Aaron A. King and William M. Schaffer. 2001. The geometry of a population cycle: a mechanistic model of snowshoe hare demography. Ecology 82: 814-830.
Appendix A. Nondimensionalization of the minimal model.
A.1 Rescaling
to reveal the Hamiltonian limit
For a wide range of parameter values, the autonomous dynamical system obtained
by setting in the minimal
model (10) admits an interior equilibrium (B*,H*,P*),
B*, H*, P* > 0. When
such an equilibrium exists, it is invariably unique. We rescale the phase variables
B, H, P around this equilibrium by defining new state variables
and time
![]() |
(A.1) |
Defining the dimensionless parameters
![]() |
(A.2) |
we arrive at the rescaled system
![]() |
(A.3) |
In terms of the dimensionless parameters (A.2), the conditions for co-existence of the three trophic levels are simply
![]() |
(A.4) |
Whether or not these conditions are satisfied, in the autonomous case, the
equilibrium of (A.3) is located at the origin. A Hamiltonian
limit is located at ,
. At the
Hamiltonian limit, the dynamics in the three dimensional phase space of (A.3)
rapidly collapse to the two-dimensional submanifold x = 0. On this limiting
submanifold, the dynamics are Hamiltonian. For a recent discussion of Hamiltonian
dynamics in predator-prey models, see . Hamiltonian dynamics generally are discussed
in , , and . For an introductory discussion of the nature and significance of
Hamiltonian limits, see . It can be shown that this is the only such limit in
(A.3). The fact that this special case lies on the boundary of
the region of biologically meaningful solutions (A.4)
neither stops us from investigating it nor prevents this singularity from making
itself felt well within the regime of interest.
![]() |
As we see in Fig. A.1, resonance
horns, each one bounded by a surface of tangent (or saddle-node)
bifurcations, emanate from the Hamiltonian limit into the dissipative
regime and terminate on the surface of Nejmark-Sacker bifurcations
upon which the annual cycle changes stability. For a more detailed
discussion of the global bifurcation structure, see King
et al. (1996). Hamiltonian dynamics generally are discussed
in Henon (1983), Tabor (1989),
and Lichtenberg & Lieberman (1992).
For an introductory discussion of the nature and significance
of Hamiltonian limits, see King & Schaffer
(1999). It can be shown that this is the only such limit
in (A.3). The fact that this special case
lies on the boundary of the region of biologically meaningful
solutions (A.4) neither stops us
from investigating it nor prevents this singularity from making
itself felt well within the
regime of interest.
A.2 Alternate
rescaling.
Although the rescaling (A.2)
reveals the geometry underlying the dynamics of Eqns. (9),
it obscures the biology. Modification of a single biological parameter
such as K traces, via the equilibrium values B*,
H*, P* and (A.2),
a curvilinear path through the full 13-dimensional parameter space.
To facilitate Interpretation of our results we introduce a simpler,
more directly meaningful, rescaling, as follows. Let
![]() |
(A.5) |
Under this rescaling, with the same dimensionless time as before, Eqns. (9) become
![]() |
(A.6) |
In performing the numerical computations leading to Figs.
3-7, we have used rescaling (A.5). We
remark that the Hamiltonian limit does not exist in model (A.6) for finite values of the parameters.
M. Henon (1983). Numerical explorations
of Hamiltonian systems. In G. Iooss, R. H. G. Helleman,
& R. Stora (eds.), Chaotic Behavior of Deterministic Systems.
North-Holland, Amsterdam.
A. A. King & W. M. Schaffer (1999).
The rainbow bridge: Hamiltonian limits and resonance in
predator-prey dynamics. J. Math. Biol. 39:439-69.
A. A. King, W. M. Schaffer, J. Treat,
C. Gordon, & M. Kot (1996). Weakly dissipative predator-
prey systems. Bull. Math. Biol. 58:835-860.
A. J. Lichtenberg & M. A. Lieberman
(1992). Regular and Chaotic Dynamics. Springer-Verlag,
New York, 3rd edn.
M. Tabor (1989). Chaos and Integrability
in Nonlinear Systems. Wiley, New York.