Multiplicative Population Growth Simulation

In Concept 42.1 we spelled out how the size of a biological population will change through time:

The number of individuals in a population at some time in the future =
the number now + the number that are born – the number that die

We turned this word equation into symbols to arrive at Equation 42.1, the most basic mathematical model of population growth:

Nt+1 = Nt + BD

where N is population size, B is the total number of births in the population during the time interval from t to t + 1, and D is the total number of deaths in that interval.

We rearranged this equation to arrive at Equation 42.2:

ΔN / ΔT = BD

where ΔN, the change in population size during some time interval, is divided by ΔT, the change in time represented by that time interval, to yield the rate of change in population size.

Recall that B and D are births and deaths for the entire population, and can be expressed in the form B = b N and D = d N, where b and d are average per capita (per individual) birth and death rates. We can therefore rewrite the rate of change in population size as shown in Equation 42.4:

ΔN / ΔT = r N

These equations are used to describe how population size changes for organisms that reproduce only at certain times, during a restricted breeding season. Births and deaths are calculated from one breeding season to the next one, often a year later. Suppose, however, that the time interval between breeding events, ΔT, approaches zero, as would be the case for organisms (such as humans) that reproduce at any time of the year instead of in a restricted breeding season. Now population size changes continually, and it is appropriate to write a differential equation:

dN / dt = r N

Note that r is now scaled so that it represents the per capita rate of birth minus death at an instant in time.

Finally, we would like to know what the population size Nt will be at some future time t in terms of the initial population size N0 at time “zero.”

For an organism with seasonal breeding, the equation is:

Nt = N0 (1 + r)t

For an organism with continuous growth, the equation is:

Nt = N0 e rt

The value of r in each case is scaled to the appropriate time interval. The Deriving the Growth Equations tab shows how we got these equations.

Let's use the continuous-growth equation, known as exponential growth, to explore the properties of multiplicative population growth.

Property 1: r determines whether a population will grow, shrink, or not change in size.

Recall from Concept 42.4 that if r > 0 the population will increase in size, if r = 0 there will be no change, and if r < 0 it will decrease. To see this property, click the Graph tab. Enter 10 for the starting population size, N0, and 1 for r. Click on New Plot. How does the population size change through time? Now enter a value of 0 for r and click on New Plot. Does the population size change through time? Finally, enter a value of −0.5 for r and click on New Plot. How does the population size change through time? Repeat the exercise for different values of N0. How does changing the value of N0 influence population growth?

Property 2: The time to double is a constant if r does not change.

A second important property of multiplicative growth with constant r is that there is a constant doubling time. To see this property, plot growth of a population with N0 = 10 and r = 1. You can see the time at which N reaches any given value by moving your cursor to the desired location on the curve, then single click. To clear values click outside the graph. The graph shows the value of N and t at that point. Click on the curve where N = 20, twice the starting value. What is the value of t? Now click on the curve where N = 40, and calculate the time it took for the population to go from 20 to 40 by subtracting the first time from the second time. This should be the same time as it took for the population to go from 10 to 20.

The Doubling Time tab provides the mathematical proof of this second property—that there is a constant doubling time when growth is multiplicative and r is constant.

The Conclusion tab summarizes these two properties of multiplicative growth with constant r.

r =     N0 =

Property 1: r determines whether a population will grow, shrink, or not change in size.

As you graphed exponential growth with different values of r, you confirmed that the population grows if r > 0, stays a constant size if r = 0, and shrinks to zero if r < 0. You also saw that changing N0 moved the growth curve up or down, but did not affect whether the population grew, shrank, or stayed at a constant size.

This outcome may make more intuitive sense if you translate the mathematical symbols back into words! Recall that r is the average number of births per individual (b) in an interval of time, minus the average probability of death of each individual (d) in that interval of time. Thus r is the average contribution of each individual (the average per capita contribution) to growth of the population in an interval of time. If this average individual contribution is positive (r > 0) the population grows, if it is zero (r = 0) the population size does not change, and if it is negative (r < 0) the population shrinks.

You also saw for a positive value of r that the population grows at a faster and faster rate, producing a "J" shaped curve of population size through time. Even if you choose a small value of r, the "J" shape appears if you plot growth over a long enough time. In other words, the population rapidly heads toward an infinite size!

This should serve as a graphic reminder that the equation for exponential growth (or its counterpart for multiplicative growth in an organism with seasonal breeding) will not describe the size of real populations forever, because the resources needed for individuals to survive and reproduce are not infinite in the real world. On the other hand, by deriving equations for the size of populations at some future time in terms of the current size, we have gained very useful tools for short-term prediction of future size, and for seeing the intrinsic power of multiplicative growth to quickly generate large numbers.

Property 2: The time to double is a constant if r does not change.

With r = 1, the time it took the population to double from N = 10 to N = 20 was about 0.695 time units. The time it took the population to reach N = 40 was about 1.39, twice 0.695. This means that the time it took the population to double from N = 20 to N = 40 was the same as the time it took the population to double from N = 10 to N = 20. In other words, the population doubles every 0.695 time units. This property is a mathematical consequence of multiplicative growth with constant r, as the Doubling Time tab demonstrates.

Deriving Equations that Predict Future Size of a Population from its Current Size

 Seasonal Breeding N1 = N0 + rN0 = (1 + r) N0 N2 = N1 + rN1 = (1 + r) N1 = (1 + r)2 N0 N3 = N2 + rN2 = (1 + r) N2 = (1 + r)3 N0 In general at some future time that is t breeding seasons later, Nt = ( 1 + r )t N0 Continuous Breeding Rearrange the differential equation dN / dt = r N as: dN / N = r dt Now integrate both sides between the starting time (0) and the future time (t): dN / N = r dt Recall that dx / x = ln x, and that r dx = r x. Also recall that a definite integral is obtained by subtracting the value of the integral at the lower limit from the value of the integral evaluated at the upper limit. Therefore, dN / N = ln Nt - ln N0 = ln (Nt / N0), and r dt = (r × t) - (r × 0) = r t Substituting these expressions, we get: ln (Nt / N0) = r t Exponentiate both sides: Nt / N0 = e r t Express Nt as a function of N0 by multiplying both sides by N0: Nt = N0 e r t

A population that is growing with a constant per capita growth rate, r, will double in size over a certain time period td, and then will double again in each successive time period td. In other words, it will have a constant doubling time. This is true whether breeding occurs continuously, or seasonally.

Continuous Breeding

Consider the equation for continuous breeding from the Deriving the Growth Equations tab:

Nt = N0 e r t

How long does it take for the population to double?

To answer this question, realize that N at the time to double, td, equals twice N at time zero:

2N0 = N0 e r (td)

Dividing both sides by N0,

2N0/N0 = 2 = e r (td)

To solve for td, take the natural logarithm of both sides:

ln 2 = r (td)

Divide both sides by r:

td = ln (2)/ r ≈ 0.69/ r

Notice that td is a constant as long as r is constant. In other words, there is a constant value of the doubling time td. Also notice that we can calculate the doubling time if we know what r is.

Let's give an example. The most recent estimate of r for the global human population, which does not breed seasonally, is 1.14 percent per year, or 0.0114. An estimate of the current doubling time for the human population is therefore:

td ≈ 0.69/r = 0.69/0.0114 = 60.5 years.

Seasonal Breeding

Consider now the equation for seasonal breeding from the Deriving the Growth Equations tab:

Nt = (1+r)t N0

Recall that for seasonal breeding, r is the per capita growth rate per breeding episode, and t is expressed in numbers of breeding episodes. How many breeding seasons does it take for the population to double? To answer this question, again realize that N at the time to double equals twice N at time zero:

2N0 = (1+r) td N0

Again,divide both sides by N0:

2 = (1+r) td

Take the natural logarithm of both sides

ln (2) = td ln (1+r)

Solve for the doubling time, td:

td = ln (2)/ln (1+r) ≈ 0.69 / ln (1+r)

Example: If a population has one breeding season per year and grows by 3% per breeding episode, how long will it take for it to double? Answer:

Textbook Reference: Concept 42.4 Populations Grow Multiplicatively, but the Multiplier Can Change