Problem 72 - Calculus for Biology and Medicine

Question

Fish Growth von Bertalanffy's equation is used to model the growth of fish. The length of the fish, \(L(t)\), grows at a rate that depends on its current length \(L(t)\) (that is, big and small fish grow at different \mathrm{\\{} r a t e s ) . ~ The growth of a fish is modeled using the differential equation: $$ \frac{d L}{d t}=k\left(L_{\infty}-L\right) $$ where \(k\) and \(L_{\infty}\) are both positive constants. To solve the equation, we will learn in Chapter 8 that it is necessary to calculate the following integral: $$ t=\int \frac{d L}{k\left(L_{\infty}-L\right)} $$ Evaluate the integral (7.14), keeping \(k\) and \(L_{\infty}\) as unknown constants.

Step-by-Step Solution

Answer

The evaluated integral is \( t = \frac{-1}{k} \ln|L_{\infty}-L| + C \).

Step 1: Set up the integral

The goal is to find the integral \( \int \frac{d L}{k(L_{\infty}-L)} \), which relates the change in length \( L \) of the fish over time. This integral will help us understand how \( L \) varies as \( t \) changes.

Step 2: Factor out the constant \( k \)

The constant \( k \) can be factored out of the denominator to make the integration process easier: \( \int \frac{d L}{k(L_{\infty}-L)} = \frac{1}{k} \int \frac{d L}{L_{\infty}-L} \). Now the integral looks simpler and more manageable.

Step 3: Use substitution to simplify

Substitute \( u = L_{\infty} - L \), which implies \( dL = -du \). This substitution transforms the integral into \( \frac{1}{k} \int \frac{-du}{u} \).

Step 4: Integrate the expression

Integrating \( \int \frac{-du}{u} \) gives \( -\ln|u| + C \), where \( C \) is the constant of integration. Therefore, the integral is \( \frac{-1}{k} \ln|L_{\infty}-L| + C \).

Step 5: Solve for \( t \)

Remember, the integral equals \( t \) from the original problem statement. Thus, the value of \( t \) is \( t = \frac{-1}{k} \ln|L_{\infty}-L| + C \).

Key Concepts

Understanding Differential Equations

Differential equations are mathematical equations that describe how a particular quantity changes over time or with respect to another variable. They are used to model systems where change is happening. In the case of fish growth, the differential equation \( \frac{dL}{dt} = k(L_{\infty} - L) \) tells us how the length of a fish, \( L(t) \), evolves over time.

• Components: In the given equation, \( \frac{dL}{dt} \) represents the rate of change of the fish's length, meaning how fast or slow the fish is growing over time.
• Constants: The constants \( k \) and \( L_{\infty} \) are crucial here. \( k \) is a growth rate constant indicating how quickly the fish approaches its maximum length, and \( L_{\infty} \) is the maximum theoretical length the fish can reach if it were to grow indefinitely.

Differential equations like this one are essential because they help us predict future outcomes, like the size of the fish over time, based on current conditions.

To solve this differential equation, we need to move into the world of integral calculus.

Introduction to Integral Calculus

Integral calculus deals with finding the function when given its rate of change. It is effectively the reverse process of differentiation. When solving differential equations, and in our fish growth model, we often use integrals to find the solution function.

• The Role of Integrals: In the fish growth exercise, you were asked to solve the integral \( \int \frac{dL}{k(L_{\infty} - L)} \). This integral, once solved, helps us find the specific function of length over time, \( t \).
• Techniques: In our example, you need to simplify the integral for easier evaluation. This includes factoring out constants and substitutions. For instance, setting \( u = L_{\infty} - L \) allows the transformation of the integral into a more straightforward form, \( \int \frac{-du}{u} \).

Understanding integrals is key because they enable us to "undo" differentiation, giving us the function's original form, such as a fish's growth over time. Learning integral techniques is important as it allows us to compute variables like time or growth directly from a rate of change.

Exploring the von Bertalanffy Growth Model

The von Bertalanffy growth model provides a realistic framework for understanding how animals like fish grow over time. It considers biological principles and produces S-shaped growth curves that start steep, slow down, and level off as the animal approaches a maximum size.

• Growth Dynamics: In this model, different growth speeds depend on current size. Smaller fish grow rapidly as they are far from their potential maximum length \( L_{\infty} \), while larger fish grow slower.
• The Equation: The underlying equation \( \frac{dL}{dt} = k(L_{\infty} - L) \) reflects this dynamic. The term \( L_{\infty} - L \) shows that growth rate decreases as the fish approaches its maximum length.
• Applications: This model is widely used in fields like fisheries science, marine biology, and conservation. It helps scientists understand growth patterns and predict stock health or changes due to environmental factors.

The beauty of the von Bertalanffy model is its ability to reflect biological reality through mathematics. Its use of differential equations and calculus allows for detailed analysis and predictions of growth trajectories based on real-world parameters.