Logistic Growth Function and Differential Equations
The Organic Chemistry Tutor・2 minutes read
Logistic equations explain exponential versus logistic growth, with logistic growth reaching a limit known as carrying capacity. By solving for b and k, the logistic equation predicts the population growth of dogs on an island, reaching approximately 1,094 dogs by 2020.
Insights
- Logistic equations differentiate between exponential and logistic growth, with exponential growth being unrestricted and logistic growth having a limit called carrying capacity.
- Deriving the logistic curve equation involves manipulating the differential equation through partial fractions and integration, providing a mathematical framework to model population growth with resource constraints.
Get key ideas from YouTube videos. It’s free
Recent questions
What is logistic growth?
Logistic growth has a limit called carrying capacity.
Related videos
Summary
00:00
Logistic Equations: Exponential vs. Logistic Growth
- Logistic equations focus on exponential vs. logistic growth, with exponential growth being unlimited and logistic growth having a limit known as the carrying capacity.
- Logistic curves describe population growth, where resources limit growth to a certain point.
- Differential equations for exponential growth: dy/dt = ky, and for logistic growth: dy/dt = ky(1 - y/l).
- Exponential growth increases proportionally to population size, while logistic growth behaves exponentially at low populations and levels off at the carrying capacity.
- General solution for exponential growth: y = ce^(kt), and for logistic growth: y = l/(1 + be^(-kt)).
- Deriving the logistic curve equation involves manipulating the differential equation through partial fractions and integration.
- The process involves setting up partial fractions, solving for constants a and b, integrating, and simplifying to derive the general logistic equation.
- Applying the derived logistic equation to a problem involving the release of dogs on an island in 2005 and calculating the population in 2012.
21:24
"Island Dog Population Growth Analysis"
- The island can sustain a maximum of 5,000 dogs assuming logistic growth.
- The general equation for the population at any time is y = L / (1 + b * e^(-kt)).
- In 2005, the population started with 100 dogs (p(0) = 100).
- By 2012 (t = 7), the population had grown to 324 dogs (p(7) = 324).
- The carrying capacity (L) is 5,000 dogs, and the goal is to solve for b and k.
- Solving for b: b = 49.
- Solving for k: k = 0.1746.
- The population equation is p(t) = 5000 / (1 + 49 * e^(-0.1746t)).
- By 2020 (t = 15), there will be approximately 1,094 dogs on the island.
- The population will reach 1,000 dogs between 2019 and 2020.
- Approximately 94 dogs are born per year in 2016.
- To estimate population growth, the average rate of change between 2012 and 2020 is used.
- By calculating the population in 2015 and 2017, the average growth rate in 2016 is found to be around 93.85.
