In 1868, the accidental introduction into the United States of the cottony-cushion insect (Icerya purchasi) from Australia threatened to destroy the American citrus industry. To counteract this situation, natural Australian predator, a ladybird beetle (Novius cardinalis), was imported. The beetles kept the insects to a relatively low level. When DDT(an insecticide) was discovered to kill scale insects, farmers applied it in the hope of reducing the scale insect population even further. However, DDT turned out to be fatal to the beetle as well, and the overall effect of using the insecticide was to increase the number of the scale insect. Develop a mathematical model to predict the cottony-cushion and lady bird population before the application of DDT.

(a) Identify the fixed points of your model.
(b) Describe the stability of your fixed points.