18.03 Lecture Notes - Lecture 8: Logistic Function, Population Model, Bifurcation Diagram
Document Summary
Final tuesday, may 18, 9:00 - 12:00, johnson track. Hour exam next wednesday: rooms to be announced monday. Please go to the hour you are registered for. [1] back to qualitative study of differential equations. The general first order equation is y"(t) = f(t,y) Autonomous means conditions are constant in time, though they may depend on the current value of y . Eg [natural growth/decay] constant growth rate: so y" = k0 y . Eg bank account with interest rate not depending on time but possibly depending upon current balance, and constant savings rate: y" = i(y) y + q . Extended example: population model with variable growth rate k(y) depending on the current population but not on time; so y" = k(y) y. Suppose that when y is small the growth rate is approximately k0 , but that there is a maximal sustainable population p , and as y gets near to p the growth rate decreases to zero.
