Exponential Growth and Decay

Chapter 1 Review Problems

2.1 Exponential Growth and Decay

2.1.1 Modeling Population Growth in Discrete Time

t 0

t 20

t 40

Figure 2.1 Bacteria split every 20 units of time.

Imagine that we observe bacteria that divide every 20 minutes and that, at the start of the experiment, there was one

bacterium. How will the number of bacteria change over time? We call the time when we started the observation time 0. At time 0, there is one bacterium. After 20 minutes, the bacterium splits in two, so there are two bacteria at time 20.

Twenty minutes later, each of the bacteria splits again, resulting in four bacteria at time 40, and so on (Figure 2.1).

We can produce a table that describes the growth of this population:

Time (min) 0 20 40 60 80 100 120

Population size 1 2 4 8 16 32 64

We can simplify the description of the growth of the bacterial population if we measure time in more convenient units. We say that one unit of time equals 20 min- utes. Two units of time then corresponds to 40 minutes, three units of time to 60 minutes, and so on. We reproduce the table of population growth with these new units:

Time (20 min) 0 1 2 3 4 5 6

Population size 1 2 4 8 16 32 64

62

The new time units make it easier to write a general formula for the population size at timet. Denoting byN(t)the population size at timet, wheretis now measured in the new units (one unit is equal to 20 minutes), we guess from the second table that N(t)=^2t, t =⁰,¹,², . . . ^(2.1) We encountered this function in Section 1.2 when we discussed exponential functions.

There, the function was defined for allt ≥0, whereas now, the function is defined only for nonnegative integer values. Equation (2.1) allows us to determine the population size at any discrete timetdirectly, without first calculating the population sizes at all previous time steps. For instance, at timet = 5, we find thatN(⁵) = ²⁵ = ^{32, as} shown in the second table, or, at timet = ^10, N(¹⁰) = ²¹⁰ = 1024. The graph of N(t)=^2t is shown in Figure 2.2.

0 1 2 3 4 5 6

t 0

10 20 30 40 50 60 70

N(t)

Population size

N(t)

Figure 2.2 The graph ofN(t)=2^tfort=0,1,2, . . . ,6.

The function N(t) = ^2t,t = ⁰,¹,², . . ., is an exponential function, and we call the type of population growth that it representsexponential growth. The base 2 reflects the fact that the population size doubles every unit of time.

Instead ofN(t), we will often write Nt. The subscript notation is used only for functions N(t)^wheret is a nonnegative integer. So, instead of writing N(t) = ^2t, t =⁰,¹,², . . ., we can writeNt =^2t,t =⁰,¹,², . . .^.

So far, we assumed that N(⁰) = N₀ = 1. Let’s see what N_t looks like if N₀ = 100. Regardless ofN₀, the population size doubles every unit of time. We obtain the following table, where time is again measured in units of 20 minutes:

Time (20 min) 0 1 2 3 4 5 6

Population size 100 200 400 800 1600 3200 6400

We can guess the general form ofN_t withN₀=100 from the table:

N_t =¹⁰⁰·^2t, t =⁰,¹,², . . .

We see that the initial population sizeN₀=100 appears as a multiplicative factor in front of the term 2^t. If we do not want to specify a numerical value for the population sizeN₀at time 0, we can write

N_t = N₀2^t, t =0,1,2, . . .

We already mentioned that the base 2 indicates that the population size doubles every unit of time. Replacing 2 by another number, we can describe other popula- tions. For instance,

Nt =^3t, t =⁰,¹,², . . .

describes a population with N0 = 1 and that triples in size every unit of time. The corresponding table is

Time 0 1 2 3 4

Population size 1 3 9 27 81

Now that we have some experience with exponential growth in discrete time, we give the general formula:

Nt = N0R^t, t =⁰,¹,², . . . ^(2.2) The parameterRis a positive constant called thegrowth constant. The constantN₀is nonnegative and denotes the population size at time 0. The assumptionsR>^{0 and} N₀≥0 are made for biological reasons: Negative values forRorN₀would result in negative population sizes, andR=0 would be uninteresting.

EXAMPLE 1 Suppose a population of cells reproduces every 15 minutes and we measure its size every 30 minutes:

Time (min) 0 30 60 90 120 150 180

Population size 1 4 16 64 256 1024 4096

Write a formula for timen=⁰,¹,², . . . when (a) one unit of time is 30 minutes, (b) one unit of time is 60 minutes, and (c) one unit of time is 15 minutes.

Solution

(a) We see from the values listed in the table that when one unit of time is 30 minutes, the population quadruples every unit of time, withN₀=1. Thus,

Nt =^4t, t =⁰,¹,², . . .

(b) This time, we see from the values in the table that when one unit of time is equal to 60 minutes, the population grows by a factor of 16 each unit of time. Again, N₀=^{1. Hence,}

N_s =^16s, s=⁰,¹,², . . .

We could also have arrived at this answer by noting that the time step in (b) is twice that of the time step in (a). In other words, when one unit of time elapses in (b), two units of time elapse in (a):

t 0 1 2 3 4 5 6

s 0 1 2 3

We find thatt =²s. If we substitute 2sfort in (a), we find that N_t =^{4t yields} N_s =^42s =^16s fors=⁰,¹,², . . .^.

(c) When one unit of time is 15 minutes, and we use the variableu=⁰,¹,², . . . to denote time, it follows thatt =u/^{2 and}

N_t =^{4t yields} N_u =^4u/2=^2u foru=⁰,¹,², . . .^.

The function N_t = N₀R^t,t = ⁰,¹,², . . ., is an exponential function. We dis- cussed exponential functions in the previous chapter. There, we looked at f(x)=a^x, x ∈ R. To make the comparison easier, we choose N₀ = 1 in (2.2) and restrict the function f(x)=a^xtox ≥0. If we choose the same values forRanda, then the two functionsNtand f(x)use the same rule to compute their values. The difference is in the domain:Ntis defined only for nonnegative integers, whereas f(x)is defined for all nonnegative real numbers. The two functions agree where they are both defined.

This can be seen when we graphNtand f(x)in the same coordinate system forR=a (Figure 2.3).

In Chapter 1, we learned how f(x) = a^x,x ∈ R, behaves for different values ofa. We can use this behavior now to describe that ofN_t = N₀R^t,t = 0,1,2, . . .. In Figure 2.4, we show the function f(x) = a^x,x ≥ 0, for different values ofa. Superimposed are the graphs ofN_t =N₀R^t,t =⁰,¹,², . . .^{, for}R=aandN₀=^1.

We see that whenR>1, the population sizeN_tincreases indefinitely; whenR= 1, the population sizeN_t stays the same for allt =⁰,¹,², . . .; and when 0<R<^1, the population size N_t declines and approaches 0 ast increases. The behavior is the same for other positive initial population sizes (N₀ >^0).

16 14 12 10 8 6 4 2 0

0 2 4 6

f(x) or N(t)

x or t 8 10 12

N(t) f(x)

Figure 2.3 The graphs of f(x)=a^x, 0≤x≤^{10, and}N(t)=R^t, t=0,1,2, . . . ,10, whena=R=1.3.

7 6 5 4 3 2 1 0

0 2 4 6

f(x) or N(t)

x or t

8 10 12

a 0.5 a 1 a 1.2

Figure 2.4 The graphs of f(x)=a^x, 0≤x≤^{10, and}N(t)=R^t,t=⁰,¹,², . . . ,^10, for three different values ofa=R:a=R=0.5,a=R=1, anda=R=1.2.

2.1.2 Recursions

When we constructed the tables for the bacterial population size with R = ^{2 at} consecutive time steps, we doubled the population size from time step to time step.

In other words, we computed the population size at timet +1 on the basis of the population size at timet, using the equation

Nt+1=²Nt (2.3)

Equation (2.3) is a rule that is applied repeatedly to go from one time step to the next and is called arecursion. We say that Equation (2.3) defines the population size recursively.

If we want to use Equation (2.3) to find the population size, say, at timet =^{4, we} need to know the population size at some earlier time, say, timet =0. Let’s assume thatN0 =1. Then, applying the recursion (2.3) repeatedly, we find that

N₁=²N₀=² N₂=²N₁=⁴ N3=²N2=⁸ N₄=²N₃=¹⁶

We thus have two equivalent ways to describe this population: Fort =⁰,¹,², . . .^, N_t =2^t is equivalent to N_t+1 =2N_t withN₀=1

The recursion for a general value ofRis

Nt+1 =R Nt with N₀=population size at time 0 (2.4)

Applying (2.4) repeatedly, we obtain N₁= R N₀

N2= R N1 =R²N0

N₃= R N₂ =R³N₀ N₄= R N₃ =R⁴N₀

...

N_t = R N_t−1= R^tN₀ The two descriptions fort =⁰,¹,², . . .^{, namely,}

N_t = N₀R^t and N_t+1 = R N_t withN₀=population size at time 0 are equivalent. We say that N_t = N₀R^t is asolutionof the recursionN_t+1 = R N_t with initial conditionN₀at time 0, since the functionN_t =N₀R^tsatisfies the recursion with initial conditionN(⁰)= N₀.

We can visualize recursions by plottingN_ton the horizontal axis andN_t+1on the vertical axis. The exponential growth recursion

Nt+1= R Nt (2.5)

is then a straight line through the origin with slope R(Figure 2.5). SinceN_t ≥^{0 for} biological reasons, we restrict the graph to the first quadrant.

Nt1

N_t Slope R R

0

0 1

Figure 2.5 The exponential growth recursionN_t+1= R NtwhenR>0.

What does this graph tell us? For any current population sizeNt, it allows us to find the population size in the next time step, namely,Nt+1. For instance, ifR=^{2 and} N0 =1, then successive population sizes are 1,²,⁴,⁸,¹⁶,³², . . .. For this choice of N0, we will never see a population size of, say, 5 or 10. Thus, for a specific choice ofN0, only a selected number of points on the graphN_t+1 = R N_twill be realized (Figure 2.6). A different choice of initial condition would yield a different set of points.

70 60 50 40 30 20 10 0

0 5 10 15

N(t1)

N(t) t 0

t 1 t 2 t 3

t 4

t 5

20 25 30 35

Figure 2.6 Successive population sizes on the graph of the exponential growth recursion whenR=2 fort=0,1,2, . . . ,5.

We also see from Figure 2.6 that unless we label the points according to the correspondingt-value, we would not be able to tell at what time a point(Nt,Nt+1) was realized. We say that time isimplicitin this graph. Compare Figure 2.6 with Figure 2.2, in which we graphed Nt as a function oft for the same values of RandN₀; in Figure 2.2, time isexplicit.

The hallmark of exponential growth is that the ratio of successive population sizes, Nt/N_t+1, is constant. When Nt > 0 (and hence N_t+1 > 0), it follows from N_t+1 = R N_tthat

N_t N_t+1 = ¹

R

If the population consists of annual plants, we can interpret the ratioNt/Nt+1as the parent–offspring ratio. If this ratio is constant, parents produce the same number of

offspring, regardless of the current population density. Such growth is calleddensity independent.

WhenR>1, it follows that 1/R, the parent–offspring ratio, is less than 1, imply- ing that the number of offspring exceeds the number of parents. Density-independent growth withR>1 results in an ever-increasing population size. This model eventu- ally becomes biologically unrealistic, since any population will sooner or later expe- rience food or habitat limitations that will limit its growth. (We will discuss models that include such limitations in Section 2.3.)

The density independence in exponential growth is reflected in a graph ofNt/Nt+1

as a function ofNt, which is a horizontal line at level 1/R(Figure 2.7).

N_t/N_t1

Nt

1/R

Figure 2.7 The graph of the parent–offspring ratio _N^Nt

t+1 as a

function ofNtwhenNt>0. As before, only a selected number of points are realized on the graph ofN_t/N_t+1 as a function ofN_t, and time is implicit in the graph. (See Figure 2.8, withR=2 and N₀=1.)

0.6 0.5 0.4 0.3 0.2 0.1 0

0 5 10 15

N(t)/N(t1)

N(t) t 0

t 1

t 2 t 3 t 4 t 5

20 25 30 35

Figure 2.8 The graph of the parent–offspring ratio_N^Nt

t+1as a function ofNtwhen N_t=1 andR=2.