1. A Steady State Analysis of a Rosenzweig-MacArthur Predator-Prey System
Caitlin Brown and Lianne Pinsky

2. Overview We will examine this system of equations:
Without harvesting and stocking, this system has three steady states: a saddle, a saddle or stable node and a Hopf bifurcation between stable and unstable equilibria

3. The Equations r = growth rate s = growth rate K = carrying capacity
A & B are related to predator-prey interaction
G & H are stocking and harvesting terms

4. Simplified equations We use the simplified equations:
by using the following substitutions:

5. The Jacobian
The Jacobian for this system is:

6. First Steady State
(x0, y0)=(0,0)
The equilibrium is a saddle

7. Second Steady State (x1, y1)=(1,0)
This equilibrium bifurcates between a stable node and a saddle

8. Third Steady State
This equilibrium is stable then bifurcates and is unstable

9. The Hopf Bifurcation
The Hopf Bifurcation occurs when the trace is 0

10. Bifurcation Diagrams

11. Phase Portrait:

12. Phase Portrait:

13. Phase Portrait:

14. Phase Portrait:

15. Conclusions This system has three steady states
One steady state is a saddle
One steady state bifurcates between a stable node and a saddle
One steady state has a Hopf Bifurcation between a stable and an unstable equilibrium