The modified Predator-Prey model represents the population of the
predators ,v, and the population of the prey, u. We found that when
the population of the predators becomes extinct the population of the
prey reaches a limiting value of 1. We also found another point at
which both species coexist. Under certain parameters the populations
are stable at this point. At other parameters the populations of
predator and prey approach and then oscillate around the point of
coexistance, as shown here.
In the graph, when there are too many predators the prey population dies out. This causes the predator population to die out as well. When the predator population becomes small enough the prey population begins to grow again. This causes the predator population to grow as well. However, when the predator population grows too large the cycle starts again.
As for improving the model, we feel that allowing the prey ,u, to go
to zero would make the model more realistic. In reality if the
predator population grows too large they will eventually eat all of
the prey, causing the prey to become extinct. If the prey are the only
source of food for the predators then the predator population would
eventually become extinct as well.
It would also be a more interesting and realistic model if we were to add another predator to the environment. The new predator would compete with the original predator for food, environment, etc. We would then have a predator-prey as well as a competing species model.