The following is a model which describes the interaction
between two species of densities x and y.
x˙ = x −
1
2
xy
y˙ = −
1
4
y +
1
2
xy.
Sketch a phase portrait which shows a few trajectories about the co-existence equilibrium point in

We are given the system of differential equations that describe the interaction between two species with densities \( x \) and \( y \), where:

\[
\dot{x} = x - \frac{1}{2}xy
\]
\[
\dot{y} = -\frac{1}{4}y + \frac{1}{2}xy
\]

### Finding the equilibrium points

To find the equilibrium points, we set \(\dot{x} = 0\) and \(\dot{y} = 0\):

1. From \(\dot{x} = x - \frac{1}{2}xy = 0\), we get:
   \[
   x(1 - \frac{1}{2}y) = 0
   \]
   Therefore, either \( x = 0 \) or \( y = 2 \).

2. From \(\dot{y} = -\frac{1}{4}y + \frac{1}{2}xy = 0\), we get:
   \[
   y\left(-\frac{1}{4} + \frac{1}{2}x\right) = 0
   \]
   Therefore, either \( y = 0 \) or \( x = \frac{1}{2} \).

### Equilibrium points:
1. \( (0, 0) \)
2. \( \left(0, 2\right) \)
3. \( \left(\frac{1}{2}, 2\right) \)

Now we can sketch the phase portrait to show trajectories around the equilibrium point \( \left(\frac{1}{2}, 2\right) \), which is the co-existence equilibrium point.

### Interpretation:
- At \( (0, 0) \): This represents extinction of both species.
- At \( (0, 2) \): This represents the extinction of species \( x \) and a nonzero density of species \( y \).
- At \( \left(\frac{1}{2}, 2\right) \): This represents the coexistence of both species.

### Phase Portrait Sketch:
In the phase portrait, we should sketch the direction of the trajectories around the coexistence equilibrium point \( \left(\frac{1}{2}, 2\right) \) and other critical points. The nature of the equilibrium points (whether they are stable, unstable, or saddle points) can be determined by linearizing the system around these points, but we will focus on drawing the qualitative behavior for now.

Would you like me to generate a phase portrait for this system, or dive into more details such as stability analysis?

---

### 5 Relative Questions:
1. What is the biological interpretation of the interaction terms in the system of equations?
2. How do we analyze the stability of equilibrium points using the Jacobian matrix?
3. What are nullclines and how can they help in sketching the phase portrait?
4. How does the coexistence equilibrium relate to real-world predator-prey models?
5. Can you solve for the trajectories of the system analytically or numerically?

### Tip:
When analyzing equilibrium points in nonlinear systems, finding the nullclines is a useful way to visualize where the system's velocity fields change direction.

The following is a model which describes the interaction between two species of densities x and y.
x˙ = x − (1/2)xy
y˙ = −(1/4)y + (1/2)xy.
Sketch a phase portrait which shows a few trajectories about the co-existence equilibrium point.

Explore the interaction between two species through a system of differential equations. Learn how to find equilibrium points and sketch phase portraits for a predator-prey model, focusing on the coexistence equilibrium. This problem covers key topics in differential equations and nonlinear systems, suitable for undergraduate students.

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