154
Nutrients, Algae and Herbivores - the Paradox of Enrichment Revisited
higher than the level corresponding to grazer control of algal biomass in
other lakes. It is interesting to notice that the notch at the bifurcation point in
the predicted chla - P relationship appears to be reflected in the data from the
NIV A survey. The location of the notch in terms of total P [20-30 (JIS P) r1,
when assuming 42% DOP], is also roughly comparable to the phosphorus
level commonly used by limnologists to distinguish between mesotrophic and
eutrophic lakes.
5.8 Summary and Conclusions
By this chapter, we have reached the point where we can assemble the
submodels developed in the preceding chapters, describing phosphorus
cycling and plankton growth dynamics. By radical aggregation of compartments, a minimal model of pelagic phosphorus cycling with only three
state variables (algal phosphorus, algal biomass, and grazer biomass) has
been constructed. This model is sufficiently simple for many of the model
properties to be explored by analytical techniques.
The methods of local stability analysis show that the model can have only
one locally stable equilibrium point when the dilution rate is below a certain limit. If the dilution rate exceeds this limit, called the persistence
boundary, the system will have two locally stable equilibria, one of which
represents the extinction of the grazer population. The possibility of
deterministic extinction of the grazer population results from the consideration of stoichiometric relationships between predators and their prey. In
this respect, the present model is different from the models considered by
Rosenzweig (1971), where deterministic extinction of the predator population is impossible.
Isocline analysis combined with simulations showed that the model has
two principal modes of dynamic behavior for dilution rates below the persistence boundary; the system could either be attracted to the central focus
predicted by the local stability analysis, or it could end up in a limit cycle.
The possibility of a limit cycle behavior is not indicated by the local stability analysis, pointing to the limitations of this technique when the state
equations include functions with discontinuous derivatives, like Eqs. (5.5)
and (5.6).
The bifurcation sequence leading to the limit cycle corresponds to the
paradox of enrichment (Rosenzweig 1971), where a stable equilibrium is
destabilized into a limit cycle by increasing the carrying capacity of the
prey population. Below a certain phosphorus loading, called the bifurcation
level, attraction to the stable equilibrium is the only possible outcome. The
emergence of the limit cycle above the bifurcation loading level makes it
possible for the system to be attracted to two alternate stationary states,
depending on the initial conditions. This coexistence of the two attractors
Nutrients, Algae and Herbivores - the Paradox of Enrichment Revisited
higher than the level corresponding to grazer control of algal biomass in
other lakes. It is interesting to notice that the notch at the bifurcation point in
the predicted chla - P relationship appears to be reflected in the data from the
NIV A survey. The location of the notch in terms of total P [20-30 (JIS P) r1,
when assuming 42% DOP], is also roughly comparable to the phosphorus
level commonly used by limnologists to distinguish between mesotrophic and
eutrophic lakes.
5.8 Summary and Conclusions
By this chapter, we have reached the point where we can assemble the
submodels developed in the preceding chapters, describing phosphorus
cycling and plankton growth dynamics. By radical aggregation of compartments, a minimal model of pelagic phosphorus cycling with only three
state variables (algal phosphorus, algal biomass, and grazer biomass) has
been constructed. This model is sufficiently simple for many of the model
properties to be explored by analytical techniques.
The methods of local stability analysis show that the model can have only
one locally stable equilibrium point when the dilution rate is below a certain limit. If the dilution rate exceeds this limit, called the persistence
boundary, the system will have two locally stable equilibria, one of which
represents the extinction of the grazer population. The possibility of
deterministic extinction of the grazer population results from the consideration of stoichiometric relationships between predators and their prey. In
this respect, the present model is different from the models considered by
Rosenzweig (1971), where deterministic extinction of the predator population is impossible.
Isocline analysis combined with simulations showed that the model has
two principal modes of dynamic behavior for dilution rates below the persistence boundary; the system could either be attracted to the central focus
predicted by the local stability analysis, or it could end up in a limit cycle.
The possibility of a limit cycle behavior is not indicated by the local stability analysis, pointing to the limitations of this technique when the state
equations include functions with discontinuous derivatives, like Eqs. (5.5)
and (5.6).
The bifurcation sequence leading to the limit cycle corresponds to the
paradox of enrichment (Rosenzweig 1971), where a stable equilibrium is
destabilized into a limit cycle by increasing the carrying capacity of the
prey population. Below a certain phosphorus loading, called the bifurcation
level, attraction to the stable equilibrium is the only possible outcome. The
emergence of the limit cycle above the bifurcation loading level makes it
possible for the system to be attracted to two alternate stationary states,
depending on the initial conditions. This coexistence of the two attractors