A phase trajectory is shown for populations of rabbits (R)...

A phase trajectory is shown for populations of rabbits $(R)$ and foxes $(F)$.
(a) Describe how each population changes as time goes by.
(b) Use your description to make a rough sketch of the graphs of $R$ and $F$ as functions of time.

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Key Concepts

Phase Trajectory

A phase trajectory is a curve in a state-space (or phase-plane) representing the evolution of a system over time. It portrays the relationship between two or more variables (such as populations) without directly referencing time, revealing how changes in one variable relate to changes in another throughout the system’s evolution.

Predator-Prey Dynamics

Predator-prey dynamics focus on the interactions between two species: one as a predator and the other as its prey. These interactions typically form feedback mechanisms that cause the populations to fluctuate cyclically, where the rise or fall in one can drive corresponding changes in the other.

Time Series Graphs in Dynamical Systems

Time series graphs display how individual variables change over time, capturing the dynamics of the system as distinct functions of time. This approach is crucial for visualizing the periodic behavior and trends that occur within the system, such as peaks and troughs in population levels.

Oscillatory Behavior

Oscillatory behavior in dynamical systems refers to the periodic rise and fall in the state variables. In the context of interacting populations, this behavior can be observed as regular cycles where the populations alternate between high and low levels, reflecting the underlying interactions and regulatory mechanisms of the system.

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