18.03 Lecture Notes - Saint Lawrence River, British Rail Class 37, Logistic Function
Document Summary
Vector fields and nonlinear systems: autonomous systems, vector fields, equilibria, linearization. Recall that an ode is autonomous if x" depends only on x and not on t: x" = f(x) For example, i know an island in the st lawrence river in upstate new york where there are a lot of deer [slide]. When there aren"t many of them, they multiply with growth rate k ; x" = kx . So the equation is x" = k(1-(x/a))x , the "logistic equation. " On this particular island, k = 3 and a = 3 , so x" = (3-x)x . (maybe the units are hundreds of deer, and years). There are "critical points" at x = 0 and x = 3 . I drew some solutions, and then recalled the phase line: One day, a wolf [slide] swims across from the neighboring island, pulls himself up the steep rocky coast, shakes the water off his fur, and sniffs the air.
