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

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

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

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

The Jacobian The Jacobian for this system is:

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

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

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

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

Bifurcation Diagrams

Phase Portrait:

Phase Portrait:

Phase Portrait:

Phase Portrait:

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