We have discussed how population ecologists have tried to develop a model (the logistic growth model) that helps them to understand the factors that affect population growth.
We talked a lot about the graph plotting how the population size would vary over time in a population that started much smaller than the carrying capacity (the s-curve). Why does logistic growth show this pattern?
Initially, the population is growing slowly. When populations are small the per capita growth rate is large but because there are only a few individuals in the population rN is small. Over time, the population growth rate increases becasue populations are still small enough that r is still relatively large and now a larger N allows rN to be a bigger number. Population growth rate starts to slow as populations reach their carrying capacity because in large populations the per capitat growth rate is small and even though N is large rN is small. When the population reaches its carrying capacity b = d, so population growth stops.
Density Dependent Population Regulation
We notice that populations don't keep increasing in size forever. That is because populations are naturally self regulating. As population size increases the per capita birth rate declines for the biological reasons that we discused earlier. (When a parameter decreases as population size increases that parameter is said to be negatively density dependent. As population size increases the per capitat death rates increase for the biological reasons that we discussed earlier. (when a parameter increases as the population size increases that parameter is said to be positively density dependent). Thus, the per capita birth and death rates are naturally density dependent in such a way that eventually causes the population size of species to stop growing.
Here is the link to my Smart Board Notes for this topic last year.
http://www.slideshare.net/MarkMcGinley/smart-board-notes-622
Monday, June 18, 2012
Logistic Growth
We are trying to develop a mathematical model that helps us to understand patterns of population growth. So far our first attempt, the exponential growth model, did not help us to understand population growth (for reasons that I hope that you understand by now).
The "Real" world
In our attemtp to think about population growth in the real world, we attempted to examine how per capitat birth rates and per capitat death rates should vary as population size varies. The model that describes this pattern of growth is known as the logistic growth model. It is important to realize that although this model is much more realistic, and therefore useful to us, than the exponential growth model, the logistic growth model still only exmaines what I call "the theoretical real world". That is, this model applies to our ideas about how populations should generally behave and do not thus relate directly to studying the population sizes of white tailed deer in central Texas or parrot fish on a coral reef in Fiji. These real world situations are much harder to understand than the simple "idealized" populations that I am talking about in BIOL 1404. You can take an Advanced Population Biology course if you want to learn more about how to apply these models to the "real real world".
Logistic Growth
We have discussed why, in the real world, r should decrease as population sizes increase. If this is the case then there is a population size at which the per capita birth rate equals the per capita death rate. We call this population size the carrying capacity.
1) When populations are smaller than the carrying capacity we expect them to increase in size until they reach the carrying capacity.
2) When populations are larger than carrying capacity we espect them to decrease in size untile they reach the carrying capacity.
Fun With Graphs- Exponential Growth
How do I know which graph to draw?
1) In the population ecology portion of this course we will be discussing two models of population growth- exponential growth and logistic growth. Thus, you need to know which growth model you are describing before you know which graph to draw.
2) You can't draw a graph until you know what the axes are.
Hopefully, this is a review, but it is probably worth talking about. The x-axis (the horizontal axis) is known as the independent variable. The y-axis (the vertical axis) is the dependent variable. Changing the value of the independent variable results in a change in the dependent variable. Id DOES matter which variable goes on which axis so try to get it right.
In population ecology there will be two main independent variables that we are interested in studying. Because we are interested in patterns of population growth, we will often want to observe how variables change over time. Time is always the independent variable, so it always goes on the x-axis. Sometimes we are interested in how parameters depend on population size. In this case, population size is always the independent variable.
Powerpoint Presentation
This powerpoint presentation "Fun With Graphs: Exponential Growth" reviews the graphs you are expected to be able to draw, understand, and interpret.
http://www.slideshare.net/secret/mavlOD8flFs67G
Exponential Growth
From the first lesson on Population Ecology we learned that the population growth rate (dN/dt) can be calculated as the product of the per capita growth rate (r) and the population size (N).
dN/dt = rN
This is the fundamental equation describing population growth and this equation is always true.
If we want to use this equation to analyze how population sizes change over time, then it makes sense to start by examining the simplest formulation of this equation which occurs when the per capita growth rate is constant. The equation dN/dt = rN when r is constant is known as the exponential growth equation and this equation describes a patter on growth known as exponential growth.
The graph plotting how population size changes over time is shown in the Exponential Growth article. This graph shows an exponential growth curve (sometimes known as the "j-curve"). If you have questions about why the graph has this shape let me know and I will try to explain it more thoroughly.
It is important that you are able to look at this graph and determine all of the information held in the graph. The exponential growth curve allows us to discuss how two parameters change over time- 1) the population size (shown by the x-axis) and 2) the population growth rate (shown by the slope of the line). I find that it is easier to discuss only one parameter at a time so let's start with the population size.
1) Over time, the population size increases (we know this because the line has a positive slope).
Now let's think about the population growth rate.
2) Over time, the population growth rate increases (we know this becasue the line gets steeper over time.
3) Over time, the rate at which the population growth rate increases over time, increases over time (we know this because the slope increases faster and faster over time).
Thus, if populations are growing exponentially then they keep increasing in size at an ever faster rate forever and ever.
Now try this-
Can you draw the following graphs?
1) plot how the population growth rate varies over time.
(hint- we have alredy described what this pattern will look like using words- just turn these words into pictures).
2) plot how the population growth rate depends on population size.
(hint- this graph is a little trickier, but we do have an equation that relates the two variables)
3) plot how the per capita growth rate varies over time.
(hint- think about what the basic assumption we made aboiut exponential growth)
4) plot how the per capita growth rate
(see the hint from number 3)
Exponential Growth is Unrealistic
Because population sizes keep increasing at ever faster rates for ever, exponential growth does not seem to be an accurate description of population growth in most animals, plants, and microbes. If this is an unrealistic model then why did I teach it to you? I started with exponential growth becasue it is the simplest model of population growth and scientists always like to describe the world using the simplest models that they can.
Obviously, in this case we have started with a model that is too simple to realistically describe the world. What is wrong with the exponential growth model? The fundamental assumption we made about exponential growth is that the per capita growth rate is constant. This must not be a realistic assumtpion.
It is important that you understand, and are able to explain, both the mathematical reasons and biological reasons that exponential growth is an unreasonable model of population growth. I tried to explain biologically why exponential growth is unrealistic in the "Exponential Growth" article and the attached Powerpoint presentation so take a look at those.
Suggested Readings
Here are some articles you should look at from the Encyclopedia of the Earth. I wrote these so they are brilliant!!!
Population Ecology http://www.eoearth.org/article/Population_ecology
Exponential Growth http://www.eoearth.org/article/Exponential_growth
Logistic Growth http://www.eoearth.org/article/Logistic_growth
Carrying Capacity http://www.eoearth.org/article/Carrying_capacity
Intraspecific Competition http://www.eoearth.org/article/Intraspecific_competition
Powerpoint Presentation
Click here for the Powerpoint presentation "Why is Exponential Growth Unrealistic?"
http://www.slideshare.net/secret/IDPugQtl2wvONv
Expected Learning Outcomes
By the end of this course a fully engaged student should be able to
- draw and interpret the following graphs associate with exponential growth
a) how population size change over time in exponential growth
b) how population growth rate varies over time in exponential growth
c) how the population growth rate depends on the population size
d) how per capita growth rate changes over time in exponential growth
e) how per capita growth rate depends on population size
- explain why exponential growth is an unrealistic pattern of growth for most species
- define and explain the carrying capacity
Population Biology: Basic Parameters
Here is a brief introduction to some of the important parameters that we will need to understand to be able to study population ecology. For each of the parameters it is important that you know (1) the name of the parameter, (2) the algebraic symbol used to represent the parameter, (3) the units of measurement for the parameter, (4) how to calculate the parameter, and (5) how to describe (in words) what a particular value of that parameter means.
It is probably easiest for me to introduce these concepts using an example.
Imagine that in a population of 100 elephants that in one year 10 elephants are born and 5 elephants die.
1) Population Size (N) units- individuals. Measures the number of individuals in a population.
N = 100 individuals
In this population, there are 100 elephants.
2) Population Birth Rate (B) units- number of births per time. Measures the number of births per time that occur in a population.
B = 10 births/year
In this population, each year there are 10 births.
3) Population Death Rate (D) units- number of deaths per time. Measures the number of deaths per time that occur in a population.
D = 5 deaths/year
In this population, each year there are 5 deaths.
4) Population Growth Rate (dN/dt) units- number of idividuals per time. Measures the rate of change of the population size.
dN/dt = B - D
dN/dt = 10 births/year - 5 deaths/year = 5 individuals/year
In this population, the population size increases by 5 individuals each year.
5) Per Capita Birth Rate (b) units- births per time per individual. Measures the number of births per time averaged across all members of the population.
b = B/N
b = (10 births/year)/100 individuals = 0.10 births/year/individual
In this population, each year 0.10 babies are born for each individual in the population.
6) Per Capita Death Rate (d) units - deaths per time per individual. Measures the number of deaths per time averaged across all members of the population.
d = D/N
d = (5 deaths/year)/100 individuals = 0.05 deaths/year/individual
In this population, each year 0.005 individuals die for each individual in the population.
7) Per Capita Growth Rate (r) units = individuals/time/individual. Measure the rate of change in population size averaged across all individuals. The per capita growth rate can be calcuated two ways.
a) r = b - d
r = 0.10 births/year/individual - 0.05 deaths/year/individual = 0.05 ind/year/ind
b) r = (dN/dt)/N
r = (5 individuals/year)/100 individuals = 0.05 individuals/year/individual
In this population, each year 0.05 individuals are added for each individual in the population.
Practice Problem
In a population of 50 tigers, in one year 10 tigers are born and 20 tigers die. What is B, D, dN/dt, b, d, r?
Here is the link to the Smart Board Notes.
http://www.slideshare.net/MarkMcGinley/smart-board-notes-620
Wednesday, June 13, 2012
Analyzing Data Part 3- Linear Regression and Chi Square Test
Chapter 5. Correlations between quantifiable variables.
Expected Learning Outcomes
By the end of this course a fully engaged student should be able to
1) determine when you need to use regression to help you test your hypothesis
2) use Excel to conduct regression analysis
3) interpret the outcome of this test to correctly draw conclusions
Chapter 6. Associations Between Categorical Variables
Expected Learning Outcomes
By the end of this course a fully engaged student should be able to
1) determine when you need to use a chi square test of association or a chi square goodness of fit test to test your hypotheses
2) calculate the expected values in the chi square test of association or chi square goodness of fit test
3) use Excel to conduct the Chi Squre test
4) interpret the results to correctly draw conclusions
Appendix 1.
I added Appendix 1 to the lab manual for students that might have been a little bit less math phobic and who might actually benefit from understanding a bit more about how the math of the statistical tests work. Because you are all comfortable with math, I hope you take a look at this short section.
Process of Science- Final Thoughts
After you have finished reading the entire book, don't forget that the three most important summaries are found on pages 30 - 32. Enjoy.
Here is a link to the Smart Board Notes from last year.
http://www.slideshare.net/MarkMcGinley/smart-board-notes-617
Analyzing Data Part 2- Comparing Means
The link below contains the Smart Board notes from last year when we were covering this topic. I hope these notes are helpful.
http://www.slideshare.net/MarkMcGinley/lecture-statistical-tests
Chapter 4. Comparing Means
Expected Learning Outcomes
By the end of this course a fully engaged student should be able to
1) determine when you need to use a t-test to help you test your hypothesis
2) determine which type of t-ttest you should use (i.e., one-tailed vs two-tailed test, paired vs unpaired test)
3) use Excel to perform a t-test on the computer
4) use the output of the statistical tests to correctly draw conclusions
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