Problem 11 - Calculus for Biology and Medicine

Question

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-0.2 x_{1}+0.1 x_{2}\) \(\frac{d x_{2}}{d t}=-0.1 x_{2}\)

Step-by-Step Solution

Answer

Diagram has arrows: from \(x_2\) to \(x_1\), and outflows from \(x_1\) and \(x_2\).

Step 1: Analyze the Differential Equations

Look at the given system of differential equations:1. \(\frac{d x_{1}}{d t}=-0.2 x_{1}+0.1 x_{2}\)2. \(\frac{d x_{2}}{d t}=-0.1 x_{2}\)These equations represent how the quantities \(x_1\) and \(x_2\) change over time.

Step 2: Identify Flow Terms in Equations

In \(\frac{d x_{1}}{d t}=-0.2 x_{1}+0.1 x_{2}\), the term \(-0.2 x_{1}\) represents an outflow from \(x_1\), and \(0.1 x_{2}\) represents an inflow to \(x_1\) from \(x_2\).In \(\frac{d x_{2}}{d t}=-0.1 x_{2}\), the term \(-0.1 x_{2}\) represents an outflow from \(x_2\). There are no terms indicating inflow to \(x_2\).

Step 3: Construct the Compartment Diagram

Create a diagram with two compartments:- Compartment 1: \(x_1\)- Compartment 2: \(x_2\)Draw an arrow from \(x_2\) to \(x_1\) representing the inflow term \(0.1 x_{2}\). Indicate outflow from \(x_1\) with a self-loop arrow labeled \(-0.2\), and outflow from \(x_2\) with a self-loop arrow labeled \(-0.1\).

Key Concepts

Compartment Diagram

A compartment diagram is a useful visual tool used to represent systems of differential equations, often seen in biology and engineering. Essentially, it shows how different compartments or groups interact with each other over time through the exchange of quantities. In our case, these compartments are represented by \( x_1 \) and \( x_2 \).

To construct a compartment diagram:

• Draw a box for each variable in your system. So, draw one box for \( x_1 \) and another for \( x_2 \).
• Use arrows to denote flows between compartments. The arrow from \( x_2 \) to \( x_1 \) represents the inflow to \( x_1 \) from \( x_2 \) as described by the term \( 0.1x_2 \).
• Self-loop arrows are used to indicate outflows from the compartments. For instance, an arrow that starts and ends in \( x_1 \) would represent the outflow \( -0.2x_1 \) from \( x_1 \). Similarly, \( x_2 \) would have a self-loop arrow demonstrating the outflow term \( -0.1x_2 \).

These elements help visualize the dynamic changes and relationships involved for each variable within the system.

Inflow and Outflow Terms

Inflow and outflow terms define how quantities move in and out of compartments over time, crucial for understanding the dynamics of a system described by differential equations.

In differential equations, the inflow and outflow terms specify the rate of change. They help us see which variables interact and how.

• Consider \( \frac{d x_{1}}{d t}=-0.2 x_{1}+0.1 x_{2} \): here, \(-0.2 x_{1}\) is an outflow, indicating the quantity within \( x_1 \) is decreasing over time. This decrease is relative to \( x_1 \) itself.
• The term \(0.1 x_{2}\) is an inflow for \( x_1 \). It shows that part of \( x_2 \) is contributing to the quantity within \( x_1 \), essentially flowing from \( x_2 \) to \( x_1 \).
• In the second equation, \( \frac{d x_{2}}{d t}=-0.1 x_{2} \), \(-0.1 x_{2}\) indicates an outflow. Here, \( x_2 \) is losing quantity at this rate, and no inflow into \( x_2 \) is present.

These terms are pivotal in tracking how each compartment or variable is influenced over time.

System of Equations

Differential equations are equations that relate a function with its derivatives, and they are fundamental in modeling how a system evolves over time. When multiple equations are combined, we have a system of equations.

A system of equations, such as given:

• \( \frac{d x_{1}}{d t}=-0.2 x_{1}+0.1 x_{2} \)
• \( \frac{d x_{2}}{d t}=-0.1 x_{2} \)

provides a framework to analyze how two or more related variables change together. Each equation describes the dynamics of a different variable, and these variables are interconnected through their respective inflow and outflow terms.

Understanding a system of differential equations often involves decomposing the system to see how each component influences the others over time, allowing us to predict future behavior or find equilibrium points. Knowing their behavior through compartment diagrams and inflows/outflows enables comprehensible simulations and predictions of complex real-world systems.