1. Chapter 3 Discrete Models Introduction –Independent variables are chosen discrete values Interest is accumulated monthly World population is collected yearly Number of tables Number of population –Paradigm for state variables Future value = present value + change Change = future value – present value –Difference equation

2. World population growth Midyear World Population 1950 – 1995 Year Population Year Population 1950 2,555,898,461 1951 2,593,043,325 1952 2,635,100,581 1953 2,680,437,616 1954 2,728,297,382 1955 2,779,658,083 1956 2,832,536,024 1957 2,888,278,511 1958 2,944,698,513 1959 2,996,946,986 1960 3,038,930,391 1961 3,079,552,761 1962 3,135,560,616 1963 3,204,953,986 1964 3,275,941,217 1965 3,344,855,925 1966 3,414,981,586 1967 3,484,617,262 1968 3,556,354,911 1969 3,630,875,051 1970 3,705,987,692 1971 3,783,442,817 1972 3,860,605,413 1973 3,937,265,173 1974 4,013,371,391 1975 4,087,382,478 1976 4,159,863,245 1977 4,232,579,098 1978 4,305,305,520 1979 4,380,967,472 1980 4,457,593,483 1981 4,534,279,227 1982 4,614,560,605 1983 4,694,861,502 1984 4,774,079,669 1985 4,854,659,097 1986 4,937,099,145 1987 5,022,946,463 1988 5,109,495,298 1989 5,194,326,719 1990 5,281,672,973 1991 5,365,725,046 1992 5,448,141,769 1993 5,529,370,029 1994 5,609,678,819 1995 5,691,012,889 -------------------------

3. World population growth Variables – n --- years to 1950 – P n ---population at n th year Differences:

4. World population growth Models –Linear model –Analytical solution r>0 & b=0 or r=0 & b>0: population grow r<0 & b=0 or r=0 & b<0: population decay r=0 & b=0: no change

5. World population growth –Nonlinear models –Logistic model

6. Discrete models Equilibrium (or steady state) –When Examples –Linear model Equilibrium –Logistic model Equilibrium

7. An example: A car rental company The problem– Consider a car rental company with distributorships in Orlando and Tampa. The company specializes in catering to travel agents who want to arrange tourist activities in both Orlando and Tampa. Consequently, a traveler will rent a car in one city and drop the car off in the second city. Travelers may begin their itinerary in either city. The company is trying to determine how much to charge for this drop-off convenience. Because cars are dropped off in both cities, will a sufficient number of cars end up in each city to satisfy the demand for cars in that city? If not, how many cars must the company transport from Orlando to Tampa or from Tampa to Orlando?

8. An example: A car rental company Assumptions –By analyzing the historical records and we determined that 60% of the cars rented in Orlando are returned to Orlando, whereas the other 40% end up in Tampa. Of the cars rented from the Tampa office, 70% are returned to Tampa, whereas 30% end up in Orlando.

9. An example: A car rental company The model –Variables –The equations

10. An example: A car rental company Equilibrium values: –Examples O=3000 & T=4000 O=300 & T=400 O=6000 & T=8000

11. Typical solution

15. Voting tendencies of the political parties The problem – Consider a three-party system with Republicans, Democrats, and Independents. Assume that in the next election, 75% of those who voted Republican again vote Republican, 5% vote Democrat, and 20% vote Independent. Of those who voted Democrat before, 20% vote Republican, 60% vote Democrat, and 20% vote Independent. Of those who voted Independent, 40% vote Republican, 20% Democrat, and 40% again vote Independent. Assume these tendencies continue from election to election and that no additional voters enter or leave the system.

16. Voting tendencies of the political parties

17. The model –Variables –The equations

18. Voting tendencies of the political parties Equilibrium values: –Examples R=222,221, D=77,777 & I=100,000

19. Typical solution

23. Voting tendencies of the political parties Model refinement –New voters –Different strategies –Advertisement –Attack counter-parties Story of racing

24. Equilibrium & stability Discrete model Equilibrium Stability –Asymptotic stable: there exists s.t. –Stable: for any, there exists s.t.

25. Equilibrium & stability Thm: Suppose is an equilibrium, then –If then it is asymptotic stable –If then it is asymptotic unstable Example –Discrete linear model -2<r<0 --- asymptotic stable r>0 --- asymptotic unstable

26. Equilibrium & stability –Discrete logistic model --- 0< r<1 For equilibrium 0 --- asymptotic unstable For equilibrium M --- asymptotic stable –Another discrete logistic model

27. Discrete logistic model Equilibrium – Bifurcation of nonnegative steady states –0<r<1:, stable, –First bifurcation: r=1 :unstable :stable when 1 3 –Second bifurcation: r=3

28. Discrete logistic model