A research team collected population data of a species at different time intervals. The data collected is as follows:
| Time (years) |
Population |
| 0 |
500 |
| 2 |
800 |
| 4 |
1200 |
| 6 |
1500 |
| 8 |
1700 |
Using this dataset:
(a) Estimate the instantaneous rate of change of the population at t = 4 years using a central difference method.
(b) Assuming the population follows a logistic model given by \(P(t)=\frac{2000}{1+Ae^{-rt}}\) with a carrying capacity of 2000, use your derivative estimate from part (a) to refine the estimate for the growth rate \(r\) at t = 4 years.
Question
Using the provided data, estimate the instantaneous rate of change of the population at t = 4 years using the central difference method with h = 2 years.
Hint
Hint
Hint
Apply \(\frac{f(t+h)-f(t-h)}{2h}\) with h = 2 using the values f(2)=800 and f(6)=1500.
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Answer
Answer
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Explanation
Explanation
- Formula Booklet
Complete Answer
Approximately 175 individuals per year
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Question
Assuming the logistic model \(P(t)=\frac{2000}{1+Ae^{-rt}}\) applies, use your derivative estimate from part (a) to refine the estimate for the growth rate \(r\) at t = 4 years. Hint: First, use the initial condition P(0) = 500 to find the value of A. Then, differentiate the logistic model to relate \(P'(t)\) with \(r\).
Hint
Hint
Hint
First, use P(0) = 500 to find A. Differentiate \(P(t)\) and equate \(P'(4)\) to 175, then solve approximately for \(r\).
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Answer
Answer
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Mark Scheme
Mark Scheme
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Explanation
Explanation
- Formula Booklet
Complete Answer
Approximately 0.3 per year
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Answer Explanation
