A Steady State Analysis of a
Rosenzweig-MacArthur
Predator-Prey System
Caitlin Brown and Lianne Pinsky
Overview
• We will examine this system of equations:
dx
x
bxy

 rx  1   
G


dt
K
xA
dy
B 
 x
 sy 

H

 x  A B  A
dt
• 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
dx
x
bxy

 rx  1   
G

dt
K x A
dy
B 
 x
 sy 

H

 x  A B  A
dt
•
•
•
•
•
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:
dx
y 
 
   x 1  x 

 
dt
x    
dy
 (   1)x 
 y
 1


dt
x 
by using the following substitutions:
x
by
Bst
x*  , y* 
, t* 
k
rK
B A
r(B  A)
A
B A
 
,   ,  1
Bs
K
A
G
bH (B  A)

, 
where G and H are 0
Kr
KBsr
The Jacobian
• The Jacobian for this system is:
 
y 
x

  1  2x 
2 
(


x)
x

 

    y
x  

(  x)2
x







First Steady State
• (x0, y0)=(0,0)
  0 
J

 0 1 
•    or  1
• The equilibrium is a saddle
Second Steady State
• (x1, y1)=(1,0)
 
   or
 

 
J

 0




  
  

  

• This equilibrium bifurcates between a
stable node and a saddle
Third Steady State
 

1 
(x2 , y2 )   ,   1    1   
 

 
 
2


1





 

 
J 
 
 
 

 




 
 


0








• This equilibrium is stable then
bifurcates and is unstable
The Hopf Bifurcation

2    
tr J    1


    
• The Hopf Bifurcation occurs when the
trace is 0
2

1
Bifurcation Diagrams
H
2
1
Phase Portrait:   
Phase Portrait:     H
Phase Portrait:
H
Phase Portrait:   H
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