Fun With Graphs- Quiz Yourself

Here are some questions that I have designed to let you know if you are understanding the graphs well enough to meet the course expected learning outcomes. I suggest that you do not try to answer these questions until you have thoroughly reviewed all of the information about the population ecology graphs. (I will put the answers for the multiple choice questions at the bottom of this post, for the others you need to find out whether your answers are correct or not).
1. What are the correct axes for a graph showing how population growth rate depends on population size in logistic growth?
a) x- N y- t
b) x- N y- dN/dt
c) x- dN/dt y- N
d) x- dN/dt y- t
e) x- N y- r

2. Which of the following best describes the graph that shows how the per capita growth rate varies over time in exponential growth?

a) the per capita growth rate decreases over time
b) the per capita growth rate increases over time
c) the per capita growth rate does not change over time
d) the per capita growth rate increases until it reaches a maximum and then decreases to zero when the population reaches the carrying capacity
e) the per capita death rate is initially very negative and gets less negative over time.

3. What would I ask to make you draw this graph

a) show how the population size varies over time in logistic growth when the initial population size is much smaller than the carrying capacity

b) show how the population growth rate depends on the population size in logistic growth when the intitial population is much smaller than the carrying capacity

c) show how the population size depends on population size in logistic growth when the initial population size is much smaller than the carryuing capacity

d) show how the population size varies over time in logistic growth when the intitial population is much larger than the carrying capacity

4. What are the axes of a graph showing how the per capita growth rate depends on the population size in logistic growth?
a) x- logistic y- exponential

b) x- logistic y- r

c) x-N y-r

d) x-r y-N

e) x-N y-dN/dt

5. Which of the following is true when populations are at their carrying capacity?

a) N = 100 individuals

b) dN/dt = 0

c) b > d

d) b = d

e) b and d

6. Describe how the population growth rate varies over time in logistic growth when the intial population size is much larger than the carrying capacity.

7. Draw the graph that shows how the population size varies over time in logistic growth when the initial population size is much smaller than the carrying capacity.

Answers. 1.c, 2.c, 3.b, 4.c, 5.e

Fun With Graphs- 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 we need to cover in this class (it is definitely more complex than you need to be able to explain to your students).

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.

3) When the population size equals the carrying capacity we expect no change in the size of the population.

The logistic growth equation is a mathematical equation developed by biologists to describe patterns of population growth consistent with the ideas above. Before focusing on the biological isights that we can gain from the logistic growth model (the real purpose of everything we have been doing) it is important to really understand patterns of logistic growth. Hopefully, this powerpoint presentation will help you understand these patterns better.

Powerpoint Presentation

Click here for a powerpoint presentation entitled "Fun With Graphs- Logistic Growth"
http://www.slideshare.net/secret/gyB3cjnSplLw41

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. It 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

Population Ecology II

This section on population ecology is the topic that I am most concerned about teaching via distance. It seems easier to show you in person how I work problems than it is to try to explain in writing how to work the problems. In addition, I have always found it easier to teach math and graphing when I can give students some problems to solve and I can walk around the classroom peering over their shoulders while they work (it is fun to freak out freshmen that way!).

After you are comfortable with the paremeters that I introduced in teh Population Ecology I. blog, then I would read the articles from the EoE in the following order.

Population ecology
Exponential growth
Logistic growth
Carrying Capacity
Intraspecific competition

These articles (which I have written) attemtp to introduce the readers to the two most important mathematical models that have been used to describe simple population growth.

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 varies over time.

(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 describ 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.

Powerpoint presentation "Why is Exponential Growth Unrealistic?" http://www.slideshare.net/secret/IDPugQtl2wvONv


Final Thoughts

Many students find using math to think about biological concepts and using graphs to illustrate pattersn to be difficult. It is probably difficult for students because they have not had very much practice doing it. If you are comfortable using math and graphs, then most of what we are doing will not be too difficult. However, if you lack a lot of experience using math and graphs, this section might be a bit frustrating. My advice to you is to keep plugging away. Once you learn how to approach these problems, then you will find that you have developed a skill that you can use over an over again. It will, however, require some practice to develop these skills. Please let me know if you are having any problems or questions. You can post on the blog, send me an email, or if you think it will help to actually talk, then we can talk over the phone (it is frustratingly difficult to quickly and easily show you graphs electronically). I need to go educate the masses of Texas Tech, but I will be back on-line soon to talk about logistic growth (a more realistic and useful model of population growth).

Population Ecology I. 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 (r) 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?

Adaptations to Desert Environments

Selection Thinking

The true power of the process of natural selection is that it provides us way of thinking about the diversity in the world around us. If we expect that organisms will be adapted to the condition in their environment the we can think like engineers and ask the questions- how would I design the a trait in that environment? We call this approach "selection thinking". This approach has been highly successful in allowing scientists to understand aspects of physiology, morphology, life history, and behavior in all sorts of environments in all sorts of species. In my research I have used this approach to study behavior in sparrows, woodrats, and beavers and reproduction in deer and plants.

Often, ecologists use mathematical models (often models ripped off from economists and engineers) as tools to help them understand traits of organisms. Although explicitly using mathematical models to study adaptations is probably too advanced for most middle school and high school science classes, I do think that it is important for you as teachers to know that math is an extremely important tool for scientists and be able to express that to your students as often as possible. It is unfortunate that math and science are usually taught as separate topics. As a Zoology Major at UCSB I took calculus during my Freshman year because that is what my advisor told me to do. During the final quarter of my senior year when in a graduate level Reproductive Ecology the professor used calculus to solve a problem that I had the "Oh, now I understand why I was supposed to learn all of that math!" moment.

Using mathematical models forces scientists to do very important things. First, we must clearly state our assumption. Second, it forces us to formailze out logic. When scientists don't use mathetical models they are often forced to rely upon what we call "arm waving" verbal arguments (you should be familiar with these arguments because we see them all of the time when we watch politicians on TV). Often, conclusions that seem reasonable based on verbal arguments actually are incorrect because they are based on either unrealistic assumptions or faulty logic.

I came across an example of a faulty verbal argument while I was working on my Ph.D. I was interested in understanding how parents should invest resources to their offspring, specifically, how big should plants make their seeds. This is a relatively simple problem to think about. When plants reproduce they should be selected to make as many surviving offspring as possible. The number of surviving offspring should be the product of the number of seeds produced and the probability that a seedling survives after it germinates. The number of seeds produced depends on seed size; you can make fewer larger seeds or more smaller seeds. Because the size of a seed is influenced by how many resources that seed contains, the probability that a seedling survives is positively correlated with the size of the seed. The original models predicted that fitness would be maximized if a maternal plant made all of here seeds exactly the same size. However, when you actually measure sizes of individual seeds (and I measured tens of thousands of seeds during my Ph. D.) you see that there is a lot of variation in the size of seeds produced by the same plant. The focus of my Ph.D. research was to try to figure out why plants produced seeds of different sizes.

Several years earlier a scientist named Dan Janzen (a very famous tropical biologist) had published a theory suggesting that producing different sized seeds was an adaptation. His theory was based on a "hand-waving" verbal arguement. In a class I took in graduate school I developed a model to try to see if Janzen's arguement really made sense. My model suggested that Janzen's conclusions were wrong because the verbal logic he used was faulty. My professor suggested that I tried to publish my model. While I was writing that paper, another professor from Orgegon published a matehmatical model that came up with the completely different conclusions than my model. When I compared our two models, I saw that his conclusions were based on an unrealistic assumption and when you used the correct assumption in his model we drew similar conclusions.

Selection Thinking in Arid Environments

Because the environmental conditions in arid environments are particualarly severe, deserts offer an interesting location to study adaptations to local environmental conditions. Hopefully, the readings will give you a broad exposure to how natural selection can mold physiology, morpology, reproduction, and behavior in arid environments.

Powerpoint Presentation

Click here to see a powerpoint presentation "Introduction to Desert Flora and Fauna"
http://www.slideshare.net/secret/pw2UrKumkR7KRT

Expected Learning Outcomes

At the end of this course a fully engaged student should be able to

- identify and discuss the unique challenges associated with living in arid environments (TEKS 112.43 12C)
- explain adaptations of animals and plants for water uptake and water conservation (TEKS 112.43. 7B)
- explain adaptations of animals and plants for dealing with high temperatures (TEKS 112.43. 7B
- develop curricular materials to teach students about adaptations to arid environments TEKS 112.43. 7B)
- develop curricular materials to teach how animals or plants are adapted to a different (non-desert) environment ((TEKS 112.43. 7B & 112.43.12B)

Practice Assignment

To test your understanding of how natural selection affects traits, I suggest that you try to develop a lesson to teach your students how the traits that you observe depends on the environmental conditions. In about one page, outline the lesson you would use to explain how and why the same trait varies between two very different environments. I suggest that you choose an adaptation to life in the desert and compare that train in an very different environment such as a tropical rainforest (much wetter) or the arctic (much colder). If you post your answers here I, and hopefully your classmates, will provide you some feedback.

Natural Selection

The EoE article on Natural Selection (that I wrote) is still under review, so I will give you a copy of the info in that article here. The Evolution article in the EoE has some useful information about natural selection as well.

Natural Selection

Biologists are interested in understanding the amazing diversity of life that we observe around us. Fortunately, we have the process of natural selection to aid us with this task.

It is my experience that most people have a poor understanding of how natural selection works, so it might be useful to briefly discuss how natural selection can cause organisms to become adapted to their environment. First, natural selection is not best defined as “survival of the fittest” (it is a shame that the one thing that apparently every student remembers from school is wrong). Instead, natural selection is best defined as a process.

Natural selection is a process where if-

1) there is variation in traits among individuals in a population,

2) this variation in traits is heritable (i.e., there is a resemblance in traits between parents and offspring), and

3) this variation in traits affects survival, fecundity (the number of babies produced), or mating ability,

then the trait frequency varies between the parent and offspring generation.

Variation in traits among individuals in a population can occur because different organisms have different genes or because they are found in different environments. Genes are molecules (deoxyribose nucleic acids, DNA) that are found in chromosomes. Genes play an important role in determining phenotypes because (1) genes influence which proteins are produced inside cells, (2) proteins can act as enzymes or “biological catalysts” (catalysts act by speeding up the rate of chemical reactions), and (3) phenotypes are influenced by which chemical reactions are taking place in the cells. Thus, if two individuals have different genes then they can produce different proteins that act differently as enzymes. Differences in enzymes leads to differences in the types, or rates, of chemical reactions occurring in the cell which can produce distinctly different phenotypes.

Genes also cause traits to be heritable. We tend to resemble our parents because we receive genes from both of our parents. We are not exactly like our parents because we only get half of our genes from our Mom and the other half from our Dad. When organisms reproduce sexually they produce gametes (eggs – female gametes, sperm- male gametes) by the process of meiosis. The male gamete is mobile (usually sperm are able to swim) so they move to the egg where fertilization occurs to produce a zygote. Crossing over of chromosomes during meiosis and the random combination of gametes during fertilization results in the production of offspring that are genetically different from both of their parents and all of their siblings.

Genes get passed on from one generation to the next by reproduction. Obviously, genes that produce traits that allow organisms to be good at surviving and reproducing should get passed on more often than genes that produce traits that make organism bad at surviving or reproducing. Thus, over time we would expect genes that produce traits that make organisms better at surviving and reproducing to become more common in the population (this is what is meant by the change in trait frequency over time in the definition of natural selection). We might expect that these genes would get more and more common until all individuals in the population have these genes (the gene is “fixed” in the population). If this occurred there would be no more heritable phenotypic variation (assumptions 1 and 2 would not be met) so natural selection would cease.

The creation of new genetic variation by mutation will be needed for natural selection to continue. Mutations are changes in the genes (that typically occur as the result of mistakes produced during replication of chromosomes or the production of gametes) that lead to changes in the phenotypes. Mutations are random. If a mutation occurs that causes an individual to have higher survival or reproductive success, then we would expect the frequency of that gene to increase in the population. A sequence of selection followed by the introduction of new mutations repeated over time should produce organisms that are good at surviving and reproducing in their environments. We call traits that make organisms good at surviving and reproducing “adaptations”. We expect that over time, natural selection should cause organisms to be adapted to their environments.

Because conditions vary between different environments, it is not surprising that the traits that maximize survival and reproduction in differ between different environments. Because of the differences in environmental characteristics of aquatic and terrestrial environments, it is not at all surprising that we see very different types of organisms living in the water and on land.

Expected Learning Outcomes

At the end of this course a fully engaged student should be able to

- explain how the process of natural selection has produced a trait that has increased an organism's survival or reproduction in a particular environment (TEKS 112.43. 7B).

- identify and describe behavioral, physiological, and morphological adaptations to a particular environment (TEKS 112.43. 7B).

- develop curricular materials to teach students how and why the traits of similar organisms can be different across different environments (TEKS 112.43. 7B & 12C).