Ecological Archives E082-007-A1

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 $\varepsilon_1 = \varepsilon_2 = 0$ 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

\begin{displaymath}\begin{gathered}x = \log\frac{B}{B^*} \qquad y = \log\frac{H......quad z =\log\frac{P}{P^*} \qquad t = \Omega T.\end{gathered}\end{displaymath} (A.1)

Defining the dimensionless parameters

\begin{displaymath}\begin{gathered}\gamma_b = \frac{\bar{G} K}{\Omega B^*} \qq......frac{\bar{R}_P}{\bar{R}_P + \delta_2 \bar{D}_P},\end{gathered}\end{displaymath} (A.2)

we arrive at the rescaled system

\begin{displaymath}\begin{aligned}\frac{dx}{dt} &= \gamma_b\,\left[ (e^{-x}-\a......c{(1 + \eta_h) e^y}{1 + \eta_h e^y}- 1 \right].\end{aligned}\end{displaymath} (A.3)

In terms of the dimensionless parameters (A.2), the conditions for co-existence of the three trophic levels are simply

\begin{displaymath}\begin{gathered}\alpha_b < 1 \qquad \alpha_h < 1\\\eta_b > 0 \qquad \eta_h > 0.\end{gathered}\end{displaymath} (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 $\alpha _b = 1$$\eta _h = 0$. 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.

Figure A.1: Resonance horns emanating from the Hamiltonian limit at $\alpha _b = 1$$\eta _h = 0$ in the rescaling of Eqns.  (A.3). The labels A-E designate regimes of different periodicities as in Figs.  3-7. Other parameters are $\gamma _b = 0.2806$$\gamma _h = 0.53496$$\gamma _p = 0.25465$$\alpha _h = 0.80327$$\beta _h = \beta _p = 1$$\eta _b = 6.23915$$\varepsilon _1 = 1$$\varepsilon _2 = 0$.
\includegraphics[width=100mm]{figs/og2}

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

\begin{displaymath}\begin{gathered}b = \frac{B}{K} \qquad h = \frac{k_B H}{X_B......_B} \qquad\eta_2 = \frac{X_B \Omega}{k_B X_H}.\end{gathered}\end{displaymath} (A.5)

Under this rescaling, with the same dimensionless time as before, Eqns.  (9) become

\begin{displaymath}\begin{aligned}\frac{db}{dt} &= \gamma_1 \phi_s(t) (1 - b) ......ight)\frac{\eta_2 h p}{1 + \eta_2 h} - \mu_2 p.\end{aligned}\end{displaymath} (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.
 
 

Literature cited

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.
 


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