Crystal Ball Math: Predicting Population Growth with Models
Abstract
When is an animal considered endangered? When will it become extinct? What happens when a population has reached the limits of its resources? What happens to a population when a habitat changes? These are some of the questions population biologists try to answer. They use population models, created from math equations, to predict what will happen to a population over time. If you want to see how modeling is done and make some predictions of your own, you certainly won't need a crystal ball—just try this environmental science fair project!Objective
To learn some of the ways in which animal population growth is modeled, and then use the logistic model to determine how a population grows when it is far below, at, or far above its carrying capacity.
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Introduction
Animal life is all around you, in beauty and abundance. Tiny microbes thrive on your healthy, normal skin; jellyfish float like lacy parachutes through the seas; redfaced turkey vultures cruise the skies, barely flapping at all as they scavenge for carcasses; and beadyeyed rodents scurry nervously through the fields. Such differentlooking creatures, but they all have similar biological characteristics:
 They do not have rigid cell walls.
 They have a type of cell called eukaryotes.
 Their cells are specialized into different types of tissues, like skin and muscle.
 They try to keep their internal environments within a normal range (they use feedback control for homeostasis).
 They eat plants or animals to get the carbon and energy needed to survive (they are heterotrophs).
 Most of them move.
 All of them reproduce, usually with two sets of genetic material inside each of their cells.
Figure 1. There are many types of animals, but they have similar biological characteristics. (Turkey vulture, Callie Bowdish, 2006; Rat, Kristin Kearns, 2009).
All animals also interact with their external environments, and their ability to grow and increase in numbers is limited by the amount of resources that are available to them. Resources include things they use to survive, things they eat and drink, and their habitat. Natural predators also influence an animal's numbers, or population. If an environment changes, due to human or natural factors, an animal's habitat and numbers of natural predators can be affected, and an animal's population can increase or decrease. An oil spill is a wellknown example of how a human factor can impact the environment, affect a habitat, and change the populations of certain species. Another example in North America is the deer population. Many natural predators of deer—like wolves, mountain lions, and coyotes—have smaller populations than they did centuries ago, due to city growth and the resulting loss of predator habitats. Consequently, the deer population has increased.
Biologists sample and model populations to see if animals are present in healthy numbers, and if those numbers will remain healthy in the future. Modeling is a way of using math equations to predict what will happen to a population over time. The first and simplest model was developed in the late 1700's by Thomas Malthus. He noted that with unlimited resources, most populations will grow exponentially.
Equation 1:
P(t) =  P_{0}e^{rt} 
 P(t) is the population as a function of time, represented by a number.
 P_{0} is the initial population, represented by a number.
 e is Euler's number, which is approximately 2.718.
 r is the growth rate.
 t is time, often given in years.
For example, if you start with an initial population of 25 individuals, you can see in Figure 2 what will happen to the population in later years, at growth rates of 1 and 2 percent, if the animals have unlimited resources (unlimited habitat, unlimited food, and no predators). You can see from the curves that a 1 percent growth rate results in a doubling of the population about every 70 years. A 2 percent growth rates results in a doubling about every 35 years.
Figure 2. These graphs show how a population will grow, according to the simple exponential model, over a 100year time span, starting with an initial population of 25, and with growth rates of 1 and 2 percent.
If you look at even faster growth rates, like 3, 4, and 5 percent, you can see in Figure 3 how quickly the population grows, but the shape of all the curves is the same. These exponential Jcurves have been said to hold true over the short term. However, in the real world, resources are generally limited, so over the long term, the Malthus model, which assumes unlimited resources, does not hold true.
Figure 3. These graphs show how a population will grow, according to the simple exponential model (the Malthus model), over a 100year time span, starting with an initial population of 25, and with growth rates of 3, 4, and 5 percent. Note that as time increases, all the curves begin to take on a "J" shape.
In the 1800's PierreFrançois Verhulst tried to correct the model. He refined Malthus' model to include information about the carrying capacity, which is the size of a population that a habitat can support. The carrying capacity is the level at which the birth rate matches the death rate, resulting in a constant population over time. It is affected by factors such as food, number of predators, and competition for resources. Verhulst's model is given below.
Equation 2:
P(t) = 

 P(t) is the population as a function of time, represented by a number.
 P_{0} is the initial population, represented by a number.
 e is Euler's number, which is approximately 2.718.
 r is the growth rate.
 K is the carrying capacity.
 t is time, often given in years.
You can compare Equations 1 and 2 to see the different terms. Verhulst modified the Malthus model to show that if the starting population is well below the carrying capacity, K, the population will initially grow rapidly (like the Jcurve), but the growth will slow as the population reaches its carrying capacity. This logistic model produces a population curve with an "S" shape, and so is called an Scurve. The model also predicts a different kind of growth behavior if the starting population is well above the carrying capacity. In this environmental science fair project, you will investigate what happens to a population when it is far below, at, or far above the carrying capacity of its habitat.
Terms and Concepts
 Resource
 Predator
 Population
 Sample
 Model
 Exponential model
 Jcurve
 Carrying capacity
 Logistic model
 Scurve
 Difference equation
 Differential equation
 Variable
 Initial
 For loop
 Static
 Timevarying
 LotkaVolterra model
 Invasive species
Questions
 What characteristics are the same in all animals?
 What factors affect an animal's population?
 Why is modeling useful?
 Based on your research, what is a Malthusian catastrophe?
 How does the Verhulst model differ from the Malthus model?
Bibliography
This source describes the characteristics of all animals:
 Klappenbach, L. (2009). What Makes an Organism an Animal? Retrieved March 17, 2009, from http://animals.about.com/od/zoologybasics/a/animalfacts.htm
This source describes what makes populations grow and regulate themselves:
 Klappenbach, L. (2009). An Introduction to Population Biology. Retrieved March 17, 2009, from http://animals.about.com/cs/zoology/a/zoo101ae.htm
This source describes the simple, exponential (Malthus) model:
 Wikipedia Contributors. (2009, March 15). Exponential growth. Wikipedia: The Free Encyclopedia. Retrieved March 17, 2009, from http://en.wikipedia.org/w/index.php?title=Exponential_growth&oldid=277508496
This source describes the logistic model:
 Wikipedia Contributors. (2009, March 6). Logistic function. Wikipedia: The Free Encyclopedia. Retrieved March 17, 2009, from http://en.wikipedia.org/w/index.php?title=Logistic_function&oldid=275407910
This source discusses many types of population models:
 Bulaevsky, J. (2000, November 13). Interesting Facts About Population Growth Mathematical Models. Retrieved March 17, 2009, from http://www.arcytech.org/java/population/facts_math.html
This source describes difference equations:
 Teknomo, K. (2006). What is a Difference Equation? Retrieved March 17, 2009 from, http://people.revoledu.com/kardi/tutorial/DifferenceEquation/WhatIsDifferenceEquation.htm
This source discusses carrying capacity:
 (2003). Carrying Capacity: How Deep is the Barrel? Retrieved April 8, 2009 from, http://www.michigandnr.com/publications/pdfs/wildlife/viewingguide/eco_carryingcapacity.htm
This source defines the human population growth rates of different countries:
 Wikipedia Contributors. (2009, February 23). List of countries by population growth rate. Wikipedia: The Free Encyclopedia. Retrieved April 8, 2009 from, http://en.wikipedia.org/w/index.php?title=List_of_countries_by_population_growth_rate&oldid=272773977
This source shows you how population growth models can be used to predict the populations of elephants in Kruger National Park:
 University of Wisconsin Board of Regents. (2003). Topic 3: Elephant Population Growth in a Nature Reserve. Retrieved April 8, 2009 from, http://ats.doit.wisc.edu/Biology/ec/pd/t3.htm
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Materials and Equipment
Note: A programming tool, like a programmable calculator or a computer, will make the calculations in this experiment faster, but it is possible to do the experimental procedure with a simple calculator, or even pen and paper.
 Optional: Programmable calculator or computer with programmable software, like the student version of MATLAB®; available online (such as from www.mathworks.com/academia/student_version, or free to your teacher as an evaluation copy. For a student, the correct "Buy Now" link is in the right margin. You will need to prove your student status).
 Lab notebook
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Experimental Procedure
To experiment with the logistic model on a programmable calculator, computer, or in a table, you need to first convert the model into a difference equation. Equation 2, in the Introduction, is the solution to a firstorder differential equation, and when that differential equation is converted to a difference equation, you get:
Equation 3:
Population[time+1] =  Population[time] + GrowthRate × (  1  Population[time] CarryingCapacity 
) × Population[time] 
 Population[time+1] is the population at the next time increment.
 Population[time] is the population at the current time increment.
 GrowthRate is the growth rate.
 CarryingCapacity is the carrying capacity.
 time is often given in years.
This difference equation is a good approximation to what is going on in the differential equation, and is in a form that can be readily implemented on a programmable tool, like a computer or calculator.
Implementing Equation 3
You can implement Equation 3 with a programmable tool or in a table.
If you choose to use a programmable tool, a sample program of how to implement Equation 3 in software, like MATLAB, is given below. Notice that variables—like the growth rate, carrying capacity, initial population, population array, fraction of the carrying capacity array, and number of years to be evaluated—are defined and given initial (starting) values. Note: For your science fair project, you will experiment with changes to the initial values for the growth rate, carrying capacity, initial population, and number of years to be evaluated.
Following the initialization of the variables, a for loop is used to pass, in yearly steps, through time. Finally, results are plotted.
cleargrowthrate=0.15;
carryingcapacity=400;
initialpop=30;
population(1)=initialpop;
numberofyears=50;
for time=1:numberofyears,
fractioncc(time)=population(time)/carryingcapacity;
population(time+1)=population(time)+growthrate*(1fractioncc(time))*population(time);
end
fractioncc(time+1)=population(time+1)/carryingcapacity;
figure
plot(0:numberofyears,fractioncc)
figure
plot(0:numberofyears,population)
If you choose to use a table and pen and paper, create a table, so that you can see what the value of each of the variables will be with each time step. You will step through the table, just as the computer steps, in yearly increments, through the "for loop."
Evaluating Equation 3
After implementing Equation 3 on a programmable tool, or in a table, you are ready to begin evaluating Equation 3 and testing the logistic population model.
 Select a growth rate, carrying capacity, and the number of years to evaluate. You can pick any growth rate, carrying capacity, and time span that interests you, or you can do research on an animal that interests you to get its numbers. For example, states in the United States have a Department of Natural Resources website where you can get numbers on the current carrying capacity and growth rates of some species for that state. Remember that the carrying capacity and growth rate (sometimes called the intrinsic population growth rate or lambda) are not static (fixed) numbers. They depend upon the location, and change over time due to changes in habitat, predation, and competition from other species. Here are some representative numbers that you may choose to investigate:
Animal  Carrying capacity  Growth rate 
Gray wolf  300–500 (Wisconsin)  21% 
Moose  840 (Quebec boreal forest)  25% 
Woodland caribou  200 (Quebec boreal forest)  24.5% 
Elephant  7,500 (Kruger National Park)  15% 

Select three cases for an initial (starting) population:
 Case 1: Make the initial population much less than the carrying capacity.
 Case 2: Make the initial population equal to the carrying capacity.
 Case 3: Make the initial population much more than the carrying capacity.
 Make two line graphs for each case. First, plot the fraction of the carrying capacity on the yaxis and the time (in years) on the xaxis. Second, plot the population on the yaxis, and the time (in years) on the xaxis.
 If desired, repeat steps 1–3 for a different growth rate and carrying capacity.
Analyzing Your Graphs
 What was the shape of the curves for each of the cases? Sshape? Jshape? Backwards Jshape? In which case was there rapid growth in the population initially (like the Malthus model)? In which case was there rapid decline? Did the shape of the "fraction of the carrying capacity" curves match the population curves? What do you think would happen if you plotted the fraction of the carrying capacity on the xaxis and population on the yaxis? If you looked at different growth rates, did changing the growth rate change the shape of the curves?
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If you like this project, you might enjoy exploring these related careers:
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 In the logistic model given above, the carrying capacity is static—it does not change with time. Investigate what happens when the carrying capacity is timevarying. A timevarying carrying capacity can model what happens seasonally to populations (due to weather changes).
 The LotkaVolterra equations are a pair of differential equations used to model interactions between two species, like a predator and prey. Implement the predatorprey equations in differenceequation form on a programmable tool and investigate what happens to two populations simultaneously. Do the peaks and valleys in the populations occur at the same time? Or are they shifted apart from each other in time? If so, by how much are they shifted? Use classic population studies of wolves and moose, done at places like Isle Royale National Park in Lake Superior, to help you in your experiment.
 Invasive species, such as zebra mussels in the Great Lakes, have caused extensive ecological damage. Investigate, through a model, how an invasive species grows.
 Investigate, through a model, how habitat loss affects the polar bear population.
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