Extinctions, Periodic Orbits, and Domains of Attraction
135
be trapped in a stable periodic orbit, or limit cycle. This means that the
system will persist for all initial conditions inside the positive cone, but the
stationary state can either be a fIXed point or a periodic orbit, depending
on the initial conditions. By the same kind of reasoning as in the previous
case, we can expect the internal attraction basin to remain below a fixed
size as P loading is increased. Thus we should expect the probability for any
trajectory, starting from random initial conditions to end up at the internal
equilibrium, to diminish with increasing P supply.
Geometry of the Limit Cycle. Sudden, qualitative changes in dynamic behavior as result of changes in model parameters are usually called bifurcation phenomena in the terminology of nonlinear dynamic systems. Many
textbooks in theoretical ecology (e.g., May 1975; Roughgarden 1979) emphasize a mechanism called the Hopf bifurcation when discussing transitions
from stable equilibria to limit cycling behavior in prey-predator models.
The Hopf bifurcation results when a change in one or more model
parameters causes a local stability change at an equilibrium point (a change
from negative to positive real part in a pair of complex eigenvalues).
Therefore, the appearance of a limit cycle through a Hopf bifurcation
implies the disappearance of a stable equilibrium, and excludes the possibility of coexistence between the two. In the present model, a stable equilibrium is found to coexist with a stable periodic orbit (Fig. 5.7); furthermore,
the local stability analysis in Appendix A6 shows that if the internal equilibrium exists, then it will be unconditionally locally stable. In other words, the
limit cycle in the present model cannot be generated by a Hopfbifurcation.
Still, the Hopf bifurcation is not the only mechanism by which periodic
orbits can be generated in nonlinear dynamic systems (see, for example,
Thompson and Stewart 1986 for a general review). If the system possesses
two or more unstable stationary points that are saddle points (that is, they
have eigenvalues with opposite signs), the system can have closed periodic
orbits that are saddle cycles. Saddle points have the property that they will
be attracting to state trajectories approaching from some directions, while
trajectories approaching from other directions will be repelled. The set of
points formed by trajectories attracted to or repelled from a stationary
point is called the outstructure of the point. The subset of the outstructure
where trajectories are attracted is called the inset of the stationary point,
while the subset where trajectories are repelled is called the outset of the
point. It can be shown (e.g., Thompson and Stewart 1986) that eigenvectors
corresponding to eigenvalues with negative real parts are tangent to the
inset at the stationary point, while eigenvectors corresponding to eigenvalues with positive real parts are tangent to the outset. For an asymptotically
stable stationary point, the outstructure will consist only of an inset, which
will be identical to the basin of attraction.
135
be trapped in a stable periodic orbit, or limit cycle. This means that the
system will persist for all initial conditions inside the positive cone, but the
stationary state can either be a fIXed point or a periodic orbit, depending
on the initial conditions. By the same kind of reasoning as in the previous
case, we can expect the internal attraction basin to remain below a fixed
size as P loading is increased. Thus we should expect the probability for any
trajectory, starting from random initial conditions to end up at the internal
equilibrium, to diminish with increasing P supply.
Geometry of the Limit Cycle. Sudden, qualitative changes in dynamic behavior as result of changes in model parameters are usually called bifurcation phenomena in the terminology of nonlinear dynamic systems. Many
textbooks in theoretical ecology (e.g., May 1975; Roughgarden 1979) emphasize a mechanism called the Hopf bifurcation when discussing transitions
from stable equilibria to limit cycling behavior in prey-predator models.
The Hopf bifurcation results when a change in one or more model
parameters causes a local stability change at an equilibrium point (a change
from negative to positive real part in a pair of complex eigenvalues).
Therefore, the appearance of a limit cycle through a Hopf bifurcation
implies the disappearance of a stable equilibrium, and excludes the possibility of coexistence between the two. In the present model, a stable equilibrium is found to coexist with a stable periodic orbit (Fig. 5.7); furthermore,
the local stability analysis in Appendix A6 shows that if the internal equilibrium exists, then it will be unconditionally locally stable. In other words, the
limit cycle in the present model cannot be generated by a Hopfbifurcation.
Still, the Hopf bifurcation is not the only mechanism by which periodic
orbits can be generated in nonlinear dynamic systems (see, for example,
Thompson and Stewart 1986 for a general review). If the system possesses
two or more unstable stationary points that are saddle points (that is, they
have eigenvalues with opposite signs), the system can have closed periodic
orbits that are saddle cycles. Saddle points have the property that they will
be attracting to state trajectories approaching from some directions, while
trajectories approaching from other directions will be repelled. The set of
points formed by trajectories attracted to or repelled from a stationary
point is called the outstructure of the point. The subset of the outstructure
where trajectories are attracted is called the inset of the stationary point,
while the subset where trajectories are repelled is called the outset of the
point. It can be shown (e.g., Thompson and Stewart 1986) that eigenvectors
corresponding to eigenvalues with negative real parts are tangent to the
inset at the stationary point, while eigenvectors corresponding to eigenvalues with positive real parts are tangent to the outset. For an asymptotically
stable stationary point, the outstructure will consist only of an inset, which
will be identical to the basin of attraction.