AP Biology: Unit 8, Population Ecology (Drill 36)

Question 1. Using the logistic growth equation dN/dt = rN((K - N) / K), what is the approximate growth rate of the deer population in Year 3?

• A) 4.1 deer/yr
• B) 6.8 deer/yr ✓
• C) 16.5 deer/yr
• D) 8.7 deer/yr

Explanation: Substituting Year 3 values: dN/dt = 0.14 x 118 x ((200 - 118) / 200) = 0.14 x 118 x (82 / 200) = 0.14 x 118 x 0.41 = 6.77, approximately 6.8 deer/yr. A (4.1) incorrectly applies only the limiting factor term without multiplying through by r and N correctly. C (16.5) applies only rN without the logistic limiting term: 0.14 x 118 = 16.5 -- this is the exponential growth rate, ignoring carrying capacity. D (8.7) is the Year 2 logistic growth rate, not Year 3.

Question 2. Between Year 1 and Year 6, the net growth rate (r) declines from 0.20 to 0.00. Which calculation best explains why r approaches zero by Year 6?

• A) The net per-capita growth rate (r = b - d) approaches zero as birth and death rates converge to equal values (16 per 100 deer each), producing zero net growth. ✓
• B) The population exceeds carrying capacity, causing death rate to exceed birth rate and driving r below zero.
• C) The birth rate drops all the way to zero by Year 6, which eliminates all new population growth and drives the net growth rate r to zero.
• D) The limiting factor term ((K - N) / K) reaches 1.0 as N approaches K, maximizing the effect of r.

Explanation: By Year 6, birth rate and death rate are both 16 per 100 deer. The net per-capita growth rate r = b - d approaches zero as these rates converge, and the population stabilizes at carrying capacity. B is incorrect -- the population (181) has not exceeded K (200), and death rate does not exceed birth rate. C is incorrect -- birth rate at Year 6 is 16, not zero. D is incorrect -- as N approaches K, the limiting factor term approaches zero, not 1.0, which is what reduces dN/dt to zero.

Question 3. Based on the table, during which year is the absolute increase in population size greatest, and what does this indicate about logistic growth?

• A) Year 1, because the per-capita growth rate r is highest when the population is smallest.
• B) Year 2, because the combination of moderately high r and increasing N produces the greatest absolute growth. ✓
• C) Year 4, because the population is closest to K/2, where logistic growth rate is theoretically maximized according to this interpretation.
• D) Year 6, because the population is largest even though r has reached zero.

Explanation: Absolute population increase = r x N x ((K - N) / K). Year 1: 6.4 deer/yr. Year 2: 8.7 deer/yr. Year 3: 6.8 deer/yr. Year 4: 2.3 deer/yr. Year 2 produces the greatest absolute increase because it best reflects the balance between a still-high r (0.19) and a substantially larger N (72) than Year 1. This illustrates that maximum absolute growth in logistic models occurs at an intermediate population size, not at lowest N where r is highest nor near K where the limiting term dominates. A is incorrect -- high r at low N does not produce maximum absolute growth. C is incorrect -- Year 4 (N = 158) is well past K/2 and produces only 2.3 deer/yr. D is incorrect -- r = 0 means no absolute growth.

Question 4. A wildlife manager proposes harvesting 15 deer per year from the preserve beginning in Year 4 to prevent overpopulation. Based on the data, which evaluation of this proposal is most accurate?

• A) The proposal is unnecessary because the population will stabilize at K naturally without intervention, as shown by the declining r values.
• B) The proposal is well-supported because removing 15 deer per year when dN/dt is only about 2.3 deer per year would deliberately cause a sharp population decline as intended, even if the population is already very close to the model's predicted carrying capacity for that model.
• C) The proposal is unnecessary and potentially destabilizing -- at Year 4 the population is already decelerating rapidly toward K, and removing 15 deer per year exceeds the natural growth rate of approximately 2.3 deer/yr, pushing the population below its current trajectory. ✓
• D) The proposal is well-supported because harvesting near K/2 maximizes sustainable yield and will increase long-term population productivity.

Explanation: By Year 4, dN/dt is approximately 2.3 deer/yr -- the population is already decelerating sharply toward K. Harvesting 15 deer per year far exceeds natural growth at this stage, pushing the population downward rather than stabilizing it. The data show the population self-regulating through declining birth rates and rising death rates. A is partially correct in noting natural stabilization but does not evaluate the harvest rate quantitatively. B correctly identifies that 15 deer/yr exceeds natural growth but overstates the consequence -- a sharp decline is possible but not certain from this removal rate alone. D misapplies maximum sustainable yield, which applies when the population is near K/2, not near K.

Question 5. A second deer preserve has identical K and initial population size but shows no decline in r over six years, maintaining r = 0.20 throughout. Which condition most likely explains this difference?

• A) The second preserve has a higher birth rate because the deer in that preserve are genetically superior to those in the first preserve.
• B) The second preserve lacks density-dependent limiting factors, so population growth continues unchecked regardless of N.
• C) The second preserve has a larger carrying capacity than estimated, so the population has not yet experienced resource limitation.
• D) The second preserve experiences density-independent mortality such as regular culling that keeps N well below K, preventing the feedback between population density and per-capita growth rate. ✓

Explanation: If r remains constant regardless of increasing N, density-dependent regulation is not operating. The most mechanistically sound explanation is that density-independent factors keep N well below K, so the population never reaches densities where resource competition or predation pressure intensify. B identifies the absence of density-dependent factors but does not explain why they are absent -- it restates the observation rather than explaining it. A conflates genetic fitness with population-level growth rate and is unsupported. C is plausible but the question states K is identical, making D the more parsimonious explanation.