Question

A version of von Bertalanffy growth can be used to estimate the age of an individual in a homogeneous species from its length if the annual increase in year \(n+1\) satisfies \(a_{n+1}=k\left(S-S_{n}\right)\), with \(S_{n}\) as the length at year \(n, S\) as a limiting length, and \(k\) as a relative growth constant. If \(S_{1}=3, S=9\), and \(k=1 / 2\), numerically estimate the smallest value of \(n\) such that \(S_{n} \geq 8\). Note that \(S_{n+1}=S_{n}+a_{n+1} .\) Find the corresponding \(n\) when \(k=1 / 4\)

Short answer

\( n = 4 \) for \( k = \frac{1}{2} \), \( n = 8 \) for \( k = \frac{1}{4} \).

Step-by-step solution

Initialize Given Values

Start by setting up the given initial conditions, which are: \( S_1 = 3 \), \( S = 9 \), and \( k = \frac{1}{2} \). The formula for growth is given by \( a_{n+1} = k(S - S_n) \) and \( S_{n+1} = S_n + a_{n+1} \).

Calculate Annual Increase for \(k = \frac{1}{2}\)

To find the smallest \( n \) such that \( S_n \geq 8 \), calculate each year's growth starting from \( n = 1 \). \( a_2 = \frac{1}{2}(9 - 3) = 3 \), so \( S_2 = 3 + 3 = 6 \).Calculate \( a_3 = \frac{1}{2}(9 - 6) = 1.5 \), so \( S_3 = 6 + 1.5 = 7.5 \).Next, \( a_4 = \frac{1}{2}(9 - 7.5) = 0.75 \), so \( S_4 = 7.5 + 0.75 = 8.25 \).

Verify Condition for \(k = \frac{1}{2}\)

Since \( S_4 = 8.25 \) and \( S_4 \geq 8 \), the smallest \( n \) for \( k = \frac{1}{2} \) is 4.

Recalculate for \(k = \frac{1}{4}\)

Now set \( k = \frac{1}{4} \) and repeat the calculations. Start from \( n = 1 \):\( a_2 = \frac{1}{4}(9 - 3) = 1.5 \), so \( S_2 = 3 + 1.5 = 4.5 \).Compute \( a_3 = \frac{1}{4}(9 - 4.5) = 1.125 \), so \( S_3 = 4.5 + 1.125 = 5.625 \).Then, \( a_4 = \frac{1}{4}(9 - 5.625) = 0.84375 \), so \( S_4 = 5.625 + 0.84375 = 6.46875 \).Next, \( a_5 = \frac{1}{4}(9 - 6.46875) = 0.6328125 \), so \( S_5 = 6.46875 + 0.6328125 = 7.1015625 \).Then, \( a_6 = \frac{1}{4}(9 - 7.1015625) = 0.474609375 \), so \( S_6 = 7.1015625 + 0.474609375 = 7.576171875 \).Finally, \( a_7 = \frac{1}{4}(9 - 7.576171875) = 0.35595703125 \), so \( S_7 = 7.576171875 + 0.35595703125 = 7.93212890625 \).Finally, \( a_8 = \frac{1}{4}(9 - 7.93212890625) = 0.2669677734375 \), so \( S_8 = 7.93212890625 + 0.2669677734375 = 8.1990966796875 \).

Verify Condition for \(k = \frac{1}{4}\)

Since \( S_8 = 8.1990966796875 \) and \( S_8 \geq 8 \), the smallest \( n \) for \( k = \frac{1}{4} \) is 8.

Key concepts

von Bertalanffy Growth Model

Understanding the von Bertalanffy growth model begins by recognizing its application in biological contexts, particularly in assessing the growth rates of organisms. This model helps estimate how individuals from a homogeneous species grow over time. It's especially useful for species where length is a practical measure of biological development.
The model expresses growth by how much an organism grows in a given year, using a growth constant and the organism's current length relative to a limiting length. In this specific exercise, the formula used is: \( a_{n+1} = k(S - S_n) \).

• \( S \): Limiting length, indicating the maximum potential size.
• \( S_n \): The length at year \( n \).
• \( k \): Relative growth constant, reflecting how quickly the organism approaches its limiting size.

The mechanism behind this model is that each year, the growth amount \( a_{n+1} \) decreases as the organism gets closer to its maximum size \( S \). This decrease is linear, thanks to the subtraction \( (S - S_n) \), making the model mathematically feasible and conceptually straightforward for estimating growth initially and over time. The exercise applies this model using mathematical calculations to determine specific growth intervals by varying the growth constant \( k \).

Numerical Estimation

Numerical estimation in this context is about determining the minimum number of years—denoted as \( n \)—necessary for an organism to reach a specified size, using the von Bertalanffy growth model. Instead of finding a perfect analytical solution, we rely on iterative computations year by year.
This approach is practical when growth parameters like the rate \( k \) change, or when we're computing over successive stages in growth where manual calculation is labor-intensive.
The process for numerical estimation involves:

• Starting with an initial size and applying the growth formula to simulate each year's growth.
• Updating the size after each estimated growth yearly.
• Continuing until the desired size or condition is met.

These iterative calculations give us an estimate of \( n \), showing how long it takes to reach the targeted length. This method is especially handy as small adjustments to the parameters can be easily incorporated without a full recalibration of the problem, showcasing its flexibility.

Growth Rate Calculation

Growth rate calculation is a crucial part of understanding and predicting organism growth patterns over time. In the von Bertalanffy model, the growth rate is determined by the formula \( a_{n+1} = k(S - S_n) \), where each of the parameters plays a specific role.This approach underscores how modifying growth rate constants (\( k \)) can dramatically change the time it takes for an organism to grow near its limiting size. For example, in the exercise, two different values for \( k \), \( \frac{1}{2} \) and \( \frac{1}{4} \), were used.

• A higher \( k \): Results in faster yearly growth, reaching closer to \( S \) sooner as evident by the smaller \( n \).
• A lower \( k \): Slows down the approach to \( S \), requiring more years to reach a similar length, thus increasing \( n \).

By calculating the growth rate annually, students can observe the dynamic relationship between the growth rate constants and the resulting biological growth over time, providing a useful contrast in how different organisms might grow under various conditions.