Problem 45 - Calculus for Biology and Medicine

Question

Many fish join with others of their species to form schools, large groups that swim together in a coordinated way. It is thought that schooling helps the fish evade predators (predators are unable to pick out a single individual in the school to prey upon), and may also allow the fish to swim more efficiently by slip-streaming off each other. There is a lot of interest in how individual fish within the school interact to produce the complex swimming patterns seen in real schools. (a) Fish tend to prefer not to be too close or too far from their neighbors. D'Orsogna et al. (2006) propose that interactions between neighbors can be modeled by incorporating an energy of interaction \(U(r)\), that depends on the distance \(r\) between the fish. The force between the two fish can be derived from this energy from the formula: \(F(r)=-\frac{d U}{d r} .\) A positive force means that the fish repel each other, and a negative force means that they attract each other. D'Orsogna et al. assume the following form for the energy of interaction: $$ U(r)=a_{1} e^{-k_{1} r}-a_{2} e^{-k_{2} r} $$ Where \(a_{1}, a_{2}, k_{1}\), and \(k_{2}\) are all positive constants. Let's assume that for one particular species of fish \(a_{1}=3, a_{2}=2, k_{1}=2\), \(k_{2}=1 .\) Show that the fish repel each other when \(r<\ln 3\) and that they attract each other with \(r>\ln 3\). (b) Based on your answer to (a) why do you think that \(r=\ln 3\) is referred to as the equilibrium spacing of the fish? (c) In addition to maintaining their distance from each other, fish must also follow the movements of their neighbors.

Step-by-Step Solution

Answer

Fish repel one another when \( r < \ln 3 \) and attract when \( r > \ln 3 \). At \( r = \ln 3 \), the force is zero, indicating equilibrium.

Step 1: Understand the Force Formula

The force between two fish, \( F(r) \), is given by the derivative of the energy of interaction \( U(r) \) with respect to the distance \( r \). Specifically, \( F(r) = -\frac{dU}{dr} \).

Step 2: Differentiate the Energy Expression

Given the energy expression \( U(r) = a_{1} e^{-k_{1}r} - a_{2} e^{-k_{2}r} \), differentiate it with respect to \( r \). The result is: \[ \frac{dU}{dr} = -a_{1}k_{1}e^{-k_{1}r} + a_{2}k_{2}e^{-k_{2}r}. \]

Step 3: Calculate the Force Expression

Plug the differentiated expression into the force formula: \[ F(r) = -\left(-a_{1}k_{1}e^{-k_{1}r} + a_{2}k_{2}e^{-k_{2}r}\right) = a_{1}k_{1}e^{-k_{1}r} - a_{2}k_{2}e^{-k_{2}r}. \] Substituting the given values \( a_{1}=3, a_{2}=2, k_{1}=2, k_{2}=1 \), the force becomes: \[ F(r) = 3 \cdot 2 \cdot e^{-2r} - 2 \cdot 1 \cdot e^{-r} = 6e^{-2r} - 2e^{-r}. \]

Step 4: Determine the Sign of the Force

To find the turning point (equilibrium spacing), set \( F(r) = 0 \): \[ 6e^{-2r} = 2e^{-r}. \] Simplify this equation by dividing both sides by \( 2e^{-2r} \): \[ 3 = e^r. \] Thus, \( r = \ln3 \). For \( r < \ln3 \), \( 6e^{-2r} > 2e^{-r} \) implying \( F(r) > 0 \) (repulsion). For \( r > \ln3 \), \( 6e^{-2r} < 2e^{-r} \), implying \( F(r) < 0 \) (attraction).

Step 5: Discuss Equilibrium Spacing

At \( r = \ln 3 \), the force \( F(r) \) is zero, meaning the fish neither attract nor repel each other. This spacing corresponds to a state of equilibrium, as the fish are balanced at this distance without exerting any force on each other.

Key Concepts

Energy of Interaction

In the context of fish schooling behavior, the concept of energy of interaction is pivotal in understanding how fish position themselves relative to one another. The energy of interaction, denoted as \( U(r) \), is a function that describes how much energy is required to maintain a certain distance \( r \) between two fish. This value changes depending on how close or far the fish are from each other, and it's modeled mathematically to understand these dynamics better.
In this scenario, the energy of interaction is expressed as:\[ U(r) = a_1 e^{-k_1 r} - a_2 e^{-k_2 r} \]where \( a_1, a_2, k_1, \) and \( k_2 \) are positive constants specific to a species. This expression is composed of two exponential terms, representing repulsive and attractive interactions, respectively.

• The first term, \( a_1 e^{-k_1 r} \), can be thought of as the repulsion energy that increases when fish are closer, discouraging them from being too near.
• The second term, \( -a_2 e^{-k_2 r} \), represents the attraction energy that becomes significant when fish are separated by a certain distance.

This balance between attraction and repulsion helps maintain the fish's cohesion in schools, ensuring they stay coordinated.

Force Derivative

To understand how fish adjust their positions, it is crucial to consider the force derivative. The force between two fish at a distance \( r \) is derived from the energy of interaction. It is calculated using the derivative of the energy function and is represented as \( F(r) = -\frac{dU}{dr} \).
This operation involves differentiating the energy expression to understand how the force acting on a fish changes with the distance \( r \). The derivative is:\[ \frac{dU}{dr} = -a_1 k_1 e^{-k_1 r} + a_2 k_2 e^{-k_2 r} \]By applying the negative sign from the force formula, we get:\[ F(r) = a_1 k_1 e^{-k_1 r} - a_2 k_2 e^{-k_2 r} \]

• A positive force \( F(r) > 0 \) suggests that the fish repel each other, indicating that they are too close.
• A negative force \( F(r) < 0 \) implies attraction, meaning the fish are too far and will move closer together to reach an optimal distance.

Thus, the derivative serves as a tool to calculate and predict how fish will move in response to their neighbors, maintaining the school formation efficiently.

Equilibrium Spacing

Equilibrium spacing is a critical concept when examining how fish maintain their positions within a school. It refers to the distance at which the forces of repulsion and attraction between fish balance each other out. At this point, the net force \( F(r) \) acting on the fish is zero, meaning neither attraction nor repulsion prevails.
In mathematical terms, this occurs at:\[ F(r) = 0 \]From the given energy model, solving this yields:\[ 6e^{-2r} = 2e^{-r} \]Simplifying gives:\[ 3 = e^r \] leading to \( r = \ln 3 \).
This equilibrium distance ensures that fish maintain a stable position relative to each other, which is essential for the collective dynamics of schooling. When fish are at equilibrium spacing, they conserve energy and enjoy a balance that helps prevent collision or scattering.

Fish Schooling Behavior

Fish schooling is a fascinating behavior that provides numerous benefits beyond just physical spacing advantages. The complex patterns and synchronized swimming seen in these schools often arise from simple interaction rules such as those modeled by energy and force equations.
Here are some key aspects:

• Predator Avoidance: By swimming in coordinated groups, fish can confuse predators, making it harder for them to single out individual targets.
• Increased Efficiency: Schooling reduces hydrodynamic drag for individuals, similar to the slipstreaming seen in cyclist pelotons, allowing fish to conserve energy.
• Information Sharing: School members can quickly communicate changes in direction or speed, enabling rapid responses to environmental changes.

Overall, schooling involves a finely tuned balance of social interaction signals, allowing these aquatic creatures to thrive in challenging environments by promoting survival and efficiency.