Equilibrium Points and Local Stability
125
The Internal Equilibrium. If the input concentration is above the critical
level (P L > P'L)' and the dilution rate is below the persistence boundary
(D < D: ), it is shown in Appendix A6 that there will be only one internal
stationary point (that is, with all state variables being nonzero). This
stationary point is characterized by an algal biomass below the incipient
limiting level (C < C), such that zooplankton growth will be food carbonlimited. At input P concentrations below a critical level P"v defined by Eq.
(A6.24), zooplankton growth will be simultaneously C- and P- limited at
the internal stationary point. In Appendix A7 it is shown that the internal
stationary point will be unconditionally locally stable, and can thus be
called an internal equilibrium.
The analytical expressions developed in Appendix A6 imply that the
equilibrium point traces out a continuous curve when the input phosphorus concentration P L is increased from P'L and upward. When the steady
state biomasses are viewed as functions of P L (Fig. 5.1), it is seen that C
decreases while Z increases with increasing phosphorus enrichment. At the
same time, the algal growth rate increases in proportion with grazer
z'
Cf.)
Cf.)
e
0
.- ..c
9
::t
·0
..c
:..::=
.- ::t
t:r
~
!...ft---~-------====---.Algae - - - - - - - - - l
crt
0
Pi
PE
InputP
Fig. 5.1. Phyto- and zooplankton biomasses at the internal equilibrium point as function of
input phosphorus concentration