Problem 12 - Calculus for Biology and Medicine

Question

Find the corresponding compartment diagram for each system of differential equations. \(\frac{d x_{1}}{d t}=-0.2 x_{1}+1.1 x_{2}\) \(\frac{d x_{2}}{d t}=0.2 x_{1}-1.1 x_{2}\)

Step-by-Step Solution

Answer

Diagram includes \(x_1\) and \(x_2\) with bidirectional arrows showing inflows and outflows based on rates.

Step 1: Identify Variables and Their Rates

The system consists of two variables \(x_1\) and \(x_2\). The rates of change for these variables are given by the equations: \(\frac{d x_1}{d t} = -0.2 x_1 + 1.1 x_2\) for \(x_1\) and \(\frac{d x_2}{d t} = 0.2 x_1 - 1.1 x_2\) for \(x_2\). These rates show how each variable affects the other.

Step 2: Determine Incoming and Outgoing Connections

For \(x_1\), the term \(-0.2 x_1\) indicates loss proportional to its own quantity, and \(1.1 x_2\) indicates an incoming contribution from \(x_2\). For \(x_2\), \(0.2 x_1\) indicates an incoming contribution from \(x_1\), while \(-1.1 x_2\) indicates a loss proportional to its own quantity.

Step 3: Draw the Compartment Diagram

Create a diagram with two compartments labeled \(x_1\) and \(x_2\). Draw an arrow from \(x_2\) to \(x_1\) indicating an inflow to \(x_1\) with a rate of 1.1. Draw another arrow from \(x_1\) to \(x_2\) for an inflow to \(x_2\) with a rate of 0.2. Each compartment should also show its own outflow, \(0.2 x_1\) for \(x_1\) and \(1.1 x_2\) for \(x_2\). This visualizes the interaction: \(x_1\) and \(x_2\) both contribute to each other while also having self-limiting factors.

Key Concepts

System of Differential Equations

A system of differential equations is a collection of two or more interrelated differential equations. These equations involve derivatives of functions and are used to describe complex dynamic systems. In our exercise, we have two equations that form the system:

• \( \frac{d x_{1}}{d t} = -0.2 x_{1} + 1.1 x_{2} \)
• \( \frac{d x_{2}}{d t} = 0.2 x_{1} - 1.1 x_{2} \)

This system outlines how the rates of change of the variables \( x_1 \) and \( x_2 \) are dependent on each other. These equations model a scenario in which two compartments, represented by the variables, both influence and are influenced by one another. The primary goal of solving these systems is to understand how such variables evolve over time under given conditions. Solving them can involve various methods, such as matrix techniques or numerical solvers, especially when tackling more complex systems.

Rates of Change

The rate of change in a differential equation describes how a variable changes with respect to time or another variable. In simple terms, it's the speed at which something changes. In our system, each variable's equation reflects its rate of change:

• For \( x_1 \), the rate is \( \frac{d x_{1}}{dt} = -0.2 x_1 + 1.1 x_2 \)
• For \( x_2 \), the rate is \( \frac{d x_{2}}{dt} = 0.2 x_1 - 1.1 x_2 \)

The rate \( -0.2x_1 \) for example indicates how much \( x_1 \) decreases by itself over time, while \( 1.1x_2 \) shows how \( x_2 \) contributes to the growth of \( x_1 \). Rates of change are crucial because they offer insights into the dynamic behavior of systems. Understanding these can help predict future behaviors or balance interactions within the system.

Interaction between Variables

The interaction between variables in a system of differential equations is fundamental in understanding how one variable influences another over time. This interaction is represented in our system by the following terms:

• The term \( 1.1 x_{2} \) in the first equation shows that \( x_2 \) has a positive influence on \( x_1 \), contributing to its increase.
• Similarly, the term \( 0.2 x_{1} \) in the second equation reveals that \( x_1 \) positively affects \( x_2 \).

These kinds of interactions are typical in systems describing ecological, biological, or economic relationships where one entity's growth or decline can directly influence another's. Recognizing and correctly diagramming these interactions in a compartment diagram can offer a visual insight, helping better grasp these mathematical relationships and forecast any dynamic system's behavior.