Problem 10 - Calculus for Biology and Medicine

Question

Mountain Gorilla Conservation You are trying to build a mathematical model for the size of the population of mountain gorillas in a national park in Uganda. The data in this question are taken from Robbins et al. (2009). (a) You start by writing a word equation relating the population of gorillas \(t\) years after the study begins, \(N_{t}\), to the population \(N_{t+1}\) in the next year: \(N_{t+1}=N_{t}+\begin{array}{l}\text { number of gorillas } \\ \text { born in one year }\end{array}-\begin{array}{l}\text { number of gorillas } \\\ \text { that die in one yeai }\end{array}\) We will derive together formulas for the number of births and the number of deaths. (i) Around half of gorillas are female, \(75 \%\) of females are of reproductive age, and in a given year \(22 \%\) of the females of reproductive age will give birth. Explain why the number of births is equal to: \(\quad 0.5 \cdot 0.75 \cdot 0.22 \cdot N_{t}=0.0825 \cdot N_{t}\) (ii) In a given year \(4.5 \%\) of gorillas will die. Write down a formula for the number of deaths. (iii) Write down a recurrence equation for the number of gorillas in the national park. Assuming that there are 300 gorillas initially (that is \(N_{0}=300\) ), derive an explicit formula for the number of gorillas after \(t\) years. (iv) Calculate the population size after \(1,2,5\), and 10 years. (v) According to the model, how long will it take for population size to double to 600 gorillas? (b) In reality the population size is almost totally stagnant (i.e., \(N_{t}\) changes very little from year to year). Robbins et al. (2009) consider three different explanations for this effect: (i) Increased mortality: Gorillas are dying sooner than was thought. What percentage of gorillas would have to die each year for the population size to not change from year to year? (ii) Decreased female fecundity: Gorillas are having fewer offspring than was thought. Calculate the female birth rate (percentage of reproductive age females that give birth) that would lead to the population size not changing from year to year. Assume that all other values used in part (a) are correct. (iii) Emigration: Gorillas are leaving the national park. What number of gorillas would have to leave the national park each year for the population to not change from year to year?

Step-by-Step Solution

Answer

Gorilla population doubles in about 19 years. Births are 0.0825\(N_t\) and deaths 0.045\(N_t\). For no growth, increase deaths to 8.25%, reduce births to 12%, or emigrate 11 gorillas/year.

Step 1: Derive the number of births

We are given that half of the gorillas are female, hence 0.5 of the population. Additionally, 75% of these females are of reproductive age, so 0.75 times 0.5 gives us those able to reproduce. Finally, 22% of these females will give birth in a year, so we multiply by 0.22. Thus, the number of births is \( 0.5 \times 0.75 \times 0.22 \times N_t = 0.0825 \times N_t \).

Step 2: Derive the number of deaths

The problem states that 4.5% of the gorilla population will die each year. Therefore, the number of deaths is \( 0.045 \times N_t \).

Step 3: Set up the recurrence relation

We can establish a recurrence relation by combining the birth and death rates. The formula becomes \( N_{t+1} = N_t + 0.0825 \times N_t - 0.045 \times N_t \), which simplifies to \( N_{t+1} = (1 + 0.0375)N_t = 1.0375N_t \).

Step 4: Derive the explicit formula for population size

Given the recurrence relation \( N_{t+1} = 1.0375 N_t \), the general explicit solution can be modeled as \( N_t = N_0 \times (1.0375)^t \). Starting with \( N_0 = 300 \), the formula becomes \( N_t = 300 \times (1.0375)^t \).

Step 5: Calculate population sizes over time

Using the explicit formula, calculate the population sizes as follows: - After 1 year: \( N_1 = 300 \times (1.0375)^1 = 311.25 \approx 311 \)- After 2 years: \( N_2 = 300 \times (1.0375)^2 = 322.9 \approx 323 \)- After 5 years: \( N_5 = 300 \times (1.0375)^5 = 358.44 \approx 358 \)- After 10 years: \( N_{10} = 300 \times (1.0375)^{10} = 428.55 \approx 429 \).

Step 6: Determine time to double population

To find how long it takes to double the population from 300 to 600, solve \( 600 = 300 \times (1.0375)^t \). Dividing both sides by 300 gives \( 2 = (1.0375)^t \). Taking the natural logarithm of both sides, solve for \( t \) using \( t = \frac{\ln(2)}{\ln(1.0375)} \approx 18.94 \). Thus, it takes about 19 years.

Step 7: Adjust death rate for stagnant population

For a stagnant population where growth is zero, the birth rate \( 0.0825 \) must equal the death rate. Setting up the equation \( 1 - d = 1 - 0.0825 \), solving yields \( d = 0.0825 \) or 8.25%.

Step 8: Adjust birth rate for stagnant population

Under the same scenario of no change, the number of births must equal the adjusted death rate. Therefore, the birth rate for females of reproductive age should be \( \frac{0.045}{0.5 \times 0.75} \approx 0.12 \) or 12%.

Step 9: Calculate emigration rate for stagnant population

If no population growth occurs, set the equation \( N_t = N_t + B - D - E \) where \( B = D \). Assume emigration \( E = N_t (B - D) \) because births equal deaths achieved at \( E = 0.0375 N_t \). Thus, approximately 11 gorillas (as 0.0375 of 300) must leave annually.

Key Concepts

Mountain Gorilla Conservation

Mountain gorillas are an endangered species, commonly found in the national parks of Uganda. Protecting these magnificent creatures requires a thorough understanding of their population dynamics. Population studies help in creating conservation strategies that ensure their survival.
One major factor affecting mountain gorilla conservation is the balance of birth and death rates within their population. By monitoring these rates, conservationists can predict changes in population size and strategize accordingly. Efforts are made to maintain or increase the gorilla population by protecting their habitats and minimizing threats such as poaching and habitat destruction.
Conservation initiatives also focus on research and monitoring. This allows scientists to continuously update the data model used in predicting population changes, ensuring that conservation strategies can be effectively adapted to real-world conditions.

Recurrent Equations

Recurrent equations are mathematical expressions used to describe sequences, where the next term depends on one or more of the previous terms. In the context of mountain gorilla population dynamics, a recurrent equation helps track how the population size changes from one year to the next.
For example, the recurrent equation used in this scenario is:

• \(N_{t+1} = N_{t} + 0.0825N_{t} - 0.045N_{t} = 1.0375N_{t}\)

This equation shows that the gorilla population grows by an annual rate of 3.75%, factoring in birth and death rates. This makes recurrent equations essential in modeling biological populations and predicting future changes.
Solving recurrent equations often involves breaking them down to find explicit formulas that predict long-term population dynamics over several years. These formulas are invaluable for conservation efforts, offering clear insights into whether populations should grow, shrink, or remain stable.

Mathematical Model

A mathematical model is a representation using mathematical concepts to describe the real-world behavior of a system—in this case, the population of mountain gorillas. Such models simplify the complexity of ecological systems and help in making predictions about future events.

• The model here considers factors like birth and death rates to define a relationship over time.
• The basic form of the mathematical model is expressed as a recurrent equation or explicit formula.

This gorilla population model considers:

• Each year, a proportion of females give birth.
• Annually, a certain percentage of gorillas die.
• The initial population size starts at 300.

The explicit formula derived becomes \(N_t = 300 \times (1.0375)^t\), allowing researchers to predict how long it takes for the population to double or halve. Models like this guide in determining conservation needs and the adequacy of current strategies.

Ecological Impact

The ecological impact of population changes in mountain gorillas extends beyond the species itself and influences the entire ecosystem. As herbivores, gorillas play a significant role in the seed dispersal of forest plants, aiding in maintaining the health and diversity of their habitats.
Large fluctuations in population can disrupt these processes. For example, a declining population would mean fewer seeds are dispersed, potentially affecting plant regeneration and canopy structure. Conversely, a stable or increasing gorilla population supports a thriving ecosystem by ensuring continual plant growth and reproduction.
Therefore, maintaining an ecological balance is crucial. Changes are monitored not only through direct observation of species numbers but also by examining broader ecological indicators like plant diversity and forest health. This interconnected approach highlights the intrinsic value of each species within an ecological framework and confirms why mountain gorilla conservation is a pivotal aspect of preserving biodiversity.