6.3—Logistic Growth L The limit L is called the carrying capacity. t y Here is the basic shape of a logistic curve: The growth starts out slowly. After a little while, the function increases rapidly. Finally, things begin to level out. dy/dt = ky (1- ) Most of the logistic growth problems will be given to you in this form: y L

6.3—Logistic Growth Solving this differential equation takes about 20 steps, is cumbersome, and involves partial fractions along with some fairly tricky algebra. The answer comes out to be: L y = 1 + C e -kt This PowerPoint will show you the steps needed to solve the differential equation. You will probably find it much easier to simply memorize the answer than to do this each time! dy/dt = ky (1- ) Most of the logistic growth problems will be given to you in this form: y L

6.3—Logistic Growth dy/dt = ky (1 – y/L) dy = ky (1 - y/L) dt = k dt y (1-y/L) dy (requires partial fractions) = kt + C

6.3—Logistic Growth = kt + C y (1-y/L) dy y A 1-y/L B dy +dy A(1-y/L) + By = 1 1 When y = 0, A = 1 When y = L, B = 1/L 1 1/L

6.3—Logistic Growth y A 1-y/L B dy +dy 1 1/L

6.3—Logistic Growth y A 1-y/L B dy +dy 1 1/L ln y ln L - y- Multiply top & bottom by L. y A L-y B dy +dy 1 1 ln y ln L - y- = kt + C

6.3—Logistic Growth ln y ln L - y-= kt + C L - yL - y y ln L - yL - y y = e (kt + C) y L - yL - y = e (-kt – C)

6.3—Logistic Growth y L - yL - y = e (-kt – C) y L - yL - y = Ce -kt L y - 1 = Ce -kt L y = 1 + Ce -kt

6.3—Logistic Growth L y = 1 + Ce -kt y L = 1 y (1 + Ce -kt ) = L (1 + Ce -kt ) L y =

6.3—Logistic Growth Solving this differential equation takes about 20 steps, is cumbersome, and involves partial fractions along with some fairly tricky algebra. The answer comes out to be: L y = 1 + C e -kt This PowerPoint will show you the steps needed to solve the differential equation. You will probably find it much easier to simply memorize the answer than to do this each time! dy/dt = ky (1- ) Most of the logistic growth problems will be given to you in this form: y L