How Did I Know What the Exponential Growth Curve Looked Like?

In class I showed you the shape of the exponential growth curve.  Now I would like to explain to you how I knew to draw that particular shape for that graph.

The exponential growth curve is produced anytime

dN/dt = rN and r is constant.

Remember, that making r a constant makes this equation the simplest that it can be.

If you want to know how to draw the graph then we can simply plug some numbers in to the equation and plot the results.

We will have to make up some value for r.  The simplest value  is to assume

r = 1 individual/year/individual.

All I need to do now is to calculate dN/dt for different values of N.  The simplest values of N that I can imagine are 1, 2, 3, 4, etc.  Let's start by assuming that in Year 1 N = 1 and plugging that into dN/dt = rN.

dN/dt = (1 individual/year/individual)(1 individual) = 1 individual/year

We can now add this value to the table below.


Year        N  (individual)                            dN/dt (individuals/year)
1              1                                                 1
2
3
4
5

If we started with a population size of one individual and the population increased in size by one individual during the first year then at the start of the second year the population should contain 2 individuals.  We can add that value to the table.


Year        N  (individual)                            dN/dt (individuals/year)
1              1                                                 1
2              2
3
4
5

Following the same logic that we used above you should be able to calculate N and dN/dt for both years 1 through 6.  (make sure you can do this yourself before looking at my calculations).


Year        N  (individual)                            dN/dt (individuals/year)
1              1                                                 1
2              2                                                 2
3              4                                                 4
4              8                                                 8
5             16                                                16
6             32                                                32

You should now be able to plot the following graphs using the information held in this table.

1.  How does the population size vary over time?

2.  How does the population growth rate vary over time?

3.  How does the population growth rate depend on the population size?

Clarifying Natural Selection: Individual and Inclusive Fitness

There are so many more things to talk about in BIOL 1404 then we possibly have time to discuss.  Thus, every semester I have to decide what to mention and what to leave out of lectures.  This year I decided not to discuss a couple of terms during lecture and after talking to a couple of students I see that it might have been helpful to talk about these terms.

When we talked about natural selection we concluded that natural selection should produce selfish traits (those that maximize the survival and reproduction of individuals).  Thus, natural selection should maximize an organism's "individual fitness" (the number of genes that an individual passes on by reproducing itself).

Kin selection suggests that sometimes we can pass on genes by helping our close relative to reproduce more than they would have without out help.  Genes that are passed on by helping your close relatives to reproduce are known as "inclusive fitness".

Thus, "total fitness" (the total number of genes passed on by an individual) is he sum of "individual fitness" and "inclusive fitness".

Total fitness = individual fitness + inclusive fitness.

When we reexamine the process of natural selection we see that natural selection maximizes total fitness.  Thus, sometimes an organism can pass on more genes by increasing its inclusive fitness at the expense of its individual fitness.

Expected Learning Outcomes

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

- define and distinguish between individual fitness, inclusive fitness, and total fitness
- discuss the role of inclusive fitness in the selection of altruistic acts via kin selection.

More Cool Info about Sexual Selection

Hello Everyone,

I am pleased that I have received some feedback that some of you found the info on sexual selection and mate choice to be interesting.  I wish we had more time to talk about this fascinating topic, but I know that you are so excited about learning the math and graphs related to population biology that we need to move on.

Some of your classmates have sent me links to some interesting online info about the topic that you might like to take a look at.  Thanks for sending them to me.

1.  The spider that loses its penis during sex to make it more fierce in battle against love rivals.

http://www.dailymail.co.uk/sciencetech/article-2158818/Nephilengys-malabarensis-spider-loses-penis-sex-make-fierce-battle-love-rivals.html

2.  The Science of Sex Appeal

http://dsc.discovery.com/tv-shows/other-shows/videos/other-shows-science-of-sex-appeal-videos.htm

This site contains several short videos that discuss some of the topics that we were only able to touch on that relate sexual selection and mate choice in humans.  These videos are pretty interesting so take a look when you get a chance.

Male-male Competition in Species With External Fertilization

I didn't have time to talk about male-male competition in species with external fertilization in class, but I have decided that this is an interesting enough topic to deserve its own post (and the subject is addressed in the Expected Learning Outcomes for Sexual Selection).

Most of the animals that we are used to thinking about have internal fertilization which means that the male mates with the female and then deposits sperm into the body of the female.  Other species, including many marine and aquatic species have external fertilization.  For example, in most fishes females deposit eggs either in a nest or in the open water column and then males swim by and deposit sperm in the water which will find the eggs and fertilize them.

Because it is possible for many males to attempt to fertilize the same eggs, in many cases the male that fertilizes the most eggs is simply the one that produces the most sperm. Thus, male-male competition in many species of fishes has led to the production of large amounts of sperm. In some cases the testes can make up almost half of the body mass of a male fish.

The Amazing Sex Life of Fish!!

Fish have some of the craziest and most interesting sex lives of all of the species on earth!  One of the interesting aspects of the sex lives of some of the coral reef fishes is that they change sex during their lives. Usually, individuals will start their lives as females and then change sex to males when they are older and larger (sequential hermaphroditism).

In some species, such as the blue-head wrasse, males defend territories that can contain several several females that he mates with.  In some populations there are fish with an interesting strategy.  There are fish that look and act just like females, but are actually males (these fish are known as "sneakers" - now the ridiculous photo at the top of the post should make sense).  The territorial male accepts the sneakers in his territory because he thinks that they are females.  When a female lays eggs in the territory the sneaker males sneak in, deposit a cloud of sperm, and fertilize some eggs! Pretty cool!!

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 that relate to exponential growth.

http://www.slideshare.net/secret/mavlOD8flFs67G

NOTE:
Any graphs that contain the incorrect axes will be considered to be completely wrong on all exams and assignments!!

Population Biology 2. Exponential Growth

Lecture Video- http://mediacast.ttu.edu/Mediasite/Play/b8c64d66f62a4747b7983398113f0b391d?catalog=4dc7289a-d3e0-4ae5-8fdc-5b86c027a06b


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 then take a look at the blog post entitled "How Did I Know What the Exponential Growth Curve Looked Like?".

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 1. Basic Parameters

Lecture Video: http://mediacast.ttu.edu/Mediasite/Play/20ade8c97b0b40af8eaab29ae07eee6f1d?catalog=4dc7289a-d3e0-4ae5-8fdc-5b86c027a06b


IMPORTANT NOTE!!!
For the next several lectures we will be using math and graphs to help us explore population ecology. From my experience teaching this topic in the past, many of you will experience some difficulties with this material because you are not confident when dealing with math and graphs.

Rather than introducing the concepts to you in lecture and then having you work on activities to help you master the material out of class, this year I would like to "flip the class". This year I would like for you to study the material before coming to class so that we can use the class time to answer your questions and to help you master the material.

Assignment- Before Wed December 30th, I expect that you will have read the following post and are able to meet all of the expected learning outcomes listed below. If you have not mastered the material in this blog, then you will find that you will be hopelessly lost in the lectures that follow!!

Expected Learning Outcomes

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

- define b, d, r, B, D, dN/dt.

- identify and use the proper units associated with each parameter

- use the correct algebraic equations to calculate each of these parameters

- be equally comfortable referring to these concepts verbally or via their algebraic symbols.

Basic Parameters of Population Ecology

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 of elephants, there are 100 individuals.

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

1. 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?

2. List the equation/equations for calculating the following parameters
a) b
b) the population growth rate
c) r