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Global Qualitative Analysis of Coupled
Three-Level Food Chains
Chuang-Hsiung Chiu
Center for General Education, Southern Taiwan University
2008/12/26
Contents:
 Predator-prey system
 Three-level food Chain
 Coupled three-level food Chains
1. Predator-prey system
x
mx
 dx
 dt  rx (1  k )  a  x y  xg( x)  p( x) y

 dy  ( mx  d ) y  ( p ( x)  d ) y
 dt
ax
x(0)  x0  0, y (0)  y0  0
Where   0, k  0, a  0, m  0, d  0
x(t )  the population density of prey at time t
y (t )  the population density of predator at time t
r  intrinsic growth rate of prey.
k  carrying capacity of prey.
d  the death rate of predator.
‧ Some typical forms of p( x) are
mx n
(1) p ( x ) 
, n
n
ax
mx 2
(2) p ( x ) 
( a  x )(b  x )
Main results
x
mx

r (1  ) 
y0


k
ax
Let 
( mx  d ) y  0

 ax
We can find the nontrivial equilibriu m E  ( x , y )
ad
r
x
where x 
, y  x (1  )
md
d
k