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Nutrients, Algae and Herbivores - the Paradox of Enrichment Revisited
If we consider the case where the dilution rate is below the persistence
boundary (Fig. 5.5A), we can follow the trajectory of the system from an
initial location in the vicinity of the internal equilibrium, say, somewhere in
the region labeled I in Fig. 5.SA. In region I, both algal and grazer growth
rates will be negative, so both biomasses will decrease with time. As time
evolves, the trajectory will eventually cross the algal isocline and enter
region II. In region II, algal growth rate is positive, such that algal biomass
will increase with time until the trajectory crosses the grazer isocline and
enters region III. In region III, both algal and grazer growth rates will be
positive, such that both biomasses will increase with time. This means that
the system will eventually leave the region bounded by the algal isocline
and enter region IV. In region IV, growth rates have opposite signs, such
that algal biomass will decrease while grazer biomass will increase with
time, until the system finally enters region I again. We would thus expect
the typical system behavior to be counterclockwise orbits around the internal equilibrium. Although local stability analysis indicates that some
trajectories should spiral into the internal equilibrium point, the qualitative
isocline analysis does not exclude the possibility of some trajectories
ending up in a closed periodic orbit (limit cycle).
The analysis for the case where the dilution rate is above the persistence
boundary (Fig. 5.5B) will initially be very similar to the previous case. If we
initiate the system from the vicinity of the internal eqUilibrium point in
region I, the trajectory will proceed to region III via region II, as before.
The main difference from the previous case is that the trajectory can now
follow two different directions when leaving region III. The presence of two
internal stationary points, of which one is locally stable while the other is
locally unstable, means that the isoclines will have two intersections, thus
creating the region labeled V in Fig. 5.SB. If the trajectory leaves region III
by crossing the algal isocline and entering region IV, it will eventually reenter region I and trace out the same kind of counterclockwise cyclical
orbit as in the previous case. On the other hand, if the trajectory leaves by
crossing the grazer isocline and entering region V, grazer biomass will
decrease while algal biomass continues to increase, until the system eventually is trapped at the grazer extinction point.
5.4 Extinctions, Periodic Orbits, and Domains of
Attraction
The presence of multiple stationary points means that the system can be
attracted to different steady states, depending on the initial conditions. The
set of initial conditions from which the system will be attracted to a given
stationary state is said to be its domain, or basin of attraction.
Nutrients, Algae and Herbivores - the Paradox of Enrichment Revisited
If we consider the case where the dilution rate is below the persistence
boundary (Fig. 5.5A), we can follow the trajectory of the system from an
initial location in the vicinity of the internal equilibrium, say, somewhere in
the region labeled I in Fig. 5.SA. In region I, both algal and grazer growth
rates will be negative, so both biomasses will decrease with time. As time
evolves, the trajectory will eventually cross the algal isocline and enter
region II. In region II, algal growth rate is positive, such that algal biomass
will increase with time until the trajectory crosses the grazer isocline and
enters region III. In region III, both algal and grazer growth rates will be
positive, such that both biomasses will increase with time. This means that
the system will eventually leave the region bounded by the algal isocline
and enter region IV. In region IV, growth rates have opposite signs, such
that algal biomass will decrease while grazer biomass will increase with
time, until the system finally enters region I again. We would thus expect
the typical system behavior to be counterclockwise orbits around the internal equilibrium. Although local stability analysis indicates that some
trajectories should spiral into the internal equilibrium point, the qualitative
isocline analysis does not exclude the possibility of some trajectories
ending up in a closed periodic orbit (limit cycle).
The analysis for the case where the dilution rate is above the persistence
boundary (Fig. 5.5B) will initially be very similar to the previous case. If we
initiate the system from the vicinity of the internal eqUilibrium point in
region I, the trajectory will proceed to region III via region II, as before.
The main difference from the previous case is that the trajectory can now
follow two different directions when leaving region III. The presence of two
internal stationary points, of which one is locally stable while the other is
locally unstable, means that the isoclines will have two intersections, thus
creating the region labeled V in Fig. 5.SB. If the trajectory leaves region III
by crossing the algal isocline and entering region IV, it will eventually reenter region I and trace out the same kind of counterclockwise cyclical
orbit as in the previous case. On the other hand, if the trajectory leaves by
crossing the grazer isocline and entering region V, grazer biomass will
decrease while algal biomass continues to increase, until the system eventually is trapped at the grazer extinction point.
5.4 Extinctions, Periodic Orbits, and Domains of
Attraction
The presence of multiple stationary points means that the system can be
attracted to different steady states, depending on the initial conditions. The
set of initial conditions from which the system will be attracted to a given
stationary state is said to be its domain, or basin of attraction.