Exponential Growth Use Figure 1 in answering the questions below.

1. Write out the exponential equation and make sure you know the definition of each term.

• dN/dt = (b-d)N. dN/dt is the population growth rate, b is the average per-capita birth rate (NOT the number of births), d is the average per-capita death rate (NOT the number of deaths), and N is the population size.

2. Is the population growth rate (dN/dt) higher at time B or time C, or is it the same at both points?

• Population growth rate is higher at time C. Exponential growth is characterized by an accelerating growth rate; this means that growth rate gets higher and higher as the population gets larger and larger.

3. Show how this curve would be different if the average death rate suddenly increased at time B (but stayed lower than the average birth rate)

The population would still be growing because there would still be more births than deaths. But the growth RATE would be lower, because (b-d) would now be smaller. It would look something like this:

4. Show how this curve would be different if the average birth rate suddenly fell at time C, so that it equalled the average death rate.

When birth rate equals death rate, then (b-d) is zero, which (using the equation) means that population growth rate is zero. Zero population growth rate means that the population is neither growing or shrinking. So it would level off:

5. Imagine that at time B the population was suddenly reduced down to the size shown at time A. Draw a curve showing how population would resume growing (assume there are no lasting effects from whatever reduced their numbers).

The population would resume growing at a rate identical to the rate at which it grew the LAST time it was at that size. It would look like this:

Logistic Growth Use Figure 2 in answering the next questions.

6. Write out the logistic equation and make sure you know the definition of each term.

• dN/dt = (b-d)N (K-N)/K. The definitions of dN/dt, b, d, and N are the same as in the exponential equation; in fact, except for the last term, this is the SAME as the exponential equation. K is the carrying capacity (the maximum number of individuals that the habitat can support). (K-N)/K is a mathematical ŇtrickÓ to make population growth slow down as the population size (N) approaches the carrying capacity (K).

7. What is Ňzero population growthÓ? Be able to explain this concept using the terms in the logistic equation.

• Zero population growth is where dN/dt (the population growth rate) equals zero. It has been reached either when (b-d) is zero (as shown in the answer for #4, above), or when (K-N)/K is zero - that is, at carrying capacity.

8. At what time (D, E, or F) has the population reached its carrying capacity? What is the value of dN/dt at that time?

• The population has reached its carrying capacity at time F, at which point dN/dt is zero.

9. During which interval is the population growing exponentially: D to E, E to F, D to F, or none of these?

• This isnŐt a particularly good question. The answer is actually Ňnone of these,Ó since the population is really growing exponentially when N is really tiny (think about the equation: when N is tiny, then (K-N)/K is about 1, and the logistic equation is the same as the exponential equation). But growth will still be pretty close to the exponential rate during the period D to E.

10. Imagine that at time E the population was suddenly reduced down to the size shown at time D. Draw a curve showing how population would resume growing. Now do the same, imagining that at time F it was suddenly reduced to the level it was at time E.

The way you answer these questions is very similar to #5, above.

11. Show how the population would grow if carrying capacity of the population were G individuals.

If the population was below G when the carrying capacity changed, it would go up but level off at G instead of where it did previously (figure at left, below). If it was above G when the carrying capacity changed, it would have to go back DOWN until it reached G, where it would level off (figure at right).

Survivorship Curves Use Figure 3 in answering the next questions.

12. For which of these curves is the likelihood of survival low at young ages but fairly high at older ages?

• In Figure 3A. The likelihood of survival is low at young ages: from an original group of individuals, almost none survive beyond the first year or two (that corresponds to the large drop in the figure). For individuals that survive that period, though, most hang on and donŐt die until they are quite old.

13. Describe in words the difference in timing of mortality between curves A and B.

• In contrast, in Figure 3B, almost all individuals survive the first years of life; the big dropoff in numbers comes at a fairly advanced age.

14. Try to sketch a survivorship curve for a species that experiences highest mortality halfway through its lifespan (a midlife crisis?)

ItŐs hard to draw! You want the curve to be fairly flat at the beginning and fairly flat at the end, with a big drop in the middle. HereŐs my attempt:

15. Below is a list of AGES AT DEATH in (A) a lizard population and (B) an acanthocephalan population. Which of the survivorship curves in Figure 3 best matches these data? Be able to explain your answer.

• (A) Ages at death: 1, 3, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8.
• (B) Ages of death: 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 5, 9.

• (A) There are 15 individuals and nearly all of them died at fairly old ages; this matches Figure 3B. (B) There are 14 individuals and nearly all of them die when very young; this matches Figure 3A. Data that would match Figure 3C would be, for instance: 1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8.

16. Compare the two survivorship curves below. What biological explanation could account for the difference in shape between them?

• The curves are very similar except for maximum longevity (that is, where they hit the x axis). In both cases, most individuals live until they are near that population-specific maximum longevity. One explanation could be that there is a disease that hits elderly individuals in Population A but not in Population B. Many other explanations are possible. The correct answer is NOT that the populations represent different sexes, and females live longer than males (for instance), because populations have to include both sexes! (Why? WhatŐs the definition of a population?)

Population Estimation

17. In June 1996 you mark 30 coyotes in the Tucson Mountains. In September you go back and catch 100 individuals, of which 10 are marked. How large do you estimate the population is? What assumptions have you made, in making this estimation?

Using the mark-recapture equation from your notes, you would set up the 
following  calculation: 

 10 recaptured that have marks	 	             30 originally marked	
--------------------------------------------    =   --------------------------------------------     
          100 recaptured total	                         ??? original population size

Solving the equation by cross-multiplying:  30 (100) = 10x, x= 3000/10 = 300 in 
the   original population.

The many assumptions of this method include:  the mark doesnŐt harm or 
benefit animals or  change their behavior in any way; marked and unmarked 
individuals have a chance to fully  mix before the next sampling period; and 
there are no births or deaths between sample  periods.

18. Why is mark/recapture an ineffective method for estimating plant population sizes? What are some better measures of estimation, for plants?

• The whole point of mark/recapture is that it assumes that marked individuals move around and remix with unmarked ones. To the best of my knowledge, plants donŐt do this. You have two options for plants: either count them all, or count some of them and then estimate. One way to do the latter is to use the quadrat method: count the number of plants in small areas, then multiply by the number of such small areas that would fit into the big area.

19. What are some alternate ways of measuring the human population of the US? What are their benefits and drawbacks?

• I suppose you could use mark/recapture, but itŐs more likely for humans that you would either count them all or use one of the estimation techniques such as those mentioned in #18. The problem with counting every individual is that some are going to be harder to find (for instance, if you go door to door with census forms, you miss the homeless population). Quadrats would only make sense if everyone stopped moving for a while (you could count the number of 182 students in one row in SS 100 and then multiply by the number of rows, for instance). Other estimation techniques make other assumptions. For instance, you could count the number of people in a section of an aerial photo of a protest march, then estimate how many occupied sections of that size there are in the photo, and multiply to get an estimate of the number of people present at the march. But this assumes that the photo was taken when the crowd was at its largest, and that density in the counted section is neither lower nor higher than average.