Problem 8 - Calculus for Biology and Medicine

Question

Consider $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{2} \\ \frac{d x_{2}}{d t}=x_{1}+x_{2} \end{array} $$ Determine the direction vectors associated with the following points in the \(x_{1}-x_{2}\) plane, and graph the direction vectors in the \(x_{1}-x_{2}\) plane: \((1,0),(0,1),(-1,0),(0,-1),(1,1),(0,0)\), and \((-2,-2)\)

Step-by-Step Solution

Answer

Direction vectors are: (0,1), (1,1), (0,-1), (-1,-1), (1,2), (0,0), (-2,-4).

Step 1: Understanding Direction Vectors

Direction vectors give the direction of the system's flow at a given point in the phase plane. The system of equations provides the derivatives \( \frac{d x_{1}}{d t} = x_{2} \) and \( \frac{d x_{2}}{d t} = x_{1} + x_{2} \) as the direction vector \( (\frac{d x_{1}}{d t}, \frac{d x_{2}}{d t}) \).

Step 2: Calculate Direction Vector at (1, 0)

Substitute \( x_{1} = 1 \) and \( x_{2} = 0 \) into the derivatives: \( \frac{d x_{1}}{d t} = 0 \) and \( \frac{d x_{2}}{d t} = 1 + 0 = 1 \). Thus, the direction vector is \( (0, 1) \).

Step 3: Calculate Direction Vector at (0, 1)

Substitute \( x_{1} = 0 \) and \( x_{2} = 1 \): \( \frac{d x_{1}}{d t} = 1 \) and \( \frac{d x_{2}}{d t} = 0 + 1 = 1 \). The direction vector is \( (1, 1) \).

Step 4: Calculate Direction Vector at (-1, 0)

Substitute \( x_{1} = -1 \) and \( x_{2} = 0 \): \( \frac{d x_{1}}{d t} = 0 \) and \( \frac{d x_{2}}{d t} = -1 + 0 = -1 \). The direction vector is \( (0, -1) \).

Step 5: Calculate Direction Vector at (0, -1)

Substitute \( x_{1} = 0 \) and \( x_{2} = -1 \): \( \frac{d x_{1}}{d t} = -1 \) and \( \frac{d x_{2}}{d t} = 0 + (-1) = -1 \). The direction vector is \( (-1, -1) \).

Step 6: Calculate Direction Vector at (1, 1)

Substitute \( x_{1} = 1 \) and \( x_{2} = 1 \): \( \frac{d x_{1}}{d t} = 1 \) and \( \frac{d x_{2}}{d t} = 1 + 1 = 2 \). The direction vector is \( (1, 2) \).

Step 7: Calculate Direction Vector at (0, 0)

Substitute \( x_{1} = 0 \) and \( x_{2} = 0 \): \( \frac{d x_{1}}{d t} = 0 \) and \( \frac{d x_{2}}{d t} = 0 + 0 = 0 \). The direction vector is \( (0, 0) \).

Step 8: Calculate Direction Vector at (-2, -2)

Substitute \( x_{1} = -2 \) and \( x_{2} = -2 \): \( \frac{d x_{1}}{d t} = -2 \) and \( \frac{d x_{2}}{d t} = -2 + (-2) = -4 \). The direction vector is \( (-2, -4) \).

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They play a critical role in modeling the behavior of dynamic systems. In this context, the function typically represents a physical quantity, and its derivative represents the rate of change of that quantity. We are looking at a set of first-order differential equations for variables \( x_1 \) and \( x_2 \):\[ \frac{d x_{1}}{d t} = x_{2} \]and\[ \frac{d x_{2}}{d t} = x_{1} + x_{2} \].
This system tells us how \( x_1 \) and \( x_2 \) change with respect to time \( t \). By solving these equations, you determine how the variables evolve over time, which is essential for predicting future states of the system. Each term on the right hand side represents influences contributing to the changes in \( x_1 \) and \( x_2 \).
When solving, we often seek a general solution that encompasses the behavior across all initial conditions. This involves integrating the system, which might lead to discovering stable points, oscillations, or exponential behaviors. Differential equations like these are foundational in fields like physics, biology, and economics, allowing us to describe phenomena ranging from population growth to the movement of celestial bodies.

Phase Plane

The phase plane is a graphical representation of a dynamical system in terms of its variables and their rates of change. For our system of differential equations, the phase plane is constructed using the axes of \( x_1 \) and \( x_2 \), displaying how the variables evolve over time. Each point \( (x_1, x_2) \) in the phase plane represents a state of the system at a given time.
The idea is to plot direction vectors at various points in this plane. These vectors are derived from the differential equations and indicate the direction in which the system will evolve from that state. For example, at the point \( (1, 0) \), the direction vector \( (0, 1) \) tells us that the system state will move upward in the \( x_2 \) direction without a change in \( x_1 \).
The phase plane allows us to visualize the behavior of the system over time, showing patterns such as cycles, spirals, or node formations. It transforms the abstract formulae into tangible trajectories enabling a better understanding of long-term behaviors without solving the equations analytically. By analyzing a phase plane, students can quickly grasp stability and predict how systems respond to changes.

System of Equations

A system of equations involves multiple equations working together to define a set of relationships between variables. In this exercise, the system of equations consists of:

• \( \frac{d x_{1}}{d t} = x_{2} \)
• \( \frac{d x_{2}}{d t} = x_{1} + x_{2} \)

These describe the rates of change for two interdependent variables \( x_1 \) and \( x_2 \).
In mathematical modeling, systems of equations allow us to describe complex interactions between variables in a concise way. They are particularly useful when dealing with multi-variable systems where the dynamic of one variable affects another.
Solving systems of equations analytically or numerically can uncover critical insights, such as equilibrium points where the system doesn't change over time, as well as information about the stability or instability of these points. Direction vectors obtained from these equations, as in the given exercise, give a detailed snapshot of how the system transitions from one state to another. This is invaluable when analyzing real-world systems, ranging from electrical circuits to ecological models, to predict their future behavior and inform decision-making processes.