Problem 17 - Calculus

Question

analyze the phase plane of the differential equations for \(x, y \geq 0 .\) Show the nullclines and equilibrium points, and sketch the direction of the trajectories in each region. $$\begin{aligned} &\frac{d x}{d t}=x\left(1-x-\frac{y}{3}\right)\\\ &\frac{d y}{d t}=y\left(1-y-\frac{x}{2}\right) \end{aligned}$$

Step-by-Step Solution

Answer

The nullclines are \( x=0, y=0, 1-x-\frac{y}{3}=0, 1-y-\frac{x}{2}=0 \), with equilibrium points at \( (0, 0), (1, 0), (0, 1), (\frac{3}{5}, \frac{2}{5}) \).

Step 1: Identify Nullclines

For the system of equations given, identify the nullclines. The nullclines occur where the derivatives equal zero. For \( \frac{d x}{d t} = x(1 - x - \frac{y}{3}) = 0 \), the nullclines are \( x = 0 \) and \( 1 - x - \frac{y}{3} = 0 \). For \( \frac{d y}{d t} = y(1 - y - \frac{x}{2}) = 0 \), the nullclines are \( y = 0 \) and \( 1 - y - \frac{x}{2} = 0 \).

Step 2: Solve for Equilibrium Points

Set both derivatives to zero and solve the equations simultaneously to find equilibrium points. Solve \( 1 - x - \frac{y}{3} = 0 \) and \( 1 - y - \frac{x}{2} = 0 \), resulting in the equilibrium point \( (x, y) = (\frac{3}{5}, \frac{2}{5}) \). Including \( x=0 \) and \( y=0 \), the equilibrium points are \( (0, 0) \), \( (1, 0) \), \( (0, 1) \), and \( (\frac{3}{5}, \frac{2}{5}) \).

Step 3: Determine Direction of Trajectories

Analyze the direction of the trajectories in each region determined by the nullclines by determining the sign of \( \frac{d x}{d t} \) and \( \frac{d y}{d t} \) in those regions. If \( \frac{d x}{d t} > 0 \), then \( x \) increases; if \( \frac{d y}{d t} > 0 \), then \( y \) increases. Use these determinations to sketch the direction fields in each region.

Step 4: Sketch the Phase Plane

Plot the nullclines on the phase plane and mark the equilibrium points. Then, sketch arrows indicating the direction of change in each region. The \( x = 0 \) and \( y = 0 \) axes should be shown, along with the lines for the other two nullclines. This visualization shows trajectories approaching or moving away from equilibrium points based on calculations.

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. In biology, physics, and engineering, they are used to model systems where change depends on certain variables. In this exercise, we have two differential equations given:

• \( \frac{d x}{d t} = x\left(1-x-\frac{y}{3}\right) \)
• \( \frac{d y}{d t} = y\left(1-y-\frac{x}{2}\right) \)

These equations describe how the variables \(x\) and \(y\) change over time. Here, \(\frac{d x}{d t}\) and \(\frac{d y}{d t}\) represent the rate of change of \(x\) and \(y\) with respect to time. Understanding these rates helps us predict the behavior of the system as time progresses. The existence of both variable-dependent terms, \(x\) and \(y\), indicates that changes in one variable can affect the other.

Nullclines

Nullclines are specific curves in a phase plane where the rate of change of one of the variables is zero. These help us understand the boundaries within which the system can change. In our differential equation system:

• The nullcline for \( \frac{d x}{d t} = 0 \) happens when either \( x = 0 \) or when \(1 - x - \frac{y}{3} = 0\).
• For \( \frac{d y}{d t} = 0 \), the nullcline occurs when \( y = 0 \) or \( 1 - y - \frac{x}{2} = 0 \).

When you plot these nullclines on a phase plane, they form lines or curves that indicate areas where one of the variables remains constant. By analyzing nullclines, you can simplify understanding the system behavior and identify regions of potential equilibrium points.

Equilibrium Points

Equilibrium points are special points in the phase plane where both \(\frac{d x}{d t} = 0\) and \(\frac{d y}{d t} = 0\). These points represent conditions where the system is at rest, meaning there is no change in both variables. For the given differential equations, solving the nullcline equations simultaneously, we find:

• The equilibrium point at \((x, y) = (\frac{3}{5}, \frac{2}{5})\).
• Trivial equilibrium points such as \((0, 0), (1, 0), \) and \((0, 1)\).

At these points, the interactions between \(x\) and \(y\) balance out, rendering no net change over time. The stability of these points can further be analyzed to predict whether a system, if slightly perturbed, will return to these rest states or move away.

Direction Fields

Direction fields, or slope fields, provide a graphical representation of how solutions to differential equations behave. By looking at direction fields, you can anticipate the trajectories of the dynamical system over the phase plane. In the present system, for each region formed by the nullclines, direction fields help infer trajectories:

• If \(\frac{d x}{d t} > 0\), \(x\) increases, otherwise \(x\) decreases.
• If \(\frac{d y}{d t} > 0\), \(y\) increases, otherwise \(y\) decreases.

The lines and arrows representing these directions guide understanding about how \(x\) and \(y\) behave relative to each other across the plane. This visualization is crucial to see how variables change over time and approach or spiral away from equilibrium points. To gain a deeper understanding of a system's behavior over time, direction fields offer an intuitive visual tool.