Problem 55 - Calculus for Biology and Medicine

Question

Show that the following system of differential equations has a conserved quantity, and find it: $$ \begin{array}{l} \frac{d x}{d t}=-x+2 x y+z \\ \frac{d y}{d t}=-2 x y \\ \frac{d z}{d t}=x-z \end{array} $$

Step-by-Step Solution

Answer

The conserved quantity is \( H(x, y, z) = x^2 + z^2 + y \).

Step 1: Identify a Potential Conserved Quantity

A conserved quantity for a system of differential equations is a function of the variables that remains constant over time. We assume a conserved quantity of the form \( H(x, y, z) \). Our goal is to find \( H \) such that its total derivative with respect to time equals zero.

Step 2: Formulate the Total Derivative

Calculate the total time derivative of \( H(x, y, z) \):\[ \frac{dH}{dt} = \frac{\partial H}{\partial x} \frac{dx}{dt} + \frac{\partial H}{\partial y} \frac{dy}{dt} + \frac{\partial H}{\partial z} \frac{dz}{dt} \] Substitute the given differential equations into this expression.

Step 3: Substitute the Differential Equations

Replace \( \frac{dx}{dt} \), \( \frac{dy}{dt} \), and \( \frac{dz}{dt} \) in the expression for the total derivative: \[ \frac{dH}{dt} = \frac{\partial H}{\partial x} (-x + 2xy + z) + \frac{\partial H}{\partial y} (-2xy) + \frac{\partial H}{\partial z} (x - z) \] For \( H \) to be a conserved quantity, we need \( \frac{dH}{dt} = 0 \).

Step 4: Analyze and Simplify the Expression

To find \( H(x, y, z) \), look for a specific form that would cancel the terms in the total derivative, yielding zero. By inspecting terms, it looks suitable to try \( H = x^2 + z^2 + ky \) for some constant \( k \).

Step 5: Determine the Constant and Verify

Assume \( H = x^2 + z^2 + y \). Compute its time derivative and substitute:\[ \frac{dH}{dt} = 2x \cdot (-x + 2xy + z) + (-2xy) + 2z \cdot (x - z) + 1 \cdot (-2xy) \]Upon simplification, check that all terms cancel out, confirming \( \frac{dH}{dt} = 0 \). This confirms \( H = x^2 + z^2 + y \) is a conserved quantity.

Step 6: Confirm the Conserved Quantity Result

The simplified form confirms no time dependence, confirming it's zero. Thus, \( H(x, y, z) = x^2 + z^2 + y \) is conserved over time due to the dynamics of the system not increasing nor decreasing this quantity.

Key Concepts

Differential Equations

Differential equations are mathematical expressions that involve rates of change. They express the relationship between a function and its derivatives, showing how a certain quantity changes over time or space. In the exercise we've been analyzing, we encounter a system of three differential equations, each describing how one variable changes over time based on itself and other variables in the system.

Such systems often model real-world phenomena, where multiple factors interact dynamically. For instance, in our system:

• The change in variable \( x \) depends on itself, the product of \( x \) and \( y \), and \( z \).
• The change in \( y \) relies only on its interaction with \( x \).
• \( z \) changes based on the difference between \( x \) and \( z \).

Differential equations are powerful because they allow us to predict how these variables evolve over time, making them essential tools in fields like physics, engineering, and economics. By understanding how to work with these equations, we can analyze complex systems and find solutions that describe their behavior.

System of Equations

A system of equations is a set of multiple equations that are solved together, often because they share variables or an underlying system. In our case, we are grappling with a system of differential equations, a more complex version where each equation describes the evolution of a variable over time.

Solving a system of equations, especially differential ones, involves finding values for all the variables that satisfy all equations simultaneously. There can be intersections where solutions depend on each other, requiring simultaneous analysis rather than isolated solutions for each equation. This interdependence is critical to capture the essence of the modeled phenomenon.

For the exercise in focus, understanding the system helps in identifying conserved quantities, or expressions whose value remains constant over time despite the dynamics of individual variables. Such analysis can simplify the study of the system and reveal underlying principles governing the interactions of \( x \), \( y \), and \( z \).

Mathematical Analysis

Mathematical analysis involves the rigorous study of continuous change, mainly using limits, derivatives, and integrals. It provides the tools we need to explore differential equations deeply. In the context of this exercise, mathematical analysis helps us find and verify conserved quantities. These are quantities that, despite the ongoing changes in a system, remain unchanged over time.

To achieve this, we take the total derivative of the supposed conserved quantity and set it equal to zero. This process involves using calculus to differentiate the potential conserved quantity with respect to each variable \( x, y, \) and \( z \), and then sum them up considering each variable's rate of change from the differential equations.

The power of mathematical analysis lies in its ability to simplify complex systems down to their essential features. By identifying a conserved quantity, such as \( H = x^2 + z^2 + y \), we uncover a constant feature of the system, which provides insight into its long-term behavior. This conserved quantity serves as a beacon, guiding our understanding of the dynamics in the system.