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Systems of Discrete Dynamical Systems

In this section, we examine models of systems of DDS. For selected initial conditions, we’ll build numerical solutions to get a sense of long-term behavior of the system. We’ll find the equilibrium values of the systems we study. We’ll then explore starting values near the equilibrium values to see if by starting close to an equilibrium value, the system will:

a. Remain close to the equilibrium value

b. Approach the equilibrium value

c. Move away from the equilibrium value

What happens near the equilibrium values gives great insight into the longterm behavior of the system. We can study the resulting numerical solutions for patterns.

Example 2.13. Location Merchants Choose.

Consider an attempt to revitalize the downtown section of a small city with merchants. There are some merchants downtown, and others in the large city shopping plaza. Suppose historical records showed that 60% of the downtown merchants remain downtown, while 40% move to the shopping plaza. We find that 70% of the plaza merchants want to remain in the plaza, but 30% want to move downtown. Build a model to determine the long-term behavior of these merchants based upon the historical data. See Figure 2.11. There are initially 100 merchants in the plaza and 150 merchants downtown. We seek to find the long-term behavior of this system.

Diagram of Merchant Movement

FIGURE 2.11: Diagram of Merchant Movement

Problem.

Determine the behavior of the merchants over time to see whether the downtown will survive.

Assumptions and Variables.

Let n represent the number of months. We define

D(n) = number of merchants downtown at the end of the nth month P{n) = number of merchants at the plaza at the end of the nth month

We assume that no incentives are given to the merchants for either staying or moving.

The Model.

The number of merchants downtown in any time period is equal to the number of downtown merchants that remain downtown plus the number of plaza merchants that relocate downtown. The same is true for the number of plaza merchants: the number is equal to the number that remain in the plaza plus the number of downtown merchants that move to the plaza. Mathematically, we write the model as the system

with d(0) = 150 and p(0) = 100 merchants, respectively.

Let’s use Maple to explore this system.

> d := proc(n)

option remember;

if n < 1 then 150 else 0.60 • d(n - 1) + 0.30 • p(n - 1) end if: end proc :

p = proc(n)

option remember;

if n < 1 then 100 else 0.40 • d(n - 1) + 0.70 • p(n - 1) end if: end proc :

Numerically, we have:

And now, graphically:

Analytically, we can solve for the equilibrium values. Let X = d(n) and Y = p{n). Then, from the DDS, we have

However, both equations reduce to X = 0.75У.

There are 2 unknowns, so we need a second equation. From the initial conditions, we know that X + Y = 250. Use the equations

to find the equilibrium values

Iterating from near these values, we find the sequences (quickly) tend toward the equilibrium. We conclude the system has stable equilibrium values. Change the initial conditions and see what behavior follows!

Interpretation.

The long-term behavior shows that eventually (without other influences) of the 250 merchants, about 107 merchants will be in the plaza and about 143 will be downtown. We might want to try to attract new businesses to the community by adding incentives for operating either in the downtown business area or in the shopping plaza.

Competitive hunter models involve species vying for the same resources (such as food or living space) in their habitat. The effect of the presence of a second species diminishes the growth rate of the first and vice versa.[1]

Let’s consider a specific example with lake trout and bass in a small lake.

Example 2.14. Competitive Hunter Model[2].

Hugh Ketum owns a small lake in which he stocks fish with the eventual goal of allowing fishing. He decided to stock both bass and lake trout. The Fish and Game Warden tells Hugh that after inspecting his lake for environmental conditions he has a solid base for growth of his fish. In isolation, bass grow at a rate of 20% and trout at a rate of 30% given an abundance of food. The warden tells Hugh that the species’ interaction for the food affects the trout more than the bass. They estimate the interaction term affecting bass is 0.0010 • bass ■ trout, and for trout it is 0.0020 • bass ■ trout. Assume no changes in the habitat occur.

Miodel.

Define the following:

Then

The equilibrium values can be found by substitut ing B(n) = x and T(n) = ■y, then solving for x and y. We have

We rewrite these equations as

The solution is (x = 0 or у = 200) and {y = 0 or x = 150) which gives the equilibrium values as (0,0) and (150,200).

Our next task is to investigate the long-term behavior of the system and the stability of the equilibrium points.

Hugh initially considered stocking 151 bass and 199 trout in his lake. The solution is left to the student as an exercise. (Don't forget to use option remember in your programs for bass and trout.) From Hugh’s initial conditions, bass will grow without bound and trout will die out (T(29) = 0). This is certainly not what Hugh had in mind.

Example 2.15. Fast Food Tendencies.

The Student Union of a university that has 14,000 students plans to have fast food chains available serving burgers, tacos, and pizza. The chains commissioned a survey of students finding the following information concerning lunch: 75% of those who ate burgers will eat burgers again at the next lunch, 5% will eat tacos next, and 20% will eat pizza next. Of those who ate tacos last, 20% will eat burgers next, 60% will stay will tacos, and 35% will switch to pizza. Of those who ate pizza, 40% will eat burgers next, 20% tacos, and 40% will stay with pizza. See Figure 2.12.

Diagram of Fast Food Movement

FIGURE 2.12: Diagram of Fast Food Movement

Model.

We formulate the problem as follows. Let n represent the nth day’s lunch, and define

Using the values in the problem and the diagram leads us to the discrete dynamical system

The same analytic technique that we used in the bass-lake trout example lets us find any equilibria for our fast food problem. Substitute В = x, T = y, and P = г in the DDS, then solve. Thus

Since the university has 14,000 students, then x + у + z = 14000. Substitute the result above into this equation.

The equilibrium value is then (B,T,P) = (7777.8,2722.2,3500).

The campus has 14,000 students who eat lunch. The graphical results in Figure 2.13 also show that an equilibrium value is reached at a value of about 7778 burger eaters, 2722 taco eaters, and 3500 pizza eaters. This information allows the fast food establishments to plan for a projected future. By varying the initial conditions, the initial numbers of who eats where, for 14,000 students we find that these are stable equilibrium values. (Do this!)

Graphical Results of Burgers, Tacos, and Pizza

FIGURE 2.13: Graphical Results of Burgers, Tacos, and Pizza

Note the equilibrium value 7778 does not indicate that the same people always eat burgers, etc., but rather that the same number of people choose burgers, etc.

Exercises

Iterate and graph the following DDSs. Explain their long-term behavior. For each DDS, find a realistic scenario that it might explain/model.

1. What happens to the merchant problem if 200 were initially in the shopping plaza and 50 were in the downtown portion?

  • 2. Determine the equilibrium values of the bass and lake trout. Can these levels ever be achieved and maintained? Explain.
  • 3. Test the fast food models with different starting conditions summing to 14.000 students. What happens? Obtain a graphical output and analyze the graph in terms of long-term behavior.

Projects

Project 2.1. Small Birds and Osprey Hawks.

Problem.

Predict the number of small birds and osprey hawks in the same environment as a function of time. Osprey hawks will eat small birds when fish aren’t readily available along the coast. The coefficients given below are assumed to be accurate.

Variables.

Miodel.

  • (a) Find the equilibrium values of the system.
  • (b) Iterate the system from the initial conditions given in Table 2.5 and determine what happens to the hawks and small birds in the long term.

TABLE 2.5: Initial Conditions for Small Birds and Hawks

Small Birds

Hawks

150

200

151

199

149

210

10

10

100

100

Project 2.2. Winning at Racket-Ball.

Rickey and Grant play racket-ball very often and are very competitive. Their racket-ball match consists of two games. When Rickey wins the first game, he wins the second game 75% of the time. When Grant wins the first game, he wins the second only 55% of the time. Diagram the ‘movement.’ Model this situation as a DDS and determine the long-term percentages of their racket-ball game wins. What assumptions are necessary?

Project 2.3. Voter Distribution.

It is getting close to election day. The influence of the new Independent Party is of concern to both the Republicans and Democrats. Assume that in the next election that 79% of those who vote Republican vote Republican again, 1% vote Democratic, and 20% vote Independent. Of those that voted Democratic before, 5% vote Republican, 70% vote Democratic again, and 20% vote Independent. Of those who previously voted Independent, 35% will vote Republican, 20% will vote Democratic, and 45% will vote Independent again.

  • (a) Formulate the discrete dynamical system that models this situation.
  • (b) Assume that there are 399,998 voters initially in the system. How many will vote Republican, Democratic, and Independent in the long run? (Hint: you can break down the 399,998 voters in any manner that you desire as initial conditions.)
  • (c) New Scenario. In addition to the above, the community is growing:
    • 18-year-olds-|-new people moving in—deaths—current people moving out.

Republicans predict a gain of 2,000 voters between elections. Democrats also estimate a gain of 2,000 voters between elections. The Independents estimate a gain of 1,000 voters between elections. If this rate of growth continues, what will be the long-term distribution of the voters?

  • [1] See “Bass are bad news for lake trout” from the online version of nature, Internationaljournal of science [Lawrencel999],
  • [2] nThis example comes from a problem developed by Dr. Rich West, Professor Emeritus,Francis Marion University.
 
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