Download presentation

Presentation is loading. Please wait.

Published byConrad Barnett Modified over 6 years ago

1
Lesson 9-5 Logistic Equations

2
Logistic Equation We assume P(t) is constrained by limited resources so : Logistic differential equation for population growth: where small k is proportional constant of growth and big K is the population carry capacity (via resource restrictions) so if P > K the population decreases and if P < K the population increases Its solution is P(t) = K / (1 + A e –kt ) where A = (K – P 0 ) / P 0 if P is small dP ----- ≈ kP dt dP ----- ≈ kP (1 – P/K) dt

3
Example 1a Find the solution to the initial value problem dP/dt = 0.01P(1 – P/500) and P(1980) = 200,, where t is given in years since 1980. Hint: Identify P 0 and K; find A. P(t) = K / (1 + A e –kt ) where A = (K – P 0 ) / P 0 Differential Equation: k=0.01; K = 500; A = (500 – 200)/200 = 3/2 P(0)= 200 Equation: t is years since 1980 P(t) = 500 / (1 + (1.5)e -0.01t ) dP ----- ≈ kP (1 – P/K) dt

4
Example 1b Use the equation to find the population in 1985. Equation: t is years since 1980 P(t) = 500 / (1 + (1.5)e -0.01t ) P(5) = 500 / (1 + (1.5)e -0.01(5) ) = 500 /

5
Example 1c When does the population reach 450? Equation: t is years since 1980 P(t) = 500 / (1 + (1.5)e -0.01t ) 450 = 500 / (1 + (1.5)e -0.01t ) (1 + (1.5)e -0.01t ) = 500/450 1.5 e -0.01t = 1/9

6
Example 2 A biologist stocks a shrimp farm pond with 1000 shrimp. The number of shrimp double in one year and the pond has a carrying capacity of 10,000. How long does it take the population to reach 99% of the pond’s capacity? P = p 0 e ±kt General Equation: decay k < 0; p 0 = 10 grams P(5730)= 5 = 10e 5730k 0.5 = e 5730k ln 0.5 = ln e 5730k = 5730k ln 2 -1 / 5730 = k -(ln 2)/5730 = k = -0.000121 Equation: Use to solve for k P(2000) = 10e 2000(-0.000121) = 7.851 grams

7
Summary & Homework Summary: –Logistics Equations are another differential equation –Carrying capacity makes them unique Homework: –pg 629-631: 5, 6, 8

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google