MATH 4B Lecture Notes - Lecture 17: Taylor Series, Logistic Function, Nonlinear System
62
MATH 4B Full Course Notes
Verified Note
62 documents
Document Summary
Model the changing populations of two competing species with a system of differential equations. Qualitatively analyze nonlinear, autonomous systems of first order des. Stability for linear systems helps us understand stability for nonlinear systems. Logistic equation for population growth: is the growth rate. In the absence of sheep, the rabbit population obeys the logistic equation. The introduction of sheet causes the rabbit population growth to slow or become negative. Absence/intro of rabbits has a similar effect on the sheep population. For one variable function , the taylor expansion at is: is a linear approxmiation of at. For two variable function , the taylor expansion at is: are partial derivatives. Math 4b page 1 behaves like the we can approximate a linear one: Going back to the example problem with the two competing species: Near a constant solution of the dynamical system. Near the equilibrium , the nonlinear system linear system.
