Problem 24 - Calculus Concepts

Question

Northern Fur Seals There are approximately 200 thousand northern fur seals. Suppose the population is being renewed at a rate of \(r(t)=60-0.5 t\) thousand seals per year and that the survival rate is \(67 \%\). (Source: Delphine Haley, Marine Mammals, Seattle: Pacific Search Press, 1978\()\) a. How many of the current population of 200 thousand seals will still be alive 50 years from now? b. Write a function for the number of seals that will be born \(t\) years from now and will still be alive 50 years from now. c. Estimate the northern fur seal population 50 years from now.

Step-by-Step Solution

Answer

a) 134,000 seals; b) Function: \(n(t) = (60-0.5t)\times 0.67\); c) 135,598 seals.

Step 1: Calculate Future Survival of Current Population

Determine how many of the 200,000 current seals will survive into the future. Since the survival rate is 67%, multiply the current population by this probability:\[ \text{Surviving Seals} = 200,000 \times 0.67 = 134,000 \]

Step 2: Derive Future Births Function

To express the number of seals that will be born and survive in future,calculate the number born at time \(t\) and surviving 50 years from now.The survival of future births at each instant into 50 years ahead is:\[ n(t) = r(t) \times 0.67 \]Substitute \(r(t) = 60 - 0.5t\):\[ n(t) = (60 - 0.5t) \times 0.67 \]

Step 3: Integrate to Find Total Future Born Population

To find the total number of seals born and surviving up to 50 years from now, integrate the function from 0 to 50:\[ N = \int_{0}^{50} n(t) \, dt \]This becomes:\[ N = \int_{0}^{50} (60 - 0.5t) \times 0.67 \, dt \]Substitute and solve the integral:First, simplify the expression:\[ N = 0.67 \int_{0}^{50} (60 - 0.5t) \, dt \]Calculate:\[ N = 0.67 \left[ 60t - 0.5\frac{t^2}{2} \right]_{0}^{50} \]\[ N = 0.67 \left[ 60 \times 50 - 0.5 \times \frac{50^2}{2} \right] \]\[ N = 0.67 \left[ 3000 - 625 \right] = 1597.5 \]So the total future population born and surviving is approximately 1598 seals.

Step 4: Calculate Total Future Population Estimate

Sum the current surviving population with those born over the next 50 years: \[ \text{Future Population} = 134,000 + 1,598 = 135,598 \]

Key Concepts

Survival Rate

When studying population growth in ecology, the survival rate is crucial. It is the percentage of a population expected to live over a certain time period. In the case of the northern fur seals, we are given a survival rate of 67%. This means that 67% of the seals alive today are likely to still be alive in 50 years. To find out how many seals this amounts to, you start with the current population figure. Given that there are 200,000 seals today, the calculation goes as follows:

• Multiply the total population by the survival rate: \( 200,000 \times 0.67 \).
• This results in 134,000 seals expected to survive.

Understanding survival rates help predict changes and highlight survival challenges a species may face over time.

Integration in Calculus

Integration in calculus is a fundamental tool used for finding the area under a curve. It allows us to calculate the total accumulation of quantities, such as populations, over time. In this problem, we seek to estimate how many seals will be born and survive over the next 50 years.The function for seal births, modified by survival probability, is \( n(t) = (60 - 0.5t) \times 0.67 \). Here, \( r(t) = 60 - 0.5t \) represents the rate at which seals are born every year. The survival rate is included by multiplying by 0.67.To find the total number of seals born and surviving over 50 years, you integrate:

• Set up the integral: \( N = \int_{0}^{50} n(t) \, dt \).
• Simplify: \( N = 0.67 \int_{0}^{50} (60 - 0.5t) \, dt \).
• Breaking it down, solve: \( N = 0.67 \left[ 60t - 0.5\frac{t^2}{2} \right]_{0}^{50} \).

Evaluating this integral gives us \( N \approx 1598 \) seals born and surviving into the future. Integration helps us quantify expected future populations based on changing birth rates over time.

Exponential Growth Model

The exponential growth model is often used in ecology to describe how populations grow over time, assuming the rate of birth exceeds the death rate and resources are not limiting. Although the situation with northern fur seals deviates somewhat into more linear dynamics given the constant and decreasing birth rates, understanding exponential growth provides foundational insights. Here, the population grows not just from those who survive from the initial group but also from new births. Specifically:

• New individuals born are modeled by the function \( r(t) = 60 - 0.5t \),indicating birth rates may be reducing over time.
• The growth model foresees fewer births as time progresses.
Exponential growth assumes a consistent, unimpeded population increase. However, real-world scenarios often involve limitations and varying rates, underlining the importance of adapting models to reflect real circumstances, as demonstrated with the integration and specific model adjustments for our seal example.