Marathon SI Review

In addition to their regularly scheduled SI session, Suzanne and Jeffrey will hold two review sessions this week.

Suzanne: Thursday at 3:30 in Chem 49
Jeffrey: Sunday at 1 - 4  in Biol LH 100

My Group OH scheduled for 6:30 on Monday Feb 12th can be considered as my Marathon Review.  Hope to see you there.

Population Growth- Final Thoughts

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.

Past Test Questions (answers at bottom of post)

1. In logistic growth, what is the per capita growth rate when N = 1/2K?
(a) rmax
(b) 2(rmax)
(c) ½ (rmax)
(d) it is a maximum
(e) you can not answer this questions with the information provided.

2. How can you calculate the population growth rate?
(a) subtract B from D
(b) add the per capita death rate to the per capita birth rate
(c) multiply r by N
(d) divide dN/dt by N
(e) a and c

3. Why don’t we expect raccoons to show exponential growth?
(a) per capita birth rates increase as population sizes increase
(b) per capita death rates increase as population sizes increase
(c) per capita birth rates decrease as population sizes increase
(d) b and c
(e) none of the above

4. Which of the following are true when populations are at their carrying capacity?
(a) dN/dt > 0
(b) r < 0
(c) b = d
(d) B > D
(e) a and d


answers- 1.e, 2.c, 3.d, 4.c

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

Population Biology 3. Logistic Growth

Lecture Video- http://mediacast.ttu.edu/Mediasite/Play/83f60ff04a32459896329229dc3cc5fd1d?catalog=4dc7289a-d3e0-4ae5-8fdc-5b86c027a06b


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.

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

NOTE: THERE IS AN ERROR ON SLIDE 16 OF THIS PRESENTATION!!!

The title of the graph on slide 16 should read "Logistic Growth: dN/dt vs t (Not N), N initially << k"

The x-axis of the graph is TIME (please ignore the values of K on the x-axis because K does not belong on the time axis). The shape of the graph is correct. Make sure you change the x-axis to Time rather than Population Size.

Expected Learning Outcomes

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

- define the carrying capacity

- draw, and interpret the following graphs associated with logistic growth
-how population size changes over time in logistic growth when the initial population size is much smaller than the carrying capacity
-how the population size changes over time in logistic growth when the initial population size is much larger than the carrying capacity
-how population growth rate changes over time in logistic growth when the initial population size is much smaller than the carrying capacity
-how the population growth rate changes over time in logistic growth when the initial population size is much larger than the carrying capacity
-how the per capita growth rate varies over time in logistic growth
-how the population growth rate varies over time in logistic growth

- discuss the causes for the shape of the s-curve (this answer will need to include a discussion of both math and biology)

- discuss the factors that regulate population size, be able to distinguish between density dependent and density independent factors that regulate population growth and give examples

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.