Population Ecology Sample Problems answers(1) PDF

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Practice problems and answers on population ecology...

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Types of population model problems Doubling time: ln(2)/r Calculating rates based on definitions b d r dN/dt Predicting population size at time t Recognizing different types of growth curves and their properties Population growth with positive r 0r negative r

Example population model problems 1. A nearly extinct population of black-footed ferrets, now on a nature preserve, is starting to recover: In this population, the per capita birth rate, b, is 0.9 and the per capita mortality rate, m, is 0.6. If the population is at 58 animals now, assuming no restrictions on growth, what is the projected population size 1 year from now? 5 years from now? Assume continuous time.

Example problems continued… 2. In 1962, 5 individuals of a type of bird called the Mute Swan escaped from captivity and ended up on the Chesapeake Bay. Swans eat submerged aquatic vegetation. The swans found themselves with few competitors and an abundance of resources and their population grew. Biologists estimated that the per capita rate of population change (r) for the swan population was 0.17. The population was allowed to grow unchecked until 2001. At this point, the state of Maryland made plans to curtail swan reproduction (by destroying eggs) because the swans were having a devastating effect on the underwater vegetation in the Bay, vegetation on which so many other Bay creatures depend. If no efforts had been made to control swan reproduction, what would you estimate to be the population size of swans in the year 2012, 50 years after the first 5 swans arrived in the Bay? (Assuming no natural restrictions on population growth.)

3. Yellowstone National Park has one of the last remaining large populations of grizzly bears in the lower 48 states. In 1991, researchers estimated the population to have an r of -0.003. At that time, the current population size was about 50. Based on these data, what did the researchers estimate to be the population size in 2091, 100 years later?

4. Consider again the equation ΔN = rN which describes the rate of change in population Δt As population size, N, increases, what happens to rate at which the population size is changing? In other words, if we increase N, what happens to the left side of the equation, the rate of change in population size? What happens to the intrinsic rate of population growth, or the average growth per individual? To help answer this question, let's revisit the endangered population of black-footed ferrets. Above you calculated that r for this population (the per capita rate of population increase) is 0.3. a) Say you start with a population of 58 animals. What is the projected population size in 5 years? (see your answer above). b) What is the projected population size in another 5 years? Use Nt = N0ert to answer. c) How many times more animals did the population gain between the years 5 to 10 than between the years 0 to 5? (To get the answer, divide your answer for b by your answer for a).

d) Why did the population grow so much more between years 5-10 than between years 0-5? The answer can be found by looking at the equation that describes the rate of change in population size: ΔN = rN Δt What was different in the years 5-10 that made the left side of the equation so much larger? Do you see now why the exponential curve has the shape it does, with the population size growing faster and faster as time progresses? However, note that the intrinsic rate, which defines a per capita rate, remains constant. 5. Say that in 2010, while on one of TU’s Study Abroad programs, you are Lost on a small South Pacific Island with 5 other people, 2 men and 3 women (6 people total). Nobody really cares that you are lost, so nobody tries to find you. So, you make a life for yourself on the island, eating, sleeping, and, well reproducing (there is no television, no computer games, etc.). If the island’s carrying capacity for humans is 40, what will be the rate of change in population size, using the equation above? Say that r is 1.0. Use the equation for logistic population growth. 6. Back to your island. The years pass, and now there are 38 of you on the island, close to the carrying capacity of 40. Now calculate the rate of change in population size change, again using an r of 1.0.

As before, we can integrate this differential form of our equation,

(

dN K −N =rN K dt

)

to get an equation that we can use to predict population size in the future. That equation is:

[

1+

]

K − N 0 −rt e N0 ¿ K Nt= ¿

This equation is not as scary as it looks. Start with the denominator. Take K - N0 and then divide it by N0. Then multiply that times e raised to the power of -rt (r times t times a negative 1). Add 1. Then divide what you get into the numerator, K.

7. In prior questions, you were asked: Say that you have a nearly extinct population of blackfooted ferrets that is now protected and starting to recover. In this population, r is 0.3. The current population size is 58. Assuming no restrictions on growth, what is the projected population size 1 year from now? 5 years from now?

Re-write your answers here: 1 year estimate: _____ 5 year estimate: _____ Of course, there are going to be restrictions on growth. Say that the ferret preserve has a carrying capacity of 600 animals. Using the formula above, which takes into consideration environmental carrying capacity, estimate the population size 1 and 5 years into the future, if your starting population is 58. Write the adjusted answers here: 1 year estimate: _____ 5 year estimate: _____ You should find that the 1 year estimates are not that much different. However, the 5 year estimates are quite different. Briefly explain why you saw a greater difference in the 5 year estimates. Answers to Simple Math Problems - Population Growth 1. We use the formula Nt = N0ert where t is time in some kind of units, in this case, years Nt is population size at t years into the future N0 is population size at the start (“time zero”) e is a mathematical constant, which is ~2.72 r is the “per capita rate of population size change.” r is difference between the birth rate, b, and the mortality rate, d (or m) in size between the birth rate and the death rate. So here: r = b – m = 0.9 – 0.6 = 0.3 N0 is population size at the start, so 58. And we want to know the population size 1 year in the future = N1 and 5 years in the future, N5 Answers: N1 = N0ert = (58)(2.72 0.3x1) = (58)(2.72 0.3) = (58)(1.35) = about 78 animals N5 = N0ert = (58)(2.72 0.3x5) = (58)(2.72 1.5) = (58)(4.48) = about 260 animals

2.

N50 = N0ert = (5)(2.72 0.17x50) = (5)(2.728.5) = (5)(4941) = about 24,706 swans

3. N100 = N0ert = (50)(2.72 -0.003x100) = (50)(2.72 -0.3) = (50)(0.74) = about 37 bears Note that because r is negative, the population is declining.

4. Questions concern the equation ΔN = rN which describes rate of change in population size. Δt You are asked to revisit the endangered population of black-footed ferrets. Above you calculated that r for this population is 0.3. a) Say you start with a population of 58 animals. What is the projected population size in 5 years? N5 = N0ert = (58)(2.72 0.3x5) = (58)(2.721.5) = (58)(4.48) = about 260 animals

b) What is the projected population size in another 5 years? Use Nt = N0ert to answer. Can do it this way: N10 = N0ert = (58)(2.72 0.3x10) = (58)(2.723) = (58)(20.09) = about 1165 animals Or can do it this way, starting at 260 animals (= N0) and going 5 more years: N5 = N0ert = (260)(2.720.3x5) = (58)(2.721.5) = (260)(4.48) = about 1165 animals

c) How many times more animals did the population gain between the years 5 to 10 than between the years 0 to 5? (To answer, divide your answer for b by your answer for a). 1165 divided by 260 = 4.48 so population size increased ~ 4.5 faster between years 5-10 than years 0-5. d) Why did the population grow so much more between years 5-10 than between years 0-5? The answer can be found by looking at the equation that describes the rate of change in population size: ΔN = rN Δt What was different in the years 5-10 that made the left side of the equation so much larger? r, the per capita rate of increase did not change. It was a positive 0.3 meaning that the birth rate was greater than the death rate. What changed was the is number of individuals in the population, N (the number of “capita”). The more individuals, the more individuals that could give birth. So the population size grew much faster when the population was larger.

5. Using ΔN = rN (K - N) Δt K On the island, r = 1.0

N=6

K = 40

ΔN = rN (K - N) = (1.0)(6) x (40 – 6) = 6 x 34 = 6 x 0.85 = 5 Δt K 40 40 Population is currently increasing at a rate of about 5 people per year. Why? Because the population is not near its carrying capacity, there is little competition for resources, birth rates are high and death rates are low.

6. ΔN = rN (K - N) = (1.0)(38) x (40 – 38) = 38 x 2 = 38 x 0.05 Δt K 40 40 Population is now only increasing at a rate of about 2 persons per year. This is a much smaller rate of increase than when there were only 6 people on the island. Why? Because the population is close to carrying capacity, competition for resources is high, birth rates are low and/or death rates are higher.

7. You are on your own. Good luck....