Problem 68 - Calculus for Biology and Medicine

Question

Fish are indeterminate growers; that is, they grow throughout their lifetime. The growth of fish can be described by the von Bertalanffy function $$ L(x)=L_{\infty}\left(1-e^{-k x}\right) $$ for \(x \geq 0\), where \(L(x)\) is the length of the fish at age \(x\) and \(k\) and \(L_{\infty}\) are positive constants. (a) Use a graphing calculator to graph \(L(x)\) for \(L_{\infty}=20\), for (i) \(k=1\) and (ii) \(k=0.1\). (b) For \(k=1\), find \(x\) so that the length is \(90 \%\) of \(L_{\infty} .\) Repeat for \(99 \%\) of \(L_{\infty} .\) Can the fish ever attain length \(L_{\infty}\) ? Interpret the meaning of \(L_{\infty}\) (c) Compare the graphs obtained in (a). Which growth curve reaches \(90 \%\) of \(L_{\infty}\) faster? Can you explain what happens to the curve of \(L(x)\) when you vary \(k\left(\right.\) for fixed \(\left.L_{\infty}\right)\) ?

Step-by-Step Solution

Answer

The fish never reaches \(L_{\infty}\). A larger \(k\) makes the fish grow faster toward \(L_{\infty}\).

Step 1: Understanding the Function

The von Bertalanffy function given is \(L(x) = L_{\infty}(1-e^{-kx})\). This formula describes the length \(L(x)\) of a fish at age \(x\), where \(L_{\infty}\) is the asymptotic maximum length the fish can theoretically reach, and \(k\) is the growth rate constant.

Step 2: Graph Plotting for Part (a)(i)

Use a graphing calculator or software to plot \(L(x) = 20(1-e^{-x})\) for \(k=1\). The graph will show an increasing curve that approaches 20 as \(x\) increases.

Step 3: Graph Plotting for Part (a)(ii)

Plot \(L(x) = 20(1-e^{-0.1x})\) for \(k=0.1\). This will be a slower growth curve compared to part (a)(i), approaching 20 more gradually.

Step 4: Solving for 90% Length for k=1

Set \(L(x) = 0.9 \times L_{\infty} = 18\), leading to the equation \(20(1-e^{-x}) = 18\). Solve for \(x\): 1. Divide both sides by 20: \(1 - e^{-x} = 0.9\)2. Subtract 1: \(-e^{-x} = -0.1\)3. Multiply by -1: \(e^{-x} = 0.1\)4. Take the natural log of both sides: \(-x = \ln(0.1)\)5. Solve for \(x\): \(x = -\ln(0.1)\approx 2.302\) units.

Step 5: Solving for 99% Length for k=1

Set \(L(x) = 0.99 \times 20 = 19.8\), leading to the equation \(20(1-e^{-x}) = 19.8\). Solve for \(x\): 1. Divide both sides by 20: \(1 - e^{-x} = 0.99\)2. Subtract 1: \(-e^{-x} = -0.01\)3. Multiply by -1: \(e^{-x} = 0.01\)4. Take the natural log of both sides: \(-x = \ln(0.01)\)5. Solve for \(x\): \(x = -\ln(0.01)\approx 4.605\) units.

Step 6: Interpreting L∞ and Attainability

The fish never actually reaches \(L_{\infty}\) because \(e^{-kx}\) approaches zero asymptotically. \(L_{\infty}\) represents the theoretical maximum length, but the fish gets infinitely close without attaining it.

Step 7: Comparing Graphs in (a)

The graph with \(k=1\) reaches 90% of \(L_{\infty}\) faster than the graph with \(k=0.1\). A larger \(k\) means a faster initial growth rate, making the curve approach \(L_{\infty}\) more rapidly. By varying \(k\) with a fixed \(L_{\infty}\), \(k\) affects the steepness of the curve—higher \(k\) leads to a steeper curve.

Key Concepts

Indeterminate Growth

Fish, unlike many other animals, are indeterminate growers. This means they continue to grow throughout their entire lives. Unlike a tree that might stop growing at a certain height, fish continue to increase in size as they age. Indeterminate growth allows fish to adapt to their surroundings and resources, as their size can be an advantage in avoiding predators and accessing different types of food. This concept partly explains the application of the von Bertalanffy function to model their growth. The function provides a mathematical way to represent this continuous size increase, showing how growth progresses over time. This makes it possible to better understand and predict the growth patterns of fish throughout their life cycle.

Asymptotic Maximum Length

The term "asymptotic maximum length" refers to the theoretical maximum size a fish can reach according to the von Bertalanffy function. In the formula, it is represented by the constant \(L_{\infty}\). This value is the length the fish approaches as time goes to infinity, but it never actually attains this size. Instead, the length approaches this maximum asymptotically, meaning it gets closer and closer to \(L_{\infty}\) but never quite hits it. In practical terms, \(L_{\infty}\) is an estimation of the largest size that the species is capable of reaching given ideal conditions. It is an invaluable parameter in fisheries and ecology, as it helps manage resources by setting realistic expectations for the growth potential of fish in a given environment.

Growth Rate Constant

The growth rate constant, represented by \(k\) in the von Bertalanffy function, dictates how quickly the fish approaches its asymptotic maximum length \(L_{\infty}\). A higher \(k\) value means the fish will grow at a faster rate initially, rapidly getting closer to its maximum potential size. If \(k\) is smaller, the growth process is slower, and the fish approaches \(L_{\infty}\) gradually. Understanding \(k\) is crucial for modeling and predicting growth because it captures the speed at which the fish grows at different ages. In comparisons like the exercise shows, when \(k\) is larger, the fish reaches significant size milestones, like 90% or 99% of \(L_{\infty}\), more quickly.

Graphing Calculator

Using a graphing calculator is a practical approach to visualizing the von Bertalanffy function, especially in understanding how the \(L(x)\) function behaves over time. By entering different values for \(k\) and \(L_{\infty}\), you can plot the growth curves and observe their shape. For example, in the exercise

• when \(k = 1\), the graph shows a steep curve indicating rapid growth towards \(L_{\infty}\).
• When \(k = 0.1\), the growth curve is more gradual and takes longer to approach \(L_{\infty}\).

By comparing these graphs, you can see how varying \(k\) affects the growth trajectory. This visualization aids in understanding the impact of the growth rate constant on reaching significant length milestones, like 90% or 99% of the asymptotic maximum length. It's a powerful tool that turns mathematical concepts into visual insights, making it easier to grasp complex growth patterns.