Nonlinear system of differential equations

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How do I go about solving and understanding the phase plane for a nonlinear system of predator and prey equations?
 

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go to
http://geosci.uchicago.edu/~gidon/geos31415/LV/LV.pdf [Broken]
for the basic idea because the simplest case is the Lotka voltera model.


In laymans terms as predators kill prey and breed but as the reproduce you end up with many predators and the prey decreases. The decrease in prey causes the predators to starve. The fact that the predators have just starve increases the prey. The increase in prey feed the predators so you have more predators. So there is an almost clockwork linked rise and fall in the predator and prey populations. It does actually happen with fish when there is only one predator and only one prey.
 
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