Problem 57 - Biocalculus Calculus for the Life Sciences

Question

Von Bertalanffy growth Many fish grow in a way that is described by the von Bertalanffy growth equation. For a fish that starts life with a length of 1 \(\mathrm{cm}\) and has a maximum length of \(30 \mathrm{cm},\) this equation predicts that the growth rate is 29\(e^{-a} \mathrm{cm} /\) year, where \(a\) is the age of the fish. How long will the fish be after 5 years?

Step-by-Step Solution

Answer

The fish will be approximately 29.81 cm long after 5 years.

Step 1: Understanding the Problem

We need to determine the length of the fish after 5 years using the von Bertalanffy growth equation. The fish starts at 1 cm, has a maximum length of 30 cm, and the growth rate is given by 29\(e^{-a}\) cm/year, where \(a\) is the age of the fish in years.

Step 2: Von Bertalanffy Growth Equation

The von Bertalanffy growth equation is generally expressed as \(L(a) = L_{\infty} - (L_{\infty} - L_0)\cdot e^{-ka}\), where \(L(a)\) is the length at age \(a\), \(L_{\infty}\) is the maximum length, \(L_0\) is the initial length, and \(k\) is the growth coefficient. We are given \(L_0 = 1\) cm and \(L_{\infty} = 30\) cm.

Step 3: Identify and Substitute Known Values

Identify the given values: maximum length \(L_{\infty} = 30\) cm, initial length \(L_0 = 1\) cm. We need to find \(L(5)\), the length after 5 years, using the given equation. The growth rate provided is 29\(e^{-a}\) cm/year, implying \(k = 1\) for simplification.

Step 4: Calculate Length at 5 Years

Substitute \(a = 5\), \(L_{\infty} = 30\), \(L_0 = 1\), and \(k = 1\) into the growth equation: \[L(5) = 30 - (30 - 1)\cdot e^{-5}\]. Simplify this expression to find the length.

Step 5: Simplify the Expression

Calculate \(30 - 29\cdot e^{-5}\). First, compute \(e^{-5}\). Then multiply by 29 and subtract from 30: \[L(5) = 30 - 29\cdot 0.0067 \approx 30 - 0.1943 \approx 29.8057\].

Key Concepts

Fish Growth Modeling

Fish growth modeling is essential in understanding how fish develop over time. The von Bertalanffy growth equation is a popular model that describes how fish grow from birth until they reach their maximum size. This equation takes into account different growth rates at various life stages. It is important for both scientists and fisheries managers, as it helps in managing fish populations and ensuring sustainable fishing.

In the von Bertalanffy model, the length of a fish at a certain age is expressed as:

• \(L(a) = L_{\infty} - (L_{\infty} - L_0) \cdot e^{-ka}\)

Here:

• \(L(a)\) is the length at age \(a\)
• \(L_{\infty}\) is the maximum length
• \(L_0\) is the initial length
• \(k\) is the growth coefficient

The model assumes a reduction in the growth rate as the fish approaches its maximum size, which reflects real-world biological processes where resources are redirected to reproduction or maintenance rather than growth.

Differential Equations in Biology

Differential equations play a vital role in biology, helping model dynamic systems such as the growth of organisms or populations. In fish biology, these equations describe changes in length or weight over time. The essential part is understanding that these changes are not constant; they vary with factors such as age or environment.

The von Bertalanffy growth equation is an example of a differential equation, where growth rate decreases as age increases. This decline in growth rate is expressed mathematically by the exponential term \(e^{-ka}\). Such models are crucial for studying biological phenomena and making predictions about future changes in a population, contributing significantly to ecological and evolutionary biology as well.

Exponential Functions in Growth Modeling

Exponential functions are fundamental in growth modeling because they describe how growth can accelerate or decelerate rapidly. When a fish grows, at first, the growth is quick due to the abundance of resources and fewer body demands. However, as the fish ages, resources start diverting towards other biological processes, slowing the growth.

The term \(e^{-ka}\) in the von Bertalanffy equation represents the exponential decay function. Here, \(e\) is the base of the natural logarithm, and it scales down the initial growth impact to reflect real-world conditions where growth slows over time as with fish attaining their maximum size. This usage of exponential decay provides a more realistic model of biological growth compared to linear models, capturing the essence of how living organisms allocate resources as they mature.