91Ó°ÊÓ

In phase plane analysis, nullclines are curves where the rate of change of the system variables is zero. For \( \frac{dx}{dt} = 0 \), we have two cases:

• \( x = 0 \)
• \( y = 1 - \frac{x}{3} \)

The first is a vertical line, indicating that the change in \(x\) is zero regardless of \(y\)'s value. The second is a diagonal line with a negative slope. Here, \( dx/dt \) changes sign as \( (x,y) \) crosses this line.
Similarly, nullclines for \( \frac{dy}{dt} = 0 \) emerge from:

• \( y = 0 \)
• \( x = 1 - \frac{y}{2} \)

The horizontal line \( y = 0 \) means no change in \(y\) regardless of \(x\), while the other line slopes negatively and divides the plane into regions where \(dy/dt\) changes sign. Nullclines provide pathways along which trajectories touch but do not cross, forming distinct regions in the phase plane.

Equilibrium points are the core of phase plane analysis. They are points where both system equations have zero rates of change, \( \frac{dx}{dt} = 0 \) and \( \frac{dy}{dt} = 0 \). Solving the nullcline equations together, \( y = 1 - \frac{x}{3} \) and \( x = 1 - \frac{y}{2} \), helps identify these points.
Here, the solution reveals \( x = \frac{3}{5} \) and \( y = \frac{4}{5} \), giving a single equilibrium point at \( (\frac{3}{5}, \frac{4}{5}) \). At this point, there is no dynamic movement in the system, meaning trajectories converge here if conditions align.
Equilibrium points help us predict stable or unstable dynamics, acting as destinations or points of origin for different trajectories. Understanding their location and stability is crucial for studying system behavior over time.

Differential equations describe how a dynamic system evolves over time. In our given system: \(\frac{dx}{dt}=x\left(1-y-\frac{x}{3}\right)\) and \(\frac{dy}{dt}=y\left(1-\frac{y}{2}-x\right)\),
these equations indicate how populations \(x\) and \(y\) interact. \( \frac{dx}{dt} \) refers to the change in population \(x\) dependent on its current size and \(y\). Conversely, \( \frac{dy}{dt} \) describes the same for \(y\) with an added competitive effect from \(x\).
By setting derivatives to zero, we find the nullclines and, subsequently, equilibrium points. Differential equations in this form, with variables multiplying each other, often signal predator-prey dynamics, competition, or cooperative interactions. Breaking these down into simpler components helps to predict long-term behavior and system interactions.

The direction of trajectories on the phase plane reveals dynamic pathways that solutions might follow. After identifying nullclines, understanding trajectory directions involves determining signs of \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) in each region they create.
For example, consider the line \( y = 1 - \frac{x}{3} \). Above this line, \( \frac{dx}{dt} \) is negative, meaning \(x\) is decreasing, while below, it is positive, signifying growth. Similarly, for \( x = 1 - \frac{y}{2} \), \( \frac{dy}{dt} \) is negative to the right and positive to the left.
By sketching arrows based on these signs, we can visualize how trajectories approach or move away from equilibrium points. These directions give critical insights into system behavior, stability of equilibrium points, and potential oscillations or cycles.