Problem 26 - Calculus for Biology and Medicine

Question

Love-Struck Romeo and Juliet Romeo and Juliet are both reckless lovers; being in love intoxicates each of them and causes their love to increase, regardless of the feelings of the other. So if Romeo starts off even mildly fond of Juliet (i.e., \(R>0\) ), then his love will grow regardless of her feelings. Juliet behaves similarly. Conversely, if Romeo hates Juliet, his hatred will grow by itself. We therefore anticipate that if \(R>0\), then \(d R / d t>0\), while if \(R<0\), then \(d R / d t<0\), and similarly for \(J .\) We model Romeo and Juliet's relationship by a system of differential equations. $$ \begin{array}{l} \frac{d J}{d t}=k J \\ \frac{d R}{d t}=k R \end{array} $$ where \(k\) is a positive constant. (a) Write the system (11.57) as a matrix equation, and find the eigenvalues of the corresponding matrix. Why can we not use the methods from Section \(11.1\) to classify the equilibrium \((0,0)\) ? (b) Solve the equations (11.57) directly to find \(R(t), J(t)\) if \(R(0)=1, J(0)=-1\). What is the fate of Romeo and Juliet's relationship as \(t \rightarrow \infty\) ? (c) Try to describe generally what happens to Romeo and Juliet's relationship as \(t \rightarrow \infty\) for all possible initial conditions.

Step-by-Step Solution

Answer

Their feelings grow exponentially, Juliet's negatively and Romeo's positively; the relationship diverges.

Step 1: Write the system as a matrix equation

The system of differential equations can be expressed as a matrix equation. We write the derivatives with respect to time as a vector, \( \begin{pmatrix} \frac{dJ}{dt} \ \frac{dR}{dt} \end{pmatrix} \), and the rates of change in terms of a matrix \( A \) applied to the vector \( \begin{pmatrix} J \ R \end{pmatrix} \): \[ \begin{pmatrix} \frac{dJ}{dt} \ \frac{dR}{dt} \end{pmatrix} = \begin{pmatrix} k & 0 \ 0 & k \end{pmatrix} \begin{pmatrix} J \ R \end{pmatrix}. \]

Step 2: Find the eigenvalues of the matrix

To find the eigenvalues of the matrix \( A = \begin{pmatrix} k & 0 \ 0 & k \end{pmatrix} \), we solve the characteristic equation \( \det(A - \lambda I) = 0 \). This becomes: \[ \det\left(\begin{pmatrix} k - \lambda & 0 \ 0 & k - \lambda \end{pmatrix} \right) = (k - \lambda)^2 = 0. \] Thus, both eigenvalues are \( \lambda = k \).

Step 3: Reason for inability to classify equilibrium

The methods from Section 11.1 require distinct eigenvalues to classify equilibria. Since both eigenvalues are the same \( (\lambda = k) \), we cannot use those methods to determine the behavior around the equilibrium at \( (0,0) \).

Step 4: Solve the differential equations directly

Each differential equation is separable and can be solved independently. For \( J(t) \), the solution is \( \frac{dJ}{dt} = kJ \), giving rise to: \[ J(t) = J(0)e^{kt}. \] Similarly, for \( R(t) \), \( \frac{dR}{dt} = kR \) becomes \[ R(t) = R(0)e^{kt}. \]

Step 5: Apply initial conditions

Using the given initial conditions \( J(0) = -1 \) and \( R(0) = 1 \): \[ J(t) = -e^{kt}, \] \[ R(t) = e^{kt}. \]

Step 6: Analyze the fate of the relationship as \( t \rightarrow \infty \)

As \( t \rightarrow \infty \), both expressions \( -e^{kt} \) for \( J(t) \) and \( e^{kt} \) for \( R(t) \) grow unbounded in magnitude. Juliet's feelings become strongly negative, whereas Romeo's become strongly positive, indicating increasing divergence. Their relationship is doomed to further mismatch.

Step 7: General description of the relationship evolving as \( t \rightarrow \infty \)

For any initial conditions \((J_0, R_0)\), Juliet's feelings will grow exponentially negative if \(J_0 < 0\), and positive if \(J_0 > 0\). Similarly, Romeo's feelings will grow positively if \(R_0 > 0\) and negatively if \(R_0 < 0\). No matter the initial conditions, their feelings intensify over time, leading to either strongly positive or negative feelings.

Key Concepts

Understanding Eigenvalues in Differential Equations

In the context of differential equations, eigenvalues play a crucial role in determining the behavior of dynamical systems like Romeo and Juliet's relationship. Eigenvalues are derived from a matrix associated with a system and indicate how solutions behave over time. In our exercise, this system of differential equations is represented by a matrix with a single eigenvalue, \( \lambda = k \).
This particular eigenvalue appears twice because of the form of the matrix, \( A = \begin{pmatrix} k & 0 \ 0 & k \end{pmatrix} \). This means we have a repeated eigenvalue, a condition known as a degenerate eigenvalue.

Usually, each unique eigenvalue would provide insight into the stability of equilibrium points. However, the fact that both eigenvalues are identical means that we can't distinguish unique behaviors for different variables based solely on eigenvalue analysis. Therefore, while we determine that both Romeo and Juliet's affections grow exponentially in magnitude, the identical eigenvalues limit our ability to classify the equilibrium at \((0,0)\) more precisely using traditional methods.
In summary, eigenvalues are essential for understanding how the components of a system influence each other, and identical eigenvalues present special challenges in classifying system behavior.

Role of Initial Conditions in Dynamical Systems

Initial conditions specify the state of a system at the beginning of observation, and in differential equations, knowing these values allows us to predict future behavior. In the case of Romeo and Juliet, the initial conditions are given by \( J(0) = -1 \) and \( R(0) = 1 \).
These values tell us that at the initial moment, Juliet has a slight negative affection towards Romeo, while Romeo feels positively about Juliet.

• For \( J(t) \), this negative starting point means her affection will become exponentially more negative over time.
• For \( R(t) \), starting at \( R(0) = 1 \) suggests Romeo's positive feeling will grow larger as time passes.

Initial conditions act like seeds that, when combined with the "growth rule" dictated by the differential equations, determine the trajectory of the system over time. In this particular example, the system diverges based on these initial seeds, leading to increasing mismatch in their affections, highlighting how powerful initial conditions can be in shaping the outcomes of dynamical systems.

Understanding the Role of a Matrix Equation

The transformation of a system of differential equations into a matrix equation simplifies complex computations and reveals relationships in a structured form. In our analysis of Romeo and Juliet, the matrix equation is expressed as:
\[ \begin{pmatrix} \frac{dJ}{dt} \ \frac{dR}{dt} \end{pmatrix} = \begin{pmatrix} k & 0 \ 0 & k \end{pmatrix} \begin{pmatrix} J \ R \end{pmatrix}. \]
The matrix equation elegantly encapsulates how Juliet's and Romeo's rates of change relate to their current emotional states. Here, the diagonal matrix \( A \) indicates that each variable, \( J \) and \( R \), evolves independently, as indicated by the zeros off the diagonal.
This form makes it straightforward to recognize that both lovers' feelings evolve separately according to the same exponential function \( e^{kt} \), determined by the constant \( k \).

By establishing this matrix representation, we can utilize linear algebra concepts such as eigenvalues and eigenvectors to further analyze the system. In this relationship model, the matrix equation reveals that neither lover's feelings influence the other, reinforcing their paths towards emotional divergence unless external factors intervene. It’s a prime example of how matrix equations serve as powerful tools for decoding complex interdependencies in dynamic systems.