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c37 - 18.03 Class 37 May 12 Introduction to general...

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18.03 Class 37, May 12 Introduction to general nonlinear autonomous systems. [1] Recall that an ODE is "autonomous" if x' depends only on x and not on t: x' = g(x) For example, I know an island in the St Lawrence River in upstate New York where there are a lot of deer. When there aren't many deer, they multiply with growth rate k ; x' = kx . Soon, though, they push up against the limitations of the island; the growth rate is a function of the population, and we might take it to be k(1-(x/a)) where a is the maximal sustainable population of deer on the island. 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 . There are "critical points" at x = 0 and x = 3 . When 0 < x < 3 , x' > 0. When x > 3 , x' < 0 , and, unrealistically, when x < 0 , x' < 0 too. I drew some solutions, and then recalled the phase line: ------<-----*---->------*------<-------- [2] One day, a wolf swims across from the neighboring island, pulls himself up the steep rocky shore, shakes the water off his fur, and sniffs the air.
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