“331book”2005/8/13page 13C H A P T E R 2Qualitative Modeling withFunctionsIt is often surprising that very simple mathematical modeling ideas can produceresults with added value. Indeed, the solutions may be elegant and provide qualityof understanding that obviates further exploration by more technical or complexmeans. In this chapter we explore a few simple approaches to qualitatively modelingphenomena with well-behaved functions.2.1 MODELING SPECIES PROPAGATIONThis problem concerns the factors that influence the number of species existing onan island. The discussion is adapted from [1].One might speculate that factors affecting the number of species could include• Distance of the island from the mainland• Size of the islandOf course limiting ourselves to these influences has the dual effect of making atractable model that needs to be recognized as omitting many possible factors.The number of species may increase due to new species discovering the islandas a suitable habitat. We will refer to this as the migration rate. Alternatively,species may become extinct due to competition. We will refer to this as the ex-tinction rate. This discussion will be simplified by employing an aggregate total forthe number of species and not attempting to distinguish the nature of each species,i.e., birds versus plants.Now we propose some basic modeling assumptions that appear reasonable.The migration rate of new species decreases as the number of species onthe island increases.The argument for this is straight forward. The more species on an island the smallerthe number of new species there is to migrate. See Figure 2.1 (a) for a qualitativepicture.The extinction rate of species increases as the number of species on theisland increases.Clearly the more species there are the more possibilities there are for species to dieout. See Figure 2.1 (b) for a qualitative picture.If we plot the extinction rate and the migration rate on a single plot weidentify the point of intersection as an equilibrium, i.e., the migration is exactlyoffset by the extinction and the number of species on the island is a constant. We13“331book”2005/8/13page 1414 Chapter 2 Qualitative Modeling with FunctionsNumber of speciesRate of MigrationM(a) Migration curve.Number of speciesRate of ExtinctionE(b) Extinction curveFIGURE 2.1: Qualitative form of the migration and extinction curves.will assume in this discussion that we are considering islands for which the numberof species is roughly constant over time, i.e., they are in a state of equilibrium.Now we consider whether this simple model provides any added value. Inparticular, can it be used to address our questions posed at the outset.First, what is the effect of the distance of the island from the mainland on thenumber of bird species? One can characterize this effect by a shift in the migrationcurve. The further the island is away from the mainland, the less likely a speciesis to successfully migrate. Thus the migration curve is shifted down for far islandsand shifted up for near islands. Presumably, this distance of the island from themainland has no impact on the extinction curve. Thus, by examining the shift inthe equilibrium, we may conclude that the number of species on an island decreasesas the island’s distance from the mainland increases. See Figure 2.2.Note in this model we assume that the time-scales are small enough that newspecies are not developed via evolution. While this may seem reasonable thereis evidence that in some extreme climates, such as those found in the GalapagosIslands, variation may occur over shorter periods. There have been 140 differentspecies of birds2.2 SUPPLY AND DEMANDIn this section we sketch a well-known concept in economics, i.e., supply and de-mand. We shall see that relatively simple laws, when taken together, afford inter-esting insight into the relationship between producers and consumers. Furthermore,we may use this framework to make predictions such as• What is the impact of a tax on the sale price?• What is the impact an increase in employees wages on sales price? Can theowner of the business pass this increase on to the consumer?Law of Supply: An increase in the price of a commodity will result inan increase of the amount supplied.“331book”2005/8/13page 15Section 2.2 Supply and Demand 15Number of speciesEM Near Island Far Island FIGURE 2.2: The effect of distance of the island from the mainland is to shift the migration curve.Consequently the equilibrium solution dictates a smaller number of species will be supported for islandsthat are farther away from the mainland.Law of Demand: If the price of a commodity increases, then the quantitydemanded will decrease.Thus, we may model the supply curve qualitatively by a monotonically in-creasing function. For simplicity we may assume a straight line with positive slope.Analogously, we may model the demand curve qualitatively by a monotonicallydecreasing function, which again we will take as a straight line.A flat demand curve may be interpreted as consumers being very sensitive tothe price of a commodity. If the price goes up just a little, then the quantity indemand goes down significantly. Steep and flat supply and demand curves all havesimilar qualitative interpretations (see the problems).2.2.1 Market EquilibriumGiven a supply curve and a demand curve we may plot them on the same axisand note their point of intersection (q∗, p∗). This point is special for the followingreason:• The seller is willing to supply q∗at the price p∗• The demand is at the price p∗is q∗So both the supplier(s) and the purchaser(s) are happy economically speaking.“331book”2005/8/13page 1616 Chapter 2 Qualitative Modeling with FunctionsSupply CurveQuantityPrice (a) Supply curveDemand CurveQuantityPrice (b) Demand curveFIGURE 2.3: (a) Qualitative form of supply and demand curves.“331book”2005/8/13page 17Section 2.2 Supply and Demand 170 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91(q1,p1)(q2,p1)(q2,p2)(q3,p2)DSo o o o FIGURE 2.4: The cobweb model illustrating a sequence of market adjustments.“331book”2005/8/13page 1818 Chapter 2 Qualitative Modeling with Functions2.2.2 Market AdjustmentOf course, in general markets do not exist in the perfect economic utopia describedabove. We may model the market adjustment as a sequence of points on the demandand supply curves.Based on market research it is estimated that consumers will demand a