing utility. b. What will the population be in the year $2020 ?$ When will it reach 12 billion?
Logistic growth: $P(t)=\frac{C}{1+a e^{-k t}}$ For populations that exhibit logistic growth, the population at time $t$ is modeled by the function shown, where $C$ is the carrying capacity of the population (the maximum population that can be supported over a long period of time), $k$ is the growth constant, and $a=\frac{c-P(0)}{P(0)} .$ Solve the formula for $t$, then use the result to find the value of $t$ given $C=450, a=8, P=400,$ and $k=0.075$.
Exponential and Logarithmic Functions
Properties of Logarithms; Solving Exponential/Logarithmic Equations
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD