A Steady State Analysis of a Rosenzweig-MacArthur Predator
Download
Report
Transcript A Steady State Analysis of a Rosenzweig-MacArthur Predator
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
xA
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