Unformatted text preview: 1.10 Linear Models (Diﬀerence Equations) Want time for Midterm 1 Review, so discuss diﬀerence equations but skip many applications in this section. Two basic types of dynamic mathematical models: (i) continuous diﬀerential or integral equations - involves equations with derivatives or integrals in time t (ii) discrete equations - involves discrete functions (deﬁned at a discrete, or distinct, set of t values, so changes over time) Here, we consider (ii) for linear models Some applications areas:
ecology, economics. engineering Deﬁnition: A (ﬁrst order) linear diﬀerence equation or
recurrence relation is of the form xk+1 = Axk for k = 0, 1, 2, . . . where A is a matrix. If x0 is known, then diﬀerence equation determines a sequence of values 1 x1 = Ax0, x2 = Ax1, x3 = Ax2 , x4 = Ax3 , · · · A subject of interest to demographers: movement of populations or groups of people between regions. EX: At beginning of 1990, population of California = 29, 716, 000, population of rest of U.S. = 218, 994, 000. During 1990, 509, 500 people moved from California to elsewhere in U.S., 564, 100 people moved from rest of U.S. to California. Assuming migration rate does not change from year to year, ﬁnd population of California and rest of U.S. in 2000. Use 5 decimal digits in your calculations. Solution: Fix 1990 as the initial year and denote population in California and rest of U.S. in that year by c0 and r0 , respectively. Let x0 be the population vector x0 = c0 r0 Denote populations in subsequent years by x1 = c1 r1 , x2 = c2 r2 , ··· After ﬁrst year, fraction of Californians who moved =
509,500 29,726,000 = .017146, fraction who stayed = 1 − .017146 = .982854.
2 For 5 digits, round to .98285. NEED fractions to sum to 1, so use 1 − .98285 = .01715 for the ﬁrst fraction! So after year 1, the original c0 people in California are .98285 .98285c0 = c0 distributed by .01715 .01715c0 Similarly, fraction of rest of people in U.S. moving to California is =
564,100 218,994,000 = .00258 and fraction staying in rest of U.S. = 1 − .00258 = .99742, so after year 1, r0 people in rest of U.S. are distributed by .00258r0 .99742r0 = r0 .00258 .99742 . Therefore, x1 = , c1 r1 = c0 .98285 .01715 +r0 .00258 .99742 = .98285 .00258 .01715 .99742 c0 r0 or x1 = M x0 where migration matrix M = .98285 .00258 .01715 .99742 With data in millions of people we ﬁnd x1 = M x0, x2 = M x1, x3 = M x2, · · ·, x10 = M x9, to be 29.7 219.0 , 29.8 218.9 30.223 218.49 , 29.8 218.9 , ··· , 30.1 218.6 , 30.18 218.53 , 30.223 218.49 so = x10 = gives the populations in 2000. 3 ...
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