8/11/2019 Eco 1 Handout

1/2

Some basic equations for population growth:Your textbook uses two different sets of equations to model population growth. One set is built around theper capita rate of increase (r) and the other uses the finite rate (l). To avoid the needless complexity of two

sets of equations, we will simply define r as the net rate of growth of a population per unit time, as described inthis handout.

The increase (I) of a population of N individuals is equal to births minus deaths:

I

=

births

-

deaths

It's often more convenient, however, to determine birth rate(b)and deathrate(d)for apopulation. Such a rate is simply the number of births or deaths over a certain period oftimedivided by the size of the population:

b

=

BirthsN d=

DeathsN

When you take birth rate and death rate into consideration together, it'spossible to combine them to to produce a value that reflects the overall netrate of population growth, (r). Remember: ris a rate!

r

=

b

-

d

Now life gets interesting. Suppose that we know the rate of population growth (r) and we'dlike to figure out how much that population will increase over a period of time. Todetermine the increase to be expected we simply multiply rby the size of the population(N). This gives us a value for the increase (I) of the population per unit time, which yourtextbook describes as !"/!# (change in number per unit time).

!"/!#= I = rN

The population expected at the end of the growth period, of course, is equal to the startingnumber (N) + the increase in number (I). If the period was one generation, we could figureout the population expected at the end of that generation like this:

N1= No(1 + r)Note that N1= the final population, No = the initial population, and r = rate of growth.To determine how much a population will increase over many generations, we simply placean exponent around the (1+r) term:

Nn= No(1 + r)n

Note that n= time (expressed as the number of intervals. Years, generations, whatever)

This is a great equation, and has many uses. For example, let's suppose that a population of250 animals is increasing in size by 6% a year (rate = 0.06). At that growth rate, how manyof them will be around in 10 years?

N10= 250 (1.06)10 = ?Work it out. The population should be 448 animals. How many generations until thepopulation reaches 1 million?

Admittedly, these equations assume that the rate of growth does not change over time, whichis not a realistic situation. Therefore, ecologists have tried to model growth by adjustingtheir predictions to account for the fact that as a population grows there are fewer and fewer

8/11/2019 Eco 1 Handout

2/2

resources available for it to exploit. Therefore, they first try to determine the growth ratethat would prevail under ideal conditions (unlimited resources), which is sometimes calledthe intrinsic growth rate(It's called that because it's presumed that only factors intrinsicto the organism - like how fast the organisms can reproduce - affect this rate) and it'sdenoted by the symbol ro. Therefore, the maximum increase in numbers that we shouldexpect for a population is simply the intrinsic rate times population size:

I=roN

A population would grow at this rate only if it found unlimited resources. However, in allreal situations, resources are limited. Therefore, ecologists assume that each system has alimit to the number of individuals that it can support, and this limit is called the carryingcapacityof the population, K. As a population approaches the carrying capacity,population growth should slow.We can model this effect by multiplying the equation inwhich we used the intrinsic growth rate by an expressionthat approaches zero as the population approaches thecarrying capacity:

I=roNK-N

K

This equation is known as the Logistic Growth Equation, and is widely used to modelpopulation growth and carrying capacity.

Be sure to think about the implications of this equation. For example, are there anycircumstances in which you would expect the increase in a population (I) to be negative?

Think about how the Logistic Growth Equation predicts the form of growth curve seenabove (from Figure 52.6 on p. 1199 of your textbook)

NOTE: This is a revised version of the Population GrowthHandout (posted on 2/11/07).