1. The characteristic time of exponential growth/decline for this population is A. 5 years B. 20 years C. 5 per year D. 20 per year T c = 1/r

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1 Use this information for the next four questions. A reintroduced population of wolves, starting with 20 individuals in year 0, is growing continuously at a rate of 5%/year. The instantaneous growth rate is r = 0.05/ yr. 1. The characteristic time of exponential growth/decline for this population is A. 5 years B. 20 years C. 5 per year D. 20 per year T c = 1/r 2. The half-life/doubling time of this population is A. Equal to the characteristic time B. The inverse of the characteristic time C. Shorter than the characteristic time D. Longer than the characteristic time E. Not enough information to answer In one characteristic time, the population would double if it were growing linearly. Since it is growing exponentially, it increases by a factor of e 2.7 in one characteristic time. The doubling time must be less than that. 3. If the wolf population continues to grow exponentially, approximately when will it reach 200 individuals? A. 20 years B. 46 years C. 66 years D. 100 years E. Never 4. An ecologist wants to model this population with a discrete-time generation-based model, using a time step t = 4 yr. Which of these is closest to the value of λ she should use to match the assumptions above? A B C D E. There is not enough information to answer this question Page 2

2 Use this information for the next two questions. The simple, time-delayed continuous model we studied is written dn(t) = (b(n(t τ)) d(n(t τ)))n(t) dt 5. In this model, τ represents: A. The characteristic time for exponential growth B. The time delay for the population size to affect growth rates C. The carrying capacity D. Density dependence 6. This model shows what range of behaviour? A. Smooth convergence to equilibrium only B. Smooth convergence and damped cycles only C. Damped cycles and persistent cycles only D. Persistent cycles or chaotic behaviour only E. All of the above behaviours 7. Consider your test paper, the province of Ontario, and the country of Canada. Which two are most similar in area, when considered by absolute difference (linear scale), or by proportional difference (log scale)? A. The paper is the most different from the other two, on either scale B. The province is the most different from the other two, on either scale C. The country is the most different from the other two, on either scale D. The paper is the most different on the linear scale, and the country is most different on the log scale E. The country is the most different on the linear scale, and the paper is most different on the log scale 8. An ecologist believes that a population s finite rate of growth follows λ(n) = (N/K) α. If N is measured in units of indiv/ha, then: A. K and α are also in [indiv/ha] B. K is unitless, and α is in [indiv/ha] C. K is in [indiv/ha], and α is unitless D. K and α are both unitless Page 3

3 9. A population of small plants has discrete, overlapping generations. Adults survive each year with a probability of 2/3 (and thus they have an average lifespan of three years). Each reproducing adult produces an average of 10 seeds each year, of which an average of 10% survive to reproduce in the next year. We model this population using a discrete-time model with time step of 1 year, and we count individuals just before reproduction. What are the values for survival p and fecundity f for this model? A. p = 1/3 and f = 1 B. p = 2/3 and f = 1 C. p = 1/3 and f = 2 D. p = 2/3 and f = A population of small plants has discrete, overlapping generations, with year-toyear survival probability p = 3/4 and year-to-year fecundity f = 1/2. This population has: A. λ = 2 and R = 1.25 B. λ = 1.25 and R = 2 C. λ = 0.67 and R = 0.75 D. λ = 0.75 and R = 0.67 Use the picture below for the next four questions. It shows the assumptions made for a continuous-time birth-death model. Per capita rate (1/t) Page 4

4 11. Which of the four pictures below could be generated by the same model as the picture above? A B C D ANS: C 12. The model illustrated above predicts that the population will increase: A. When the population is very small (only) B. When the population is very large (only) C. When the population is very small or very large D. When the population is between the two equilibria E. When the population is at the nonzero equilibrium 13. The highest total net growth rate (rn) in this model is seen: A. When the population is very small B. When the population is between the two equilibria C. When the population is at the nonzero equilibrium D. When the population is very large 14. This model shows: A. No density dependence B. Density dependence in the birth rate only C. Density dependence in the death rate only D. Density dependence in the birth and death rates Page 5

5 15. If I say a population is growing exponentially, I mean that A. It is growing faster and faster B. It is growing at a constant rate C. It is growing at a rate proportional to its own size D. It is growing at a rate proportional to the time that has elapsed 16. One of the four pictures below shows a population growing exponentially which one? A B C D ANS: B 17. In a linear population model, we expect: A. The reproductive number R is always > 1 B. The instantaneous growth rate r is always > 1 C. The finite growth rate λ is always > 1 D. R > 1 exactly when r > 1 E. R > 1 exactly when λ > 1 Page 6

6 18. (4 points). A population of perennial plants has reproductive adults which produce seeds. Some of these seeds survive the winter and sprout. In addition to producing seeds, the adults pass the winter as root systems underground, and some of these resprout in the spring. We estimate the following. Each reproductive plant produces an average of 60/80/120/140 seeds. Each seed has a 5% probability of sprouting. Each reproducing adult has a 80% probability of resprouting. Each sprout (whether from a seed, or a surviving adult) has a 20% probability of surviving to reproduce at the end of the year. We wish to estimate the growth rate using the formula λ = f + p. Estimate the value of p, f and λ for this population. Show your work briefly. Will the population grow or decline? f is the expected number of new individuals per existing individual: 60/80/120/140 seeds/ adults 5 sprouts/100 seeds 0.2 adults/ sprouts = 0.6/0.8/1.2/1.4 (the units all cancel). (1 point) p is the expected probability that an existing individual will be seen again: 0.8 resprouts/ adults 0.2 adults/ resprouts = 0.16 (1 point) λ = p + f = 0.76/0.96/1.36/1.56 (1 point). If λ > 1, the population will grow; if λ < 1 it will decline. Your answer may vary depending on version (1 point). 19. (4 points) A population of sea turtles was observed to decline from 1200/1300/1400/1500 breeding females in they year 2000 to 1000 in The instantaneous death rate d was estimated at 0.06/year. The sea turtle population has a 1:1 sex ratio. For the purposes of this question, assume the population is changing exponentially, on average. a) Why does d have units of [1/year] only (no turtles)? Because we are counting turtles per turtle, so those units cancel out. b) What is the instantaneous rate of change r for this population? We write N = N 0 exp(rt), and solve as r = log e (N/N 0 )/t = 0.018/0.026/0.034/0.041/ yr c) What is the instantaneous birth rate b? r = b d, so b = r + d = 0.042/0.034/0.026/0.019 / yr d) What is the lifetime reproductive number R? R = b/d = 0.70/0.56/0.44/0.32. The turtle population is doing pretty bad; it s declining relatively slowly because they live relatively long. c by Jonathan Dushoff and the 3SS teaching team. May be reproduced and distributed, with this notice, for non-commercial purposes only. Page 7