Problem 24 - Calculus for Biology and Medicine

Question

Fish Schooling One model that is used for the interactions between animals, including fish in a school, is that the fish have an energy of interaction that is given by a Morse potential: $$ V(r)=e^{-r}-A e^{-a r} r>0 $$ The fish will attract or repel each other until they reach a distance that minimizes the function \(V(r) .\) The coefficients \(A\) and \(a\) are positive numbers. (a) Assume initially that \(a=1 / 2\) and \(A=1\), what is the behavior of \(V(r)\) as \(r \rightarrow 0\). What is the behavior of \(V(r)\) as \(r \rightarrow+\infty\) ? (b) Find the value of \(r\) that minimizes \(V(r)\). (c) Explain what happens to the spacing that minimizes the energy of interaction if \(a=1 / 2\) and \(A=4\) ?

Step-by-Step Solution

Answer

(a) \( V(r) \to 0 \) as both \( r \to 0 \) and \( r \to \infty \). (b) \( r = \frac{2}{\ln(2)} \). (c) Spacing remains unchanged with \( A = 4 \) for given \( a \).

Step 1: Analyze Behavior as r Approaches 0

When \( r \to 0 \), the exponential term \( e^{-r} \) approaches 1 and the term \( e^{-ar} \) also approaches 1. Thus, the Morse potential \( V(r) = e^{-r} - A e^{-ar} \) approaches \( 1 - A \). Given that \( A = 1 \), \( V(r) \) approaches 0 as \( r \to 0 \).

Step 2: Analyze Behavior as r Approaches Infinity

As \( r \to +\infty \), both \( e^{-r} \) and \( e^{-ar} \) approach 0 because exponential functions decay to zero for large \( r \). Therefore, \( V(r) = e^{-r} - A e^{-ar} \) also approaches 0 as \( r \to +\infty \).

Step 3: Find the Value of r that Minimizes V(r)

To minimize \( V(r) = e^{-r} - A e^{-ar} \), we find the derivative \( V'(r) \) and set it to zero. The derivative is given by \( V'(r) = -e^{-r} + Aa e^{-ar} \). Setting \( V'(r) = 0 \) gives \( -e^{-r} + Aa e^{-ar} = 0 \) or \( e^{-r} = Aa e^{-ar} \). Solving for \( r \), we find \( e^{(a-1)r} = Aa \). When \( A = 1 \) and \( a = \frac{1}{2} \), \( Aa = \frac{1}{2} \) so \( e^{-rac{1}{2}r} = \frac{1}{2} \), leading to \( r = -\frac{2}{\ln{1/2}} = \frac{2}{\ln 2} \).

Step 4: Effect of Changing A to 4

When \( a = \frac{1}{2} \) and \( A = 4 \), we set \( e^{-r} = 2e^{-\frac{r}{2}} \) from Step 3. Solving \( e^{-\frac{1}{2}r} = 1/2 \) gives \( r = \frac{2}{\ln 2} \). Hence, the spacing remains the same because the equation reduces to the same mathematical form, indicating that the spacing is driven by \( a \) rather than \( A \).

Key Concepts

Morse potential

The Morse potential is a mathematical function used to model interactions between particles or bodies, such as fish in a school, or atoms in molecules. This potential function describes how entities influence each other across a distance, either attracting or repelling as needed to minimize energy. In its mathematical form, Morse potential is written as: \[ V(r) = e^{-r} - A e^{-a r} \] where \( r \) is the distance between two interacting particles. - **Exponential Terms**: The function contains two exponential terms. Each of these terms decays at different rates because of the coefficients \( A \) and \( a \). - **Attraction and Repulsion**: As the distance \( r \) changes, the values of \( e^{-r} \) and \( A e^{-a r} \) change, defining whether the particles attract or repel each other.- **Parameters**: Parameters \( A \) and \( a \) affect the depth and width of the potential well created by \( V(r) \). This potential helps in understanding how particles stabilize at a certain distance where energy is minimized, crucial for studying biological systems or molecular bonds.

exponential decay

Exponential decay describes a process where some quantity decreases at a rate proportional to its current value, leading to a rapid decline followed by a gradual slowdown. It plays a significant role in the behavior of biological systems and chemical interactions modeled by the Morse potential.- **Behavior at Small \( r \)**: At small values of \( r \), the exponential term \( e^{-r} \) approaches 1 very quickly, while \( e^{-a r} \), depending on \( a \), also nears 1. This causes the Morse potential \( V(r) \) to approach \( 1 - A \).- **Behavior at Large \( r \)**: As \( r \) becomes very large, both terms, \( e^{-r} \) and \( e^{-a r} \), shrink towards zero. Thus, \( V(r) \) simplifies to nearly zero, reflecting a state where interaction energy is minimized.- **Applications**: Exponential decay in the Morse potential allows for modeling how energies resolve to zero over large distances, affecting how entities stabilize into configurations.This understanding aids in predicting how biological organisms, like fish, adjust their distances to achieve optimal energy states in groups.

minimization problem

The minimization problem involves finding the distance at which the Morse potential \( V(r) \) achieves its smallest value. This distance represents the most energetically favorable state for two interacting particles.- **Derivative Approach**:
To determine the minimizing \( r \), one must calculate the derivative of \( V(r) \) and set it to zero: \[ V'(r) = -e^{-r} + Aa e^{-ar} = 0 \] Solving this equation isolates values of \( r \) where the interaction energy is minimized. This involves equating \( e^{-r} = Aa e^{-ar} \).- **Impact of Parameters**: In the given scenario, parameters \( A \) and \( a \) greatly influence \( r \). For example, with \( a = \frac{1}{2} \) and \( A = 1 \), the distance \( r \) that minimizes energy is established by the natural log relationship \( r = \frac{2}{\ln 2} \).- **Practical Implications**: Understanding this minimizes energy calculation benefits fields like biology. Here, it determines how organisms optimally space themselves in an ecosystem for minimal metabolic and interaction losses.Minimization offers insights into why and how systems retain stability, especially relevant in ecological and molecular behavior studies.