1 Excursions in Modern Mathematics Sixth Edition Peter Tannenbaum

2 Chapter 10 The Mathematics of Population Growth There is Strength in Numbers

3 The Mathematics of Population Growth Outline/learning Objectives To understand how a transition rule models population growth. To recognize linear, exponential, and logistic growth models. To apply linear, exponential, and logistic growth models to solve population growth.

4 The Mathematics of Population Growth Outline/learning Objectives To differentiate between recursive and explicit models of population growth. To apply the general compounding formula to answer financial questions. To state and apply the arithmetic and geometric sum formulas in their appropriate contexts.

5 The Mathematics of Population Growth 10.1 The Dynamics of Population Growth

6 The Mathematics of Population Growth The growth of a population is a dynamical process, meaning that it represents a situation that changes over time. Mathematician distinguish between two kinds of population growth: continuous and discrete.

7 The Mathematics of Population Growth Continuous Growth The dynamics of change are in effect all the time—every hour, every minute, every second, there is change. Discrete Growth The most common and natural way by which populations change. We can think of it as a stop-and-go type of situation.

8 The Mathematics of Population Growth Transition For a while nothing happens, then there is a sudden change in the population. Transition rules In the case of discrete growth models we deal with questions by finding the rules that govern the transitions.

9 The Mathematics of Population Growth The ebb and flow of a particular population over time can be conveniently thought of as a list of numbers called the population sequence.

10 The Mathematics of Population Growth Every population sequence starts with an initial population P 0 (called the seed of the population sequence) and continues with P 1, P 2, and so on,

11 The Mathematics of Population Growth P N is the size of the population in the Nth generation. The reason a population sequence starts with P 0 rather than P 1 is convenience.

12 The Mathematics of Population Growth P 1 denotes the population in the first generation, P 2 the population in the second generation, and so on. This is how a sequence is generated.

13 The Mathematics of Population Growth A very convenient way to describe the population sequence is by means of a time-series graph. In a time- series graph the horizontal axis usually represents time and the vertical axis usually represents the size of the population.

14 The Mathematics of Population Growth A time-series graph can consist of just marks (such as dots) indicating the population size at each generation or of dots joined by lines, which sometimes helps the visual effect: (a) scatter plot and (b) line graph.

15 The Mathematics of Population Growth 10.2 The Linear Growth Model

16 The Mathematics of Population Growth The linear growth model is the simplest of al models of population growth. In this model, in each generation the population increases (decreases) by a fixed amount called the common difference.

17 The Mathematics of Population Growth The line graph in (a) illustrates the growth of the garbage in the dump over the five years in question, and, by the way, so does the line graph in (b).

18 The Mathematics of Population Growth These two line graphs illustrate why linear growth is called linear growth– no matter how we scale it, stretch it, or slide it, those “red dots” will always line up.

19 The Mathematics of Population Growth The Arithmetic Sequence The numerical description of a population growing according to a linear growth model. The Common Difference Denoted by d to represent the difference between two consecutive values of the arithmetic sequence.

20 The Mathematics of Population Growth Linear Growth P N = P N-1 + d (recursive formula) P N = P0 P0 + N * d (explicit formula) The linear growth model with seed P0 P0 and common difference d is described by either of the above two formulas.

21 The Mathematics of Population Growth The above figure illustrates the argument why PN PN = P0 P0 + N + d: To get to PN PN the starting population P0 P0 goes through N transitions, each of which consists of adding d.d.

22 The Mathematics of Population Growth Arithmetic Sum Formula Given a population that grows acccording to a linear growth model, we often need to know what is the sum of the first so many terms of the population sequence. We can add up any number of consecutive terms easily with the following formula.

23 The Mathematics of Population Growth 10.3 The Exponential Growth Model

24 The Mathematics of Population Growth Exponential growth is based on the idea of a constant growth rate– in each transition the population changes by a fixed factor called the common ratio.

25 The Mathematics of Population Growth– The Power of Compounding Figure (a) plots the growth of the money in the account for the first eight years. Figure (b) plots the growth of the money in the account for the first 30 years.

26 The Mathematics of Population Growth– The Power of Compounding The key property of exponential growth is a constant rate of growth. This implies that each transition consists of multiplying the size of the population by a constant factor determined by the growth rate but not equal to the growth rate!

27 The Mathematics of Population Growth– The Power of Compounding The Geometric Sequence A sequence defined by repeated multiplication– every term in the sequence after the first is obtained by multiplying the preceding term by a fixed amount r.r. The Common Ratio It is the ratio of two successive terms in the sequence and is denoted by the constant factor r.r.

28 The Mathematics of Population Growth– The Power of Compounding Exponential Growth (r > 0) P N = r · P N-1 (recursive formula) P N = P0 P0 · r N (explicit formula) When r > 1, the terms of the sequence get bigger and we have real growth (positive growth), but when 0 < r < 1, the terms of the sequence get smaller and we have a situation known as exponential decay.

29 The Mathematics of Population Growth– The Power of Compounding Annual Compounding Formula $P N = P0 P0 · (1 + i) N If we let P0 P0 denote the principal; i the annual interest rate (expressed as a decimal); and N the number of years the money is left in the account, the explicit formula for exponential growth becomes the above formula.

30 The Mathematics of Population Growth– The Power of Compounding General Compounding Formula The above formula is a general compounding formula for computing the growth of P0 P0 left in an account that pay an annual interest rate i compounded k times a year.

31 The Mathematics of Population Growth– The Power of Compounding Geometric Sum Formula To simplify the notation we are using a instead of P0 P0 for the seed. The geometric sum formula works for all values of the common ratio r except for r = 1.

32 The Mathematics of Population Growth– The Power of Compounding Periodic Interest Rate The effective interest rate for the compounding period. Annual Yield The percentage increase of an investment over a one-year period.

33 The Mathematics of Population Growth 10.4 The Logistic Growth Model

34 The Mathematics of Population Growth The key idea in the logistic growth model is that the rate of growth of the population is directly proportional to the amount of “elbow room” available in the population’s habitat. Thus, lots of elbow room means a high growth rate, little elbow room means a low growth rate (possibly less than 1, which means that the population is actually decreasing), and if there is no elbow room at all the population becomes extinct.

35 The Mathematics of Population Growth Habitat Population is confined to a limited environment. Carrying Capacity The total saturation point of a habitat and is denoted by C.

36 The Mathematics of Population Growth Growth Parameter A constant growth rate r. The p-values The lowercase population figures p N represent the fraction (or percentage) of the carrying capacity taken up by the population.

37 The Mathematics of Population Growth The Logistic Equation The values of r must be restricted to be between 0 and 4 because for r bigger than 4 the p- values can fall outside the 0 to 1 range.

38 The Mathematics of Population Growth– A Stable Equilibrium If C = 10,000 fish and r = 2.5, we will use the logistic equation to model the growth of the fish population. You start by seeding the pond with an initial population of 2000 rainbow trout (that is, 20% of the pond’s carrying capacity, or p0 p0 = 0.2)

39 The Mathematics of Population Growth– A Stable Equilibrium After the first year, the population is given by: p1 p1 = 2.5 x (1 – 0.2) x (0.2) = 0.4 The population of the pond has doubled, and things are looking good!

40 The Mathematics of Population Growth– A Stable Equilibrium After the second year, the population is given by: p2 p2 = 2.5 x (1 – 0.4) x (0.4) = 0.6 The population is no longer doubling but the hatchery is still doing well.

41 The Mathematics of Population Growth– A Stable Equilibrium After the third year, the population is given by: p3 p3 = 2.5 x (1 – 0.6) x (0.6) = 0.6 Surprise the same result. Stubbornly, you wait for better luck the next year.

42 The Mathematics of Population Growth– A Stable Equilibrium After the fourth year, the population is given by: p4 p4 = 2.5 x (1 – 0.6) x (0.6) = 0.6 From the second year on, the hatchery is stuck at 60% of the pond capacity– nothing is going to change unless external forces come into play.

43 The Mathematics of Population Growth– A Stable Equilibrium We describe this situation as one where the population is at a stable equilibrium. The above figure shows a line graph of the pond’s fish population for the first four year.

44 The Mathematics of Population Growth- Conclusion In the linear model, the population is described by an arithmetic sequence. In the linear model, the population is described by an arithmetic sequence. In the exponential model, the population is described by a geometric sequence. In the exponential model, the population is described by a geometric sequence. The logistic model of population growth is described by the logistic equation. The logistic model of population growth is described by the logistic equation.