Bifurcations and Long-Term Averages
143
or outflow) and consumption. The fraction of primary production consumed by zooplankton rises steeply with increasing P loading, approaching
an asymptotic level >99%. At the same time, the secondary production per
unit of primary production consumed (the trophic efficiency of the
grazers) increases as a result of the improving food quality in terms of
increasing algal P content. As the input P concentration exceeds the bifurcation level, the trophic efficiency of the grazers drops steeply to approximately half of the asymptotic level (8.6%). At the same time, an increasing
fraction of primary production is lost by export processes, instead of being
consumed and channeled into secondary production.
Phosphorus Retention. The phosphorus retention (Rp, cf. Section 2.1) is
defined as the fraction of P entering the lake that is retained in the lake.
From the general expression (2.17) for the P loss rate as function of phytoplankton sinking and zooplankton mortality, the phosphorus retention can
be written as
R = (j P+o8Z
P
DP L
(5.8)
The P retention will be constrained between the two limiting cases
resulting from setting either P = 0 or Z = 0 in Eq. (5.8). If phytoplankton
sinking is the only loss process, then we find by substituting Z = 0 and P =
(1 - Rp) P L into Eq. (5.8), and solving for Rp, that Rp = a/(D + S). At the
other extreme, if zooplankton mortality is the only retention process, we
find by substituting P = 0 and ez = (J - Rp)P L into Eq. (5.8), and solving for
Rp, that Rp = 81(D + 8).
When the input P concentration P L is below the critical level P'L' so that
the system is unable to support any grazer population at all, the retention
will be a/(D + u), corresponding to phytoplankton sinking alone (Fig.
5.14). Increasing P L beyond P'L leads to a rapidly increasing zooplankton
fraction of total P, and a corresponding increase in the P retention toward
the asymptotic level 81(D + 8). As the zooplankton biomass levels off
toward the asymptotic level Z', the zooplankton fraction of total P starts to
decrease, resulting in a decreasing P retention with further increase in the P
loading. The slight increase in average zooplankton biomass accompanying
the transition to the limit cycle at the bifurcation point p' L is reflected by an
increase in the Pretention.
Depending on the partitioning of phosphorus between algae and grazers,
the P retention can thus vary by almost an order of magnitude. This result is
consistent with the observed increase in P retention following successful
biomanipulations (Stenson et a1. 1978; Shapiro and Wright 1984; Reinertsen
et al. 1989; Sanni and Wrervlgen 1990). It also suggests that some of the P
retention variability found in lakes with the same flushing rate (Fig. 2.2)
might be explained by the phosphorus partitioning in the food web.
143
or outflow) and consumption. The fraction of primary production consumed by zooplankton rises steeply with increasing P loading, approaching
an asymptotic level >99%. At the same time, the secondary production per
unit of primary production consumed (the trophic efficiency of the
grazers) increases as a result of the improving food quality in terms of
increasing algal P content. As the input P concentration exceeds the bifurcation level, the trophic efficiency of the grazers drops steeply to approximately half of the asymptotic level (8.6%). At the same time, an increasing
fraction of primary production is lost by export processes, instead of being
consumed and channeled into secondary production.
Phosphorus Retention. The phosphorus retention (Rp, cf. Section 2.1) is
defined as the fraction of P entering the lake that is retained in the lake.
From the general expression (2.17) for the P loss rate as function of phytoplankton sinking and zooplankton mortality, the phosphorus retention can
be written as
R = (j P+o8Z
P
DP L
(5.8)
The P retention will be constrained between the two limiting cases
resulting from setting either P = 0 or Z = 0 in Eq. (5.8). If phytoplankton
sinking is the only loss process, then we find by substituting Z = 0 and P =
(1 - Rp) P L into Eq. (5.8), and solving for Rp, that Rp = a/(D + S). At the
other extreme, if zooplankton mortality is the only retention process, we
find by substituting P = 0 and ez = (J - Rp)P L into Eq. (5.8), and solving for
Rp, that Rp = 81(D + 8).
When the input P concentration P L is below the critical level P'L' so that
the system is unable to support any grazer population at all, the retention
will be a/(D + u), corresponding to phytoplankton sinking alone (Fig.
5.14). Increasing P L beyond P'L leads to a rapidly increasing zooplankton
fraction of total P, and a corresponding increase in the P retention toward
the asymptotic level 81(D + 8). As the zooplankton biomass levels off
toward the asymptotic level Z', the zooplankton fraction of total P starts to
decrease, resulting in a decreasing P retention with further increase in the P
loading. The slight increase in average zooplankton biomass accompanying
the transition to the limit cycle at the bifurcation point p' L is reflected by an
increase in the Pretention.
Depending on the partitioning of phosphorus between algae and grazers,
the P retention can thus vary by almost an order of magnitude. This result is
consistent with the observed increase in P retention following successful
biomanipulations (Stenson et a1. 1978; Shapiro and Wright 1984; Reinertsen
et al. 1989; Sanni and Wrervlgen 1990). It also suggests that some of the P
retention variability found in lakes with the same flushing rate (Fig. 2.2)
might be explained by the phosphorus partitioning in the food web.