126
Nutrients, Algae and Herbivores - the Paradox of Enrichment Revisited
biomass from u + D towards the asymptotic maximum p; the channeling
of increased P loading into zooplankton biomass is accompanied by a
corresponding increase in primary production in order to support the
resulting level of secondary production. At low phosphorus loading, when
grazer growth is limited by both C and P, Eq. (A6.18) implies that algal
biomass will be linearly decreasing with increasing grazer biomass. When
the input concentration exceeds the critical level P"L' where the algal P
content becomes non-limiting to grazer growth, the algal biomass remains
a constant level C", given by Eq. (A6.19), while the grazer biomass continues to increase towards an asymptotic level Z', given by Eq. (A6.16).
Equations (A6.16) and (A6.19) show that the asymptotic biomass levels
of zoo- and phytoplankton {Z' and C', are both directly proportional to the
incipient limiting concentration {C, and inversely proportional to the
maximum ingestion rate {no In other words, decreasing the food collection
efficiency of the grazer, in terms of the maximal clearance rate {nC1,
would increase the asymptotic biomasses of both algae and grazers. The
asymptotic grazer biomass {Z, is directly proportional to the maximal algal
net growth rate [p' - (u+ D)], while the asymptotic algal biomass {C', is directly proportional to the sum of grazer loss rates (r + 8 + D) and inversely
proportional to the assimilation efficiency (e) of the grazer.
As a change in the asymptotic grazer biomass will also change the equilibrium zooplankton level over the whole domain of the internal equilibrium point, we would expect that increasing maximal algal growth rate (p)
or decreasing algal sinking loss rate (u) should increase the equilibrium
zooplankton biomass, without having any effect on algal biomass. Likewise,
increasing zooplankton respiration (r) or mortality (b), or decreasing assimilation efficiency (e) should increase equilibrium phytoplankton biomass,
without having any effect on zooplankton biomass. Such reciprocal relationships between prey parameters and equilibrium predator biomasses,
and vice versa, are so common in prey-predator models that the phenomenon is usually referred to in text books by a special name: the Volterra principle (e.g., Roughgarden 1979 or May 1981).
Summarizing the local stability analysis, it can be said that if the dilution
rate is such that the system is persistent (D < D~ ), there will be two locally
unstable stationary points both with Z = 0 and one locally stable stationary
point with Z > O. If D > D~ the stationary point corresponding to grazer
extinction will become locally stable, while an additional locally unstable
stationary point corresponding to P-limited grazer growth will come into
existence. The local stability of two remaining stationary points will be
unaffected by dilution rate crossing the persistence boundary D~ .
Nutrients, Algae and Herbivores - the Paradox of Enrichment Revisited
biomass from u + D towards the asymptotic maximum p; the channeling
of increased P loading into zooplankton biomass is accompanied by a
corresponding increase in primary production in order to support the
resulting level of secondary production. At low phosphorus loading, when
grazer growth is limited by both C and P, Eq. (A6.18) implies that algal
biomass will be linearly decreasing with increasing grazer biomass. When
the input concentration exceeds the critical level P"L' where the algal P
content becomes non-limiting to grazer growth, the algal biomass remains
a constant level C", given by Eq. (A6.19), while the grazer biomass continues to increase towards an asymptotic level Z', given by Eq. (A6.16).
Equations (A6.16) and (A6.19) show that the asymptotic biomass levels
of zoo- and phytoplankton {Z' and C', are both directly proportional to the
incipient limiting concentration {C, and inversely proportional to the
maximum ingestion rate {no In other words, decreasing the food collection
efficiency of the grazer, in terms of the maximal clearance rate {nC1,
would increase the asymptotic biomasses of both algae and grazers. The
asymptotic grazer biomass {Z, is directly proportional to the maximal algal
net growth rate [p' - (u+ D)], while the asymptotic algal biomass {C', is directly proportional to the sum of grazer loss rates (r + 8 + D) and inversely
proportional to the assimilation efficiency (e) of the grazer.
As a change in the asymptotic grazer biomass will also change the equilibrium zooplankton level over the whole domain of the internal equilibrium point, we would expect that increasing maximal algal growth rate (p)
or decreasing algal sinking loss rate (u) should increase the equilibrium
zooplankton biomass, without having any effect on algal biomass. Likewise,
increasing zooplankton respiration (r) or mortality (b), or decreasing assimilation efficiency (e) should increase equilibrium phytoplankton biomass,
without having any effect on zooplankton biomass. Such reciprocal relationships between prey parameters and equilibrium predator biomasses,
and vice versa, are so common in prey-predator models that the phenomenon is usually referred to in text books by a special name: the Volterra principle (e.g., Roughgarden 1979 or May 1981).
Summarizing the local stability analysis, it can be said that if the dilution
rate is such that the system is persistent (D < D~ ), there will be two locally
unstable stationary points both with Z = 0 and one locally stable stationary
point with Z > O. If D > D~ the stationary point corresponding to grazer
extinction will become locally stable, while an additional locally unstable
stationary point corresponding to P-limited grazer growth will come into
existence. The local stability of two remaining stationary points will be
unaffected by dilution rate crossing the persistence boundary D~ .