Equilibrium Points and Local Stability
123
and grazers, is in Appendix A7 shown to be locally unstable for all dilution
rates D < p' - u. This means that a sterile system can be invaded from an arbitrary small inoculum of living organisms, as long as the dilution losses do not
exceed the maximal net growth rate of the algae.
Persistence and the Grazer Extinction Point. The grazer extinction point is
characterized by zero zooplankton biomass and nonzero phytoplankton
biomass (that is, Z = 0, C> 0). In the absence of grazing, the algal growth rate
will end up at eqUilibrium with sinking and dilution losses (p = u + D). It is
shown in Appendix A7 that this stationary point will be locally unstable if the
grazers are able to maintain positive net growth rate when feeding on algae
growing at a specific rate p = u + D, and locally stable if not. This result is in
accordance with common intuition if it is expressed as: grazers will only be
able to invade an algal community at equilibrium with dilution and sedimentation losses if they are able to maintain positive net population growth on
this food resource.
An ecological system is said to be persistent if all state variables with
positive initial conditions remain nonzero over time. Gard and Hallam (1979)
demonstrated that in prey-predator systems, persistence of the top predator
is equivalent to entire system persistence. From this, we can infer that the
system will be persistent if it is repelled from the grazer extinction point; that
is, if this point is locally unstable. In Appendix A8 it is shown that this
stability condition can be expressed in terms of a critical dilution rate D~ ,
called the persistence boundary, so that the system will be bounded away
from the grazer extinction point for dilution rates D < D~ . Substituting the
model parameters in Table 5.1 into Eq. (A8.4) gives D~ = 0.032 day"" corresponding to a critical water residence time around 1 month. In the data set
compiled by Prairie (1988; Fig. 2.2), 70% of the lakes have average dilution
rates below the persistence boundary, although this data set can probably not
Table 5.1. Summary of model parameters
Parameter
Value
"
Maximum algal growth rate
1.2
q Algal sinking loss rate
0.0008
Q' Minimum algal P content
3.8
& Maximum assimilation efficiency 0.8
I' Maximum ingestion rate
0.81
C' Incipient limiting food concentratioi.17
r Respiration rate
0.25
o Grazer mortality loss rate
(J Grazer P content
0.02
30.0
day"1
day"1
(Jig P) (mg ct
day·1
(mgC)r l
day"1
day"1
(Jig P) (mg C) ·1
123
and grazers, is in Appendix A7 shown to be locally unstable for all dilution
rates D < p' - u. This means that a sterile system can be invaded from an arbitrary small inoculum of living organisms, as long as the dilution losses do not
exceed the maximal net growth rate of the algae.
Persistence and the Grazer Extinction Point. The grazer extinction point is
characterized by zero zooplankton biomass and nonzero phytoplankton
biomass (that is, Z = 0, C> 0). In the absence of grazing, the algal growth rate
will end up at eqUilibrium with sinking and dilution losses (p = u + D). It is
shown in Appendix A7 that this stationary point will be locally unstable if the
grazers are able to maintain positive net growth rate when feeding on algae
growing at a specific rate p = u + D, and locally stable if not. This result is in
accordance with common intuition if it is expressed as: grazers will only be
able to invade an algal community at equilibrium with dilution and sedimentation losses if they are able to maintain positive net population growth on
this food resource.
An ecological system is said to be persistent if all state variables with
positive initial conditions remain nonzero over time. Gard and Hallam (1979)
demonstrated that in prey-predator systems, persistence of the top predator
is equivalent to entire system persistence. From this, we can infer that the
system will be persistent if it is repelled from the grazer extinction point; that
is, if this point is locally unstable. In Appendix A8 it is shown that this
stability condition can be expressed in terms of a critical dilution rate D~ ,
called the persistence boundary, so that the system will be bounded away
from the grazer extinction point for dilution rates D < D~ . Substituting the
model parameters in Table 5.1 into Eq. (A8.4) gives D~ = 0.032 day"" corresponding to a critical water residence time around 1 month. In the data set
compiled by Prairie (1988; Fig. 2.2), 70% of the lakes have average dilution
rates below the persistence boundary, although this data set can probably not
Table 5.1. Summary of model parameters
Parameter
Value
"
Maximum algal growth rate
1.2
q Algal sinking loss rate
0.0008
Q' Minimum algal P content
3.8
& Maximum assimilation efficiency 0.8
I' Maximum ingestion rate
0.81
C' Incipient limiting food concentratioi.17
r Respiration rate
0.25
o Grazer mortality loss rate
(J Grazer P content
0.02
30.0
day"1
day"1
(Jig P) (mg ct
day·1
(mgC)r l
day"1
day"1
(Jig P) (mg C) ·1