## Stability Analysis

Authors: Alex Tee Neng Heng, David G. Green

This is a simple model of interactions between three variables X, Y, and Z. They are related by the equations below: Initially all but the diagonal terms are set to zero and the diagonal terms are set to 1. So the system is constant.

### Experiments

• Change the parameters relating X and Y to create a simple negative feedback loop, as in the earlier exercises. Use parameter values between -1 and +1 . Confirm that the resulting graphs show simple cycles.
• Now try to simulate the action of an automatic governor. Such devices are used to control positive feedback. Change the parameters relating X and Y to create a positive feedback loop coupled to the one above. Does the negative feedback loop restrain values in the positive loop from going off the scale? Try varying the parameters to see what happens?
• Reset the off-diagonal values to zero. This time try to create a negative feedback loop involving the three variables. That is make X affect Y, Y affect Z and Z affect X. Remember that you will need to have two positive terms and one negative term. Do cycles result again? If so how do they differ from the two variable case?