Looking Ahead (12/5/08) Today – HW#4 handed out. Monday – Regular class Tuesday – HW#4 due at 4:45 pm Wednesday – Last class (evaluations etc.) Thursday – Regular office hours 3:15-4:45 Friday – Extra Help 11-noon (HW#4 returned) Tuesday (12/16) – Regular office hours Wednesday (12/17) – Exam 9-noon

Application of DE’s: Population Growth Let P be the size of a population and let t be time. We have seen already that if the population grows at a rate proportional to its size, this say that it satisfies the DE: dP / dt = k P, k being the relative growth rate. This is separable, and we know the general solution is P = A e kt where A is the starting population. This is, naturally, called exponential growth.

The Logistic Model of Growth Many populations may grow exponentially at first, but eventually that growth rate slows as capacity (space, food, etc.) is reached. That is, as time passes, k will approach 0. If the maximum capacity of the population is denoted M, a simple expression which approaches 0 as P approaches M is 1 – P / M.

The Logistic DE Thus a DE which would model this “exponential growth at first but slowing of the growth rate as P approaches its maximum capacity” would be

Example Suppose a population growing by the logistic model has a maximum capacity of 1000 and displays an initial growth rate of 8%. Look at the slope field. Look at an Euler’s Method approximate solution assuming an initial population of 2. Can we explicitly solve this DE? Is it separable?

Assignment Regular class on Monday. Work on HW#4. Test #2 corrections will be returned Monday.