A. Contrary to popular opinion, most ecological populations are not very stable.
1. The usual explanation is that the fluctuations result from "random" environmental inputs impacting systens which would otherwise tend to equilibrium.
2. In the case of so-called "cyclic species," this is hard to accept, i.e., the motion is too strongly periodic to imagine that this is nothing more than
3. Famous examples include
a. The "ten year" cycle of the snowshoe hare (Lepus americanus) and its principal predator, the Canadian lynx (Lynx canadensis).
b.The 3-5 year cycles of lemmings and voles in North America and Fennoscandia.
B. In this chapter, we consider the possibility that such cycles are a consequence of the ecological interactions in which cyclic species engage. We focus on
1. Predator-prey interactions
2. The relationship of realistic models to much simpler schemes which can be shown to evidence non-integrable dynamics of the sort observed in the periodically forced pebdulum.
dP/dt = P(bkV - d)dV/dt = V(r - kP)
where
1. P and V are the densities of predators and victims.
2. b is the number of baby predators produced per victim harvested.
3. k is the per-predator kill rate.
4. r is the victim per capita rate of increase in the absence of predators.
5. d is the per-predator death rate.
D. Volterra's model induces dynamics analogous to those of the pendulum in the absence of friction and driving.
1. The P-V plane is densely populated with neutrally stable periodic orbits.
2. As in the case of the pendulum, this topology is a consequence of the fact that Volterra's equations define a conservative dynamical system.
3. Also, as in the case of the pendulum, Volterra's equations are structurally unstable: the slightest changes the topology of the phase plane. For example,
a. Volterra's equations assume that the victims grow exponentially in the absence of predators. Relaxing this assumption, e.g., replacing r with r(1-V/K) is analogous to adding friction to the pendulum: all trajectories, save those based at points on the axes, decay to the fixed point, (P,V) = [(d/bk),(r/k)].
b. Volterra's equations assume that per predator rates of reproduction vary linearly
with the density of victims.
On biological grounds, one expects a saturating response. Relaxing this assumption,
e.g., replacing bkV with
F(V) = bkV/(c+V)
causes orbits to sprial out from the fixed point.
4. In the presence of seasonal forcing, e.g., replace r with
r(t) = r(1 + e cos 2p wt)
one obtains the non-integrable dynamics of the periodically forced pendulum.
5. Coupling two autonomous (constant environment) systems via overlap in predator diets also
induces non-integrable dynamics.
a. In this case, one computes a "surface of section" by holding constant the value of an energy-like function, the so-called "Hamiltonian", H(P1,P2,V1,V2), and the density of one of the four species.
b. Result is a discretization analogous to the stroboscopic map computed for the two variable, non-autonomous system.
E. To what extent do the dynamics of the non-integrable phase space persist into other regions of parameter space?
1. An important question because
a. The biological assumptions corresponding to conservative dynamics are untenable.
b. Conservative systems have long been dismissed as irrelevant to the behavior of more realistic models because they are structurally unstable.
2. In the case of the
periodically forced two-sepcies system, the answer is as follows:
a. The periodic orbits about which the island chains are organized persist on an open intervals of parameter values away from the conservative limit.
1. The parameter values define structures called "resonance horns."
2. Each horn corresponds to the existence of cycles of a particular rotation number, r = p/q.
3. Margins of the horns are curves of "saddle-node" bifurcations where by pairs of p/q cycles are created or destroyed. (Saddle node bifurcations are the two dimensional anlogs of the tangent bifurcations discussed in the the context of periodic windows in the logistic map.)
4. Tips of the horns line up along a curve of "Neimark-Sacker" bifurcations whereby the annual oscillations (we imagine the period of forcing to be one year) are destabilized.
b. Quasiperiodic and chaotic motions probably also persist away from the Hamiltonian limit, but this has not yet been demonstarted.
F. A similar picture emerges from models which explicitly model the lynx-hare interaction.
1. These models include
1. Three state variables: lynx, hare and winter browse which is eaten by the hares;
2. Seasonal variation in browse growth and consumption, lynx and hare reproduction and mortality;
2. Despite these complications, one observes the existence of a Hamiltonian limit from which resonance horns emerge.
3. 8-11 year cycles are consistent with biologically reasonable parameterizations of the model.