Summary and Conclusions
155
indicates that the limit cycle is not generated by the destabilization of an
equilibrium point through a Hopf bifurcation. Instead, it is suggested that
the limit cycle results from the interaction between the outstructures of two
dynamically unstable saddle points.
The long-term averages of state variables and process rates reveal large
differences in phosphorus partitioning, biomass proportions, and organization of carbon flows, between the stable equilibrium and the limit cycle.
The emergence of the limit cycle makes the average algal biomass increase
steeply with phosphorus loading, while the zooplankton biomass shows
only a minor increase above the equilibrium level. This change in biomass
proportions is reflected in an increasing fraction of primary production
being exported through the outflow or by sedimentation, instead of being
consumed and channeled into secondary production. Changes in the partitioning of phosphorus between phyto- and zooplankton create a variation
of nearly an order of magnitude in the phosphorus retention of the system,
thus demonstrating the potential of zooplankton as a sink for phosphorus.
Due to its dramatic effects on biomass proportions and flow organization, the bifurcation loading level can be proposed as a threshold for the
establishment of stable grazer control on algal biomass, which is usually
stated as the goal of lake restoration by food web manipulation. It is shown
that the bifurcation level and the persistence boundary can be used to
construct a loading diagram for the feasibility ofbiomanipulation, in terms
of volumetric phosphorus loading rate and water renewal time of a given
lake. For lakes with water renewal times longer than 1 year, the critical
loading level is close to 0.1 (g P) m O
)
year" thus providing a mechanistic
justification for the empirical threshold level proposed by Benndorf (1987).
The loading diagram is able to classify correctly the success or failure of six
out of seven well-documented biomanipulation experiments.
The biomass-phosphorus relationships predicted by the present model
are shown to conform well with observations, in the sense that they provide
upper and lower limits to the zooplankton biomass and chlorophyll a
concentration at a given phosphorus level. This suggests that the present
model might be used to assess the degree of grazer control on algal biomass
in a given lake from the average levels of particulate phosphorus, chlorophyll a, and zooplankton biomass.
Even if the minimal model presented here has no representation of seasonality in forcing functions like light and temperature (since all model
parameters are assumed time-independent), it is still able to reproduce
some characteristic seasonal patterns like the spring bloom and the spring
clear-water phase. One can only speculate whether this may indicate that
perhaps many of the phenomena we perceive as driven by seasonal forcing
are actually created by the interplay between specific initial conditions and
simple, non-seasonal dynamics.
155
indicates that the limit cycle is not generated by the destabilization of an
equilibrium point through a Hopf bifurcation. Instead, it is suggested that
the limit cycle results from the interaction between the outstructures of two
dynamically unstable saddle points.
The long-term averages of state variables and process rates reveal large
differences in phosphorus partitioning, biomass proportions, and organization of carbon flows, between the stable equilibrium and the limit cycle.
The emergence of the limit cycle makes the average algal biomass increase
steeply with phosphorus loading, while the zooplankton biomass shows
only a minor increase above the equilibrium level. This change in biomass
proportions is reflected in an increasing fraction of primary production
being exported through the outflow or by sedimentation, instead of being
consumed and channeled into secondary production. Changes in the partitioning of phosphorus between phyto- and zooplankton create a variation
of nearly an order of magnitude in the phosphorus retention of the system,
thus demonstrating the potential of zooplankton as a sink for phosphorus.
Due to its dramatic effects on biomass proportions and flow organization, the bifurcation loading level can be proposed as a threshold for the
establishment of stable grazer control on algal biomass, which is usually
stated as the goal of lake restoration by food web manipulation. It is shown
that the bifurcation level and the persistence boundary can be used to
construct a loading diagram for the feasibility ofbiomanipulation, in terms
of volumetric phosphorus loading rate and water renewal time of a given
lake. For lakes with water renewal times longer than 1 year, the critical
loading level is close to 0.1 (g P) m O
)
year" thus providing a mechanistic
justification for the empirical threshold level proposed by Benndorf (1987).
The loading diagram is able to classify correctly the success or failure of six
out of seven well-documented biomanipulation experiments.
The biomass-phosphorus relationships predicted by the present model
are shown to conform well with observations, in the sense that they provide
upper and lower limits to the zooplankton biomass and chlorophyll a
concentration at a given phosphorus level. This suggests that the present
model might be used to assess the degree of grazer control on algal biomass
in a given lake from the average levels of particulate phosphorus, chlorophyll a, and zooplankton biomass.
Even if the minimal model presented here has no representation of seasonality in forcing functions like light and temperature (since all model
parameters are assumed time-independent), it is still able to reproduce
some characteristic seasonal patterns like the spring bloom and the spring
clear-water phase. One can only speculate whether this may indicate that
perhaps many of the phenomena we perceive as driven by seasonal forcing
are actually created by the interplay between specific initial conditions and
simple, non-seasonal dynamics.