AIHL 5.17.1 Phase portraits intro

The video introduces phase portraits, which are geometric representations of the trajectories of a dynamical system in the phase plane. Using the example of a predator-prey model involving rabbits and foxes, it explains how the populations of both species change over time, illustrating the cyclical nature of their interactions.

Key points include:

• The x-axis represents prey (rabbits) and the y-axis represents predators (foxes).
• The trajectories indicate how populations increase and decrease, leading to an equilibrium point where both populations stabilize.
• The mathematical representation involves coupled differential equations, such as $\frac{dx}{dt} = ax + by$ and , where the equilibrium point occurs when both rates of change are zero.