Extinctions, Periodic Orbits, and Domains of Attraction
133
Extinction or Persistence. If the dilution rate is above the persistence
boundary, the system has two locally stable states: the grazer extinction point
and the internal equilibrium. When the system is initiated inside a dosed
region surrounding the internal equilibrium (the shaded area in Fig. 5.6), it
will remain inside this basin of attraction and eventually settle down at the
internal steady state (Appendix A9 contains details on the computational
procedures for locating the boundary of the attraction basin). If the initial
state is chosen anywhere outside the domain of attraction for the internal
equilibrium, the system will eventually end up at the grazer extinction
point. Apparently, ending up at one of these two locally stable equilibria is
the only possible fate of any trajectory originating inside the positive cone,
when the dilution rate is above the persistence boundary.
The domain of the internal equilibrium has the form of a semilogarithmic ellipse centered at the internal equilibrium. When the input P concentration is increased, the steady-state biomasses at the internal equilibrium
will approach the asymptotic levels, e" and Z', implying that the basin of
1.0
~
~
O,g
...
~
u
co
8 0.6
'-'
'" ~
E
0
:0 0.4
c
0
s:
a
0..
0
0
N 0,2
0,0 .'--- , - - - - - - , - - - - - - + - - - - - - - r - - - - ' - - - - ' 1IJ- - - - '
0,001
0,01
0. 1
10
Phytoplankton biomass (mg C liter .1)
Fig. 5.6. Phase portrait of a nonpersistent system [dilution rate D = 0.1 day·l, input P
·1
concentration PL = 40 (l1g P) I ). Directed paths are trajectories from two different initial
conditions; shaded area is the domain of attraction for the internal equilibrium; broken lines
algal and grazer isoclines; filled circles locally stable stationary points; open circles locaJIy
unstable stationary points