Problem 9 - Calculus for Biology and Medicine

Question

Saving the Kakapo You are modeling the size of the population of kakapo (a rare flightless parrot) in an island reserve in New Zealand. You want to use the mathematical model to predict the size of the population. The data in this question are taken from Elliot et al. (2001) (a) You start by writing a word equation relating the population size \(N_{t}\) in year \(t\), that is, \(t\) years after the study began, to the population size \(N_{t+1}\) in the next year. umber of birds number of birds that \(N_{t+1}=N_{t}+\begin{array}{l}\text { number } 6 \pi \text { in } \\ \text { born in one vear }\end{array}\) die in one ve We will derive together formulas for each of these terms (i) To estimate the number of birds born, assume that half of the birds are female. A female bird lays one egg every four years. However, because of the large numbers of predators (mostly rats) only \(29 \%\) of hatchlings survive their first year. Explain how based on this data our prediction for the number of births is: \(N_{t} \cdot 0.5 \cdot 0.25 \cdot 0.29=0.03625 \cdot N_{t}\) (ii) Kakapo life expectancy is not well understood, but we will assume that they live around 50 years. That is, in a given year, one in fifty kakapo will die. What is the corresponding number of deaths? (iii) Assume that the starting population size on this island is 50 birds (i.e., \(N_{0}=50\). Calculate the predicted population size over the next five years (i.e., calculate \(N_{1}, N_{2}, \ldots, N_{5}\) ). (iv) When (if ever) will the population size reach 100 birds? What about 200 birds? (You will find it helpful to derive an ex plicit formula for the size of the population \(N_{\mathrm{t}}\). (b) Using your model from part (a) you want to evaluate the effectiveness of two different conservation strategies: (Strategy 1) If the kakapo are given supplementary food, then they will breed more frequently. If given supplementary food, then rather than laying an egg every four years, a female will lay an egg every two years. (Strategy 2) By hand-rearing kakapo chicks, it is possible to increase their one year survival rate from \(29 \%\) to \(75 \%\). (i) Write down a recurrence equation for the population size \(N_{t}\) if strategy 1 is implemented. Assuming \(N_{0}=50\), calculate \(N_{1}\), \(N_{2}, \ldots, N_{5}\) (ii) Write down a recurrence equation for the population size \(N_{t}\) if strategy 2 is implemented. Assuming \(N_{0}=50\), calculate \(N_{1}\), \(N_{2}, \ldots, N_{5}\) (iii) Which conservation strategy gives the biggest increase in population size?

Step-by-Step Solution

Answer

The population reaches 100 birds in about 43 years and 200 birds in 86 years. Strategy 2 gives a larger population increase.

Step 1: Derive Birth Rate Formula

To find out the number of birds born, note that half the birds are female: \( N_{t} \times 0.5 \). Each female lays one egg every four years: \( \times 0.25 \). The survival rate of hatchlings is \( 0.29 \). Therefore, the number of births can be calculated as: \( N_{t} \times 0.5 \times 0.25 \times 0.29 = 0.03625 \times N_{t} \).

Step 2: Derive Death Rate Formula

Assume a kakapo's life expectancy is 50 years, meaning that each year, on average, one in fifty birds will die. This implies the death rate is \( \frac{1}{50}\times N_{t} = 0.02 \times N_{t} \).

Step 3: Construct Recurrence Relation

Based on the above, the recurrence relation for population is: \( N_{t+1} = N_{t} + (0.03625 - 0.02) \times N_{t} = N_{t} + 0.01625 \times N_{t} \).

Step 4: Calculate Population for Each Year (Base Model)

Starting with \( N_{0} = 50 \):- \( N_{1} = 50 + 0.01625 \times 50 = 50.8125 \)- \( N_{2} = 50.8125 + 0.01625 \times 50.8125 = 51.6375 \)- \( N_{3} = 51.6375 + 0.01625 \times 51.6375 = 52.475 \)- \( N_{4} = 52.475 + 0.01625 \times 52.475 = 53.326 \)- \( N_{5} = 53.326 + 0.01625 \times 53.326 = 54.19 \)

Step 5: Find Year Population Reaches 100 Birds

Using the explicit formula, solve for \( t \) when \( N_{t} = 100 \):Let \( N_{t} = 50(1.01625)^{t} = 100 \), solve for \( t \): \( (1.01625)^{t} = \frac{100}{50} = 2 \)\( t = \frac{\log(2)}{\log(1.01625)} \approx 43 \).

Step 6: Find Year Population Reaches 200 Birds

Using the explicit formula, solve for \( t \) when \( N_{t} = 200 \):Let \( N_{t} = 50(1.01625)^{t} = 200 \), solve for \( t \): \( (1.01625)^{t} = \frac{200}{50} = 4 \)\( t = \frac{\log(4)}{\log(1.01625)} \approx 86 \).

Step 7: Implement Strategy 1 - Revised Recurrence

With supplementary food the birth term changes: \( 0.25 \) changes to \( 0.5 \), equating to birth term \( 0.5 \times 0.5 \times 0.29 = 0.0725 \). Recurrence relation now: \( N_{t+1} = N_{t} + (0.0725 - 0.02) \times N_{t} = N_{t} + 0.0525 \times N_{t} \).Population for 5 years: - \( N_{1} = 52.625 \),- \( N_{2} = 55.384 \), - \( N_{3} = 58.20 \), - \( N_{4} = 61.074 \), - \( N_{5} = 64.008 \).

Step 8: Implement Strategy 2 - Revised Recurrence

By hand-rearing, update the survivability: \( 0.29 \) becomes \( 0.75 \), equating to birth term \( 0.1125 \). Recurrence relation now: \( N_{t+1} = N_{t} + (0.1125 - 0.02) \times N_{t} = N_{t} + 0.0925 \times N_{t} \).Population for 5 years:- \( N_{1} = 54.625 \), - \( N_{2} = 59.696 \), - \( N_{3} = 65.211 \), - \( N_{4} = 71.196 \), - \( N_{5} = 77.675 \).

Step 9: Compare Conservation Strategies

Strategy 2 results in a higher population increase over 5 years (77.675 birds) compared to Strategy 1 (64.008 birds). Therefore, strategy 2 provides the largest increase.

Key Concepts

Recurrence Relations

Recurrence relations are mathematical expressions used to model population changes over time in a sequence. These are especially useful in population dynamics to predict future numbers based on current populations and known rates, such as birth and death rates. In the context of the kakapo population on an island, the recurrence relation helps represent how the population size evolves from one year to the next.

In our exercise, the core recurrence relation derived is:

• For each year, the population change is computed as the initial population plus the net growth rate times the initial population.

This net growth rate is found by considering the birth rate and reducing it by the death rate.

When studying population problems, this approach provides a simple way to simulate future population sizes without solving complex differential equations. By using the recurrence relation, we can step through years, predicting the number of kakapo each year by applying the relation repeatedly.

Conservation Strategies

Conservation strategies play a crucial role in maintaining and increasing the populations of endangered species like the kakapo. In our exercise, two strategies were proposed to influence the kakapo population positively:

• **Strategy 1:** Supplementary feeding aimed at increasing breeding frequency.
  This potentially doubles the egg-laying rate from one egg every four years to an egg every two years.
• **Strategy 2:** Hand-rearing chicks to improve survival rates of juveniles from 29% to 75%.

Each strategy affects the recurrence relation, thus altering the projected population numbers. By implementing Strategy 1, the rate at which newly-born kakapo enter the population increases, resulting in more rapid growth.

Similarly, Strategy 2 drastically enhances juvenile survival, which significantly sways population numbers upwards as fewer young kakapo perish.

Deciding between these strategies involves modeling each scenario, examining the results after a set period (five years in this case), and comparing outcomes. According to the results, Strategy 2 produced a greater increase in the kakapo population, demonstrating that enhanced survivability is a key factor in conservation success.

Kakapo Population Modeling

Modeling kakapo population provides insightful predictions about their future numbers, especially on the island reserved for their conservation. This is done through creating formulas based on natural life processes such as birth and death, along with human interventions like conservation strategies.

In the given exercise, the initial model accounted for:

• A baseline birth rate with only 29% survival for hatchlings due to predators.
• Death rates, calculated assuming an average lifespan of 50 years, meaning about 2% of adults die annually.

The model starts with an initial number of 50 birds, iteratively calculating population changes over the years using the recurrence relation. This simulation, applied over five years, forecasts gradual population growth under natural conditions and gives a point of comparison to gauge the effectiveness of conservation interventions.

Thus, kakapo population modeling is a critical tool for conservationists, providing a quantifiable method to test scenarios and guide real-world decisions on managing these precious birds.