Extinctions, Periodic Orbits, and Domains of Attraction
137
attracted to the unstable grazer extinction point will be repelled in a direction with increasing Z and decreasing C. We can envision two possible fates
of trajectories repelled from the saddle points; they can either encounter
the attraction basin of the stable focus, or they can get caught in the inset of
the opposite saddle point.
We can visualize geometrically the interactions among the outstructures of
these three stationary points by representing the outstructures of focal points
as ellipsoid surfaces and the outstructures of saddle points as funnel-shaped
surfaces (Fig. 5.8). One end of the funnel will correspond to the inset, such
that trajectories entering here will first be attracted to the saddle point, and
then repelled through the other end of the funnel.
When P loading is low (Fig. 5.8A), the attraction basin of the central focus
intercepts the outsets of the saddle points, so that any trajectory repelled
from a saddle point is trapped in the domain of the focus. When P loading is
increased, the extent of the state space will grow faster than the attraction
basin, meaning that more and more of the state space will be taken up by the
outstructures of the saddle points. Above a critical P loading, the focal attraction basin no longer intercepts all trajectories in the outsets of the saddle
points (Fig. 5.8B), meaning that some trajectories in the outset of one saddle
point can be caught in the inset of the opposite saddle point. In other words,
the emergence of a limit cycle with increasing P loading in the present model
is caused by the connection between the insets and outsets of the two unstable saddle points.
Sensitivity to Initial Conditions. Nonlinear systems have the generic property
that system output can be highly sensitive to the initial conditions. In a
plankton community of a temperate lake, the initial conditions will typically
be the founding populations present at spring overturn. Depending on the
species, the founding population can either be overwintering in the water
column, or emerge from sediment-dwelling cysts or resting eggs.
Figure 5.9 shows an example of how different a pair of trajectories can be
when the initial conditions are located on opposite sides of the boundary of
the focal attraction basin (even if the initial conditions differ by only
0.25%). The two simulations follow each other very closely for the first
month or so, when suddenly one of them breaks off into the limit cycle,
while the other one spirals into the focus. The transition toward the limit
cycle is seen to be accompanied by a quadrupling of the cycle period, and
an increasing asymmetry in the phytoplankton oscillations with respect to
the asymptotic steady-state value. Figure 5.9 illustrates how qualitatively
different seasonal trajectories might result, even in the same lake, depending on the recruitment to the founding populations present at the beginning of the growing season. It also offers an explanation for the observation
that Daphnia populations can alternate between cyclical and stable
dynamics in different years in the same lake (McCauley and Murdoch
1987).
137
attracted to the unstable grazer extinction point will be repelled in a direction with increasing Z and decreasing C. We can envision two possible fates
of trajectories repelled from the saddle points; they can either encounter
the attraction basin of the stable focus, or they can get caught in the inset of
the opposite saddle point.
We can visualize geometrically the interactions among the outstructures of
these three stationary points by representing the outstructures of focal points
as ellipsoid surfaces and the outstructures of saddle points as funnel-shaped
surfaces (Fig. 5.8). One end of the funnel will correspond to the inset, such
that trajectories entering here will first be attracted to the saddle point, and
then repelled through the other end of the funnel.
When P loading is low (Fig. 5.8A), the attraction basin of the central focus
intercepts the outsets of the saddle points, so that any trajectory repelled
from a saddle point is trapped in the domain of the focus. When P loading is
increased, the extent of the state space will grow faster than the attraction
basin, meaning that more and more of the state space will be taken up by the
outstructures of the saddle points. Above a critical P loading, the focal attraction basin no longer intercepts all trajectories in the outsets of the saddle
points (Fig. 5.8B), meaning that some trajectories in the outset of one saddle
point can be caught in the inset of the opposite saddle point. In other words,
the emergence of a limit cycle with increasing P loading in the present model
is caused by the connection between the insets and outsets of the two unstable saddle points.
Sensitivity to Initial Conditions. Nonlinear systems have the generic property
that system output can be highly sensitive to the initial conditions. In a
plankton community of a temperate lake, the initial conditions will typically
be the founding populations present at spring overturn. Depending on the
species, the founding population can either be overwintering in the water
column, or emerge from sediment-dwelling cysts or resting eggs.
Figure 5.9 shows an example of how different a pair of trajectories can be
when the initial conditions are located on opposite sides of the boundary of
the focal attraction basin (even if the initial conditions differ by only
0.25%). The two simulations follow each other very closely for the first
month or so, when suddenly one of them breaks off into the limit cycle,
while the other one spirals into the focus. The transition toward the limit
cycle is seen to be accompanied by a quadrupling of the cycle period, and
an increasing asymmetry in the phytoplankton oscillations with respect to
the asymptotic steady-state value. Figure 5.9 illustrates how qualitatively
different seasonal trajectories might result, even in the same lake, depending on the recruitment to the founding populations present at the beginning of the growing season. It also offers an explanation for the observation
that Daphnia populations can alternate between cyclical and stable
dynamics in different years in the same lake (McCauley and Murdoch
1987).