Main Points:
When considering the interaction of two populations, one needs a system of two differential equations. In the predator prey model, the idea that the prey will grow exponentially without the predator can be used to form a definition of the slope. Other ideas that should be considered are that the predator's population would decrease if there is too little prey, and the robins benefit from the worms' presence, but the worms do not. Also the number of prey being preyed upon should be directly proportional to the population of the predator. To form a phase plane, the two slope values for the populations must be multiplied. The chart will usually show a equilibrium at the origin and somewhere else that the slope points flow around. The slope points show a trajectory of the populations. As you move a little farther away from one point the slope changes slightly and you can trace the approximate path starting at one point. You can see that with this predator - prey model, the populations oscillate at regular intervals that are separated by one quarter of a period.
Challenges:
The hardest part for me was to figure out how they came up with the equation used to plot the phase plane. It looks like they just treated the slopes like functions and multiplied them together. I need some clarification though.
Reflections:
IT'S THE CIRCLE OF LIFE! It really represents that saying perfectly. The slopes of the population trace a circle around the equilibrium, and since we don't live in a perfect world of equilibrium, this graphic representation is really usefull.
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