1 Population Growth Models

1 1 Population Growth ModelsBack to our problem of trying to predict the future, or at least the future Population of somespecies in some region. Two ideas for how to do this come to first is to look at the historical data and see if we can identify trends. This is agreat idea, but is often very difficult. Data seldom fits a simple pattern perfectly and wemust constantly worry about what trends are real in the data, what trends are due totemporary changes in the situation and what trends are created by our human desire to seepatterns, even when there are other idea is to try to make a deterministic model of the Population that is basedon some basic assumptions of how the Population changes. We need a model that is simpleenough to use, but complicated enough to give an at least approximately reliable first idea requires we be able to interpret data from a complicated noisy world so weneed techniques and expertise in statistics (stay tuned!). So for now we follow the secondidea and construct Models based on some (simple) assumptions about biology.

2 Of course,combining both model building and data analysis using the data to motivate and check theassumptions and using the Models to tease out trends in the data is more powerful thaneither technique by of this section as practice building Models for physical world and seeing what kindsof behavior simple Models can predict. The hidden agenda is to use some of the functionswe have seen in our Exponential Growth ModelsAs fits our basic outline, we start by making a simple, abstract model of the Growth of apopulation. We are guided by a principle called Occam s Razor which states that a model (or explanation) should be the simplest possible model that works . That is, we do notwant to add complication unless we must to match , consider a small Population of some species let loose in a large area. For example, inOctober 1859, Thomas Austin released 24 rabbits on his farm in Australia (at least accordingto Wikipedia). This Population eventually grew to over 600 lettrepresent time measured in years.

3 If we were considering a Population of whaleswe might measure time in decades while for bacteria, we would measure time in hours orminutes. We letP(t) be the Population at timet. Again, we might measurePas number ofindividuals (for whales), or thousands or millions of individuals (for rabbits or people). Wecould also letPrepresent a Population density. That is, we could letPbe the number ofrabbits per square kilometer or square meter. So fractional values ofPare the notation suggests, we think ofP(t) as a function oft. Our goal is to be able topredict a value ofP(t) for any timet. We could just write down a guess for a formula ofP(t), but that isn t much more satisfying than just guessing the values ofP(t). Instead, let sthink a (very) little bit about do we know about rabbits? Well, rabbits do what they are famous for and begetmore rabbits. The more rabbits you have this year, the more baby rabbits you will havenext year. So the (very) basic biology of rabbits tells us not what the Population of rabbitsis, but rather, how the Population Models for Population size will be based on rules derived from how the populationchanges.

4 Keeping with Occam s Razor, we start with the simplest aspects of populationchange and make some explicit assumptions about how they work:1. The number of births between timetand timet+ 1 is proportional to the size of thepopulationP(t) at timet. That is, there is a constantb>0 such that the number ofbirths between timetandt+ 1 isb P(t).2. The number of deaths between timetand timet+ 1 is also proportional to the size ofthe size of the populationP(t) at timet. That is, there is a constantd>0 such thatthe number of deaths between timetandt+ 1 isd P(t).Clearly, these are just the most basic assumptions on how any Population might are many factors that effect births and deaths. These include external factors, likethe weather, and factors that depend on where the species is on the food chain. However,we start simple and ignore all other mechanisms of Population we must turn these assumptions into statements that we can use to compute fu-ture populations. While this makes our discussion look more mathy formulas instead ofsentences we emphasize that all we are doing in this step is translating the sentences aboveinto a form we can use for our two assumptions together, we can say that the Population at timet+ 1 isthe Population at timetplus the births betweentandt+ 1 minus the deaths betweentandt+ 1.

5 We can write this on one line asPopulation at timet+ 1 = ( Population at timet) + (birthsttot+ 1) (deathsttot+ 1).Now our assumptions say that the births timettot+ 1 are given bybP(t) while the deathsare given bydP(t). Hence, we can shorten the sentence above with notation, writingP(t+ 1) =P(t) +bP(t) dP(t).This completes our translation of the assumptions into a formula (and it really is only atranslation). We can now use the algebra you learned long ago to simplify things even factoring outP(t) on the right, we getP(t+ 1) = (1 +b d)P(t).We can consolidate a bit more by lettingk= 1 +b dand callingkthe Growth rate constant . Our model can now be written in the very efficientformP(t+ 1) =kP(t).If there are more births than deaths (b>d), thenk>1 and the Population at timet+ 1is larger than the Population at timet. If we know the Population at timet= 0, then attimet= 1 we haveP(1) =kP(0)2and at timet= 2 we haveP(2) =kP(1) =k(kP(0)) =k2P(0)and at timet= 3 we haveP(3) =kP(2) =k(k2P(0)) =k3P(0).

6 You can see the pattern developing here. A proof (by induction) shows the general case thatP(N) =kP(N 1) =k(kN 1P(0)) =kNP(0)and we have a formula for the Population for all future timesN provided we know thepopulation at time type of model is called an exponential Growth Population model because thepopulationP(N) is an exponential function. For example, ifP(0) = 24 andk= 2, that is,the Population starts at 24 at timet= 0 and the Population doubles each year, thenP(34) = 234 24 = 412,316,860,416or the original Population of 24 will grow to over 400 billion in only 34 years. This isremarkably fast Growth (see Fig. 1).01E+112E+113E+114E+115E+116E+117E+118 E+119E+1105101520253035 Figure 1: Exponential Growth withk= 2,P(0) = 24 Note that exponential Growth occurs even whenkis just slightly greater than one. Forexample, ifk= andP(0) = then (see Fig 2)P(N) = ( ).In order to use this model to predict future populations, we need two things. The initialpopulationP(0).

7 This can actually be the Population at any time since we get to decidewhent= 0, that is, we decide when to start the clock. We also need the value 2: Exponential Growth withk= ,P(0) = Predicting world populationSuppose we want to predict the world Population , say starting in the year 1000 and goinginto the future, that is, we lett= 0 be the year 1000 and we are interested inP(t) fort 1001. Since there are a lot of people, we use units of one billion people. So sayingP(0) = means that in the year 1000 there were about 300,000,000 , what value ofkdo we pick? We see right away possible problems using the expo-nential Growth model for this problem. Average life spans and birth rates have changed agreat deal over the past 1000 years, so choosing just one value ofkis a huge , assuming that life span over the past 1000 years has varied from 30 to 80 years,then the chance of dying in a particular year has varied form 1/30 to 1/80. So, lets say that1/50 = of the Population dies each year as a reasonable middle rate is harder to estimate and has fluctuated due to advances in health care andsocial norms.

8 Half the Population is women and each woman spends half to one third of herlife in child bearing years. We make an estimate of about 1/10 of the women in child bearingyears have a child in a given year (this is the biggest guess), then an estimate for birth rateof (1/2) (1/2) 1/10 = 1 , we estimate our Growth rate constant ask= 1 +140 150= 1 + = we get a Population Growth prediction in yearNofP(N) = gives the graph below and the prediction that the Population in the year 2010 (t= 1010)should be about 41 , this is larger than the actual Population of about 6 billion. Our estimate ofkmust be too high. If we takek= then the model predicts a Population in 2010 ofabout billion, which is a lot more 3: Exponential Growth withk= ,P(0) = 4: Exponential Growth withk= ,P(0) = Criticism of the Exponential Growth ModelAs noted above, the use of a constant Growth rate constantkis the most serious assumptionin this model . Obviously, improvements in public health, wars, changes in social attitudescan make a large difference does not mean that the exponential Growth model is useless it just means that wehave to be careful where and how we use it.

9 For the example above, it tells us that formost of the last thousand years, the Growth rate constant must have been very small. Weare forced to re-examine our assumptions about birth and death rates. A small populationin a large environment under fairly constant conditions will probably follow an exponentialgrowth model fairly accurately, at least until the Population becomes too moral is: There is no magic answer and no substitute for careful thought whenbuilding and evaluating Another View of the Exponential Growth modelBefore looking at more generally applicable Population Models , we need to use what we knowabout functions to get a different view of the exponential Growth far we have only drawn graphs of the populationP(t) as a function of timet. Thatis,tis on the horizontal axis andP(t) is on the vertical axis. A completely different way tovisualize this same model is to draw the graph of the relation ship between the populationat timetand the Population at timet+ 1.

10 Iftis in years, then we use the Population thisyear,Pthis year, along the horizontal axis and the Population next year,Pnext year, alongthe vertical axis. Our model states thatPnext year=k Pthis yearThis is the equation of a line through the origin with slopek(see ).024681012012345P(next year) =P(this year) P(next year) =2*P(this year) Figure 5: Graph of the exponential Growth model withPThis yearon the horizontal axisandPnext yearon the vertical axis. Herek= this is just basic graphing of functions, we are actually doing something prettysophisticated. We now have two ways to graph or picture the same model the time series where we graphP(t) versus timetand the graph of the model equation graphingPnext yearagainstPthis year. Each of these graphs can be used to tell us something about what themodel example, the graph ofPthis yearvsPnext yearimmediately tells us the the popu-lation next year will be greater than the Population this year except when the populationis zero.