Problem 16 - Calculus for Biology and Medicine

Question

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-0.2 x_{1}+0.4 x_{2}\) \(\frac{d x_{2}}{d t}=0.2 x_{1}-0.4 x_{2}\)

Step-by-Step Solution

Answer

The compartment diagram includes two compartments with flows: \(x_1\to x_2\) (rate 0.2), \(x_2\to x_1\) (rate 0.4), and self-loops \(x_1\to x_1\) (rate 0.2) and \(x_2\to x_2\) (rate 0.4).

Step 1: Identify Variables and Parameters

The system of differential equations involves two variables, \(x_1\) and \(x_2\), representing compartments. The terms with these variables indicate the rate of change of each compartment with respect to time \(t\). The coefficients in front of the variables represent the rates between compartments.

Step 2: Determine Flow Between Compartments

The equation \(\frac{dx_1}{dt} = -0.2x_1 + 0.4x_2\) indicates a flow out of compartment 1 with rate 0.2 times its current state, and a flow into compartment 1 from compartment 2 at 0.4 times the state of compartment 2. Similarly, \(\frac{dx_2}{dt} = 0.2x_1 - 0.4x_2\) represents a flow from compartment 1 into compartment 2 and a flow out of compartment 2 back to itself.

Step 3: Draw the Compartment Diagram

The compartment diagram consists of two compartments: one for \(x_1\) and another for \(x_2\). Draw an arrow from compartment \(x_1\) to itself with a rate of \(0.2\), indicating self-transitioning or flow out of the compartment. Draw another arrow from \(x_2\) to \(x_1\) with a rate of \(0.4\), and from \(x_1\) to \(x_2\) with rate \(0.2\). Finally, draw an arrow from \(x_2\) to itself with rate \(0.4\), indicating flow out of the compartment.

Key Concepts

Understanding the Compartment Diagram

A compartment diagram visually represents the components involved in a system of differential equations. Each variable, such as \( x_1 \) and \( x_2 \) in the problem, is depicted as a separate compartment. These compartments can represent anything that changes over time like populations, chemicals, or even fluid dynamics. In our case, they reflect the variables affected by the flows between them.
When drawing a compartment diagram for a system of equations:

• Each equation represents a compartment.
• Arrows between compartments show the movement or flow between them.
• Rates that affect the flow are depicted next to the arrows, showing the strength or speed of this transfer.

By observing the arrows and their rates between the compartments \( x_1 \) and \( x_2 \), we can easily understand how these elements interact over time. This visualization significantly helps in understanding complex systems at a glance.

Exploring the Rate of Change

The rate of change is a concept in which we measure how a particular quantity changes over time. In differential equations, this is expressed by the derivative \( \frac{dx}{dt} \). For our equation:

• \( \frac{dx_1}{dt} = -0.2x_1 + 0.4x_2 \) indicates how \( x_1 \) changes over time based on its own state and the influence from \( x_2 \).
• The negative sign before the term \( 0.2x_1 \) implies \( x_1 \) decreases by 20% of its value over a unit of time, representing a loss from \( x_1 \).
• Conversely, \( 0.4x_2 \) indicates that \( x_1 \) increases by an amount equating to 40% of \( x_2 \)'s state over the same timeframe.

This depiction of rate of change gives a clear indication of which compartments are depleting and which are gaining, forming a foundation for the dynamic changes occurring in the system.

Understanding Flow Between Compartments

The flow between compartments captures how quantities move or transfer between different sections of a system. In the given equations, we see how \( x_1 \) and \( x_2 \) interact through this flow.

• \( \frac{dx_1}{dt} = -0.2x_1 + 0.4x_2 \) shows that part of \( x_1 \) moves to \( x_2 \) or potentially out of the system, while there is also an inflow from \( x_2 \) to \( x_1 \).
• Similarly, \( \frac{dx_2}{dt} = 0.2x_1 - 0.4x_2 \) explains how \( x_2 \) receives some amount from \( x_1 \) but simultaneously experiences a loss equivalent to 40% of its own quantity.

This movement represented through arrows in a compartment diagram provides excellent insights into understanding the material or information flow within the model. It emphasizes interconnections and dependencies between compartments, reflecting the dynamic balancing act occurring in the system.