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LECTURE 13 (10/17/00)

12. ODE systems	(what's an ODE system?)
	fox-chicken:
	Study the population change of two competing species.
	A lot of foxes will eat a lot of chicken.
	When chicken are all eaten, fox will die out.
	If the number of foxes diminishes, the #chicken increases.
	When there are more chicken, #fox will grow.
	Then these foxes will eat more chicken.
	How to do this in math?
	Assum 	(1) Chicken will grow, without fox, at a rate
		      proportional to its population
		(2) chicken is eaten only when it interacts with a fox.
			dx/dt = a x + bxy
	(x=#chicken, y= #fox, a>0, b<0) 
	For fox:(3) dies at natural causes 
	             (4) #fox increases when interact with chicken dy/dt="c" y + dxy (c<0, d>0)
	For one example, a=.25, b = -.01, c = -1, d = .01
	Now, we have a pair of equations.
	Let's examine the system a little:
	a. stationary state: x'=y'=0, 
		ax + bxy = cy + dxy
	b and d are opposite signs, so we will have a quadratic system,
	this will lead to an elliptic equation (what if b=d???) 


	12.b  General concept of ODE systems:
		x1' = f1(t, x1, x2, ..., xn)
		x2' = f2(t, x1, x2, ..., xn)
		... ...
		xn' = fn(t, x1, x2, ..., xn)
	We can write this in a vector form:
	X = (x1, x2, ..., xn)
	Then, X' = F(t,X)
	For higher order ODEs, we can do the same thing:
	y(n) = f(t, y', y'', ..., y(n-1))
	We can design a system:
		x1 = y
		x2 = y'
		... ...
		xn = y(n-1)
	Thus,
		x1' = x2
		x2' = x3
		... ...
		xn' = f(t, x1, x2, ..., xn)
	which can also be written as
		X' = F(t,X)



	12.c Taylor series method for systems