9.4 The Logistic Equation Wed Jan 06 Do Now Solve the differential equation Y’ = 2y, y(0) = 1

HW Review

The Logistic Equation Like an exponential growth/decay equation, the logistic equation can be used to model population curves Unlike exponential growth/decay equations, the logistic equation takes into account environmental limitations such as food scarcity, etc

The Logistic Equation The logistic differential equation is modeled by: where k > 0 is the growth constant, and A > 0 is a constant called the carrying capacity As y -> A, y’ slows down to 0

Logistic Equation Properties The logistic equation has 3 families of solutions, depending on the initial value y(0): If y(0) > A, then y is decreasing and approaches A If 0 < y(0) < A, then y is increasing and approaches A If y(0) < 0, then y is decreasing and approaches negative infinity

Solving the Logistic Equation By using partial fraction decomposition (7.5), we can obtain the general solution to a logistic equation

Ex1 Solve y’ = 0.3y(4-y) with initial condition y(0) = 1

Ex 2 A deer population grows logistically with growth constant k = 0.4 year^-1 in a forest with carryin capacity of 1000 deer. Find the deer population P(t) if the initial population is 100 deer.

Closure Why is a logistic equation a better model for population growth than an exponential equation? HW: p.527 #1, 3, 4, 5, 7, 8, 11

Review Thurs Jan 7 Do Now

Ch 9 Quiz/Test 9.1 – Solving separable differential equations 9.3 – Slope Fields – Euler’s Method 9.4 – Logistic Differential equations

Closure How many different ways can we solve a differential equation? Describe them. XC: Ch 9 AP MC 1-4, 7-12, 14-16, FRQ 1-4