Patterns In Chaos

Chaotic solutions of simple systems of equations are noted for their frequent approximate but not exact repetitions. Sometimes more than one "theme" will be repeated____

In the previous chapter the approach taken to the issue of chaos in a biological population is based on how a population's dynamics respond to perturbations. The fundamental goal in this approach is to determine whether the responses that result from changes in certain demographic parameters conform to the bifurcations in a model-predicted route-to-chaos. From this point of view, all the treatments performed at various locations in the bifurcation diagram play a role in determining whether chaos is present in the experiment. Because the experimental populations underwent the dynamic changes indicated by the bifurcation diagram and because the deterministic LPA model predicts chaos specifically for the treatments designated by c^ — 0.05 and 0.35, we have some confidence in asserting that the beede cultures in these treatments are chaotic—or, as we would rather say, that their dynamics are highly influenced by a chaotic attractor.

In the present chapter we focus our attention on one of the chaotic treatments in the route-to-chaos experiment (located at Cpa = 0.35 in Fig. 4.1). We wish to see if certain characteristics and properties of the model predicted chaotic attractor are observable in the experimental data. In so doing, we will gather further evidence for the chaotic nature of these particular cultures and gain some new insights into several issues involved in the detection of chaos in biological populations and ecology in general.

A study of complex dynamics—in particular, chaotic dynamics — requires a sufficiently long time series of data. With this in mind we continued the three replicates of the chaos treatment Cpa = 0.35 (and the control cultures) beyond the 80 weeks reported in Chapter 4. In Fig. 4.8 147

FIGURE 5.1 I (A) Data from the three replicates of the treatment Cpa = 0.35 from week 40 to week 372 cluster around the chaotic attractor; (B) 500 points calculated from the demographic stochastic model (4.7) using parameter values from Table 4.1 and the treatment parameters c^ = 0.35, ¡j.a = 0.96 represent the stationary distribution.

of that chapter we see how the data from this treatment (with some transients removed) cluster in state space near the predicted chaotic attractor for 80 weeks. In Fig. 5.1A we see a similar figure for data gathered over a much longer time period, namely, 372 weeks.1 Rigorous comparisons of

1 At the time of writing, we are still maintaining these cultures in our laboratory.

the data distribution in state space and the deterministic chaotic attrac-tor, or the stationary distribution predicted by the stochastic LPA model appearing in Fig. 5.1B, are technically difficult. However, even without such statistical analyses one can see from the plots in Fig. 5.1 that the experimental data remained near the predicted deterministic chaotic at-tractor for a considerable length of time (the 372 weeks of data encompass more than 90 generations, in each of the three replicates). Also notice that the distribution of the data in state space is very similar to the predicted stochastic stationary distribution. These comparisons in state space show a clear relationship between the chaotic attractor and the population data over an extended length of time. However, such state-space plots do not allow comparisons of the temporal characteristics of the dynamics.

In what follows we consider some distinctive dynamic properties of the deterministic chaotic attractor and see to what extent they are exhibited by the beetle populations. We begin in Section 5.1 with a look at sensitivity to initial conditions. An experiment based on the "hot spot" of sensitivity located on the chaotic attractor not only documents the presence of this hallmark property of chaos in the dynamics of the beetle populations, but also illustrates how one can utilize this feature to control chaotic outbreaks of population numbers. In the rest of the chapter we consider recurrent temporal patterns. As the quote at the beginning of this chapter indicates, chaotic dynamics typically contain such patterns, and we will find that near-periodic motion on the chaotic attractor is in fact observable in the experimental data.

5.1 | SENSITIVITY TO INITIAL CONDITIONS

More individuals are born than can possibly survive. A grain in the balance will determine which individual shall live and which shall die, which variety or species shall increase in number, and which shall decrease, or finally become extinct.

— CHARLES DARWIN (On the Origin of Species)

A characteristic trait of chaos is the property of sensitivity to initial conditions. Recall that this refers to the tendency for nearby states to diverge rapidly and over time to become very different from one another. Do the beetle populations in the chaos treatment of the route-to-chaos experiment exhibit sensitivity to initial conditions?

In Chapter 1 we pointed out the difficulties involved in determining sensitivity to initial conditions in data—difficulties related to insufficient amount of data, the presence of noise, and the influences of deterministic entities other than the chaotic attractor (transients, unstable invariant sets, etc.). Another approach is to conduct experimental tests for sensitivity to initial conditions or for predicted consequences of this distinctive property of chaos.

An experiment to test for the presence of sensitivity to initial conditions in the beetle populations of the route-to-chaos experiment is reported in [52], The design of that study was based on the location in state space where sensitivity to initial conditions is greatest, as predicted by the deterministic LPA model for the chaos treatment case Cpa = 0.35. As one can see in Fig. 5.2, the most sensitive region on the attractor turns out to be near the "corner" where A-stage and L-stage numbers are low and P-stage numbers are high. Points in state space on or near this part of the chaotic attractor diverge rapidly when mapped forward in time by the LPA model. Computer simulations using the deterministic and stochastic LPA models reveal an interesting prediction when orbits enter this high-sensitivity

Chaotic Attractor

FIGURE 5.2 I Color coding depicts the degree of sensitivity to initial conditions at points on the chaotic attractor in the treatment cm = 0.35 from the route-to-chaos experiment. The coloring scheme is based on the logarithm of largest moduli of the three eigenvalues of the Jacobian matrix of the LPA model evaluated at the point (using the parameter estimates in Table 4.1). On the attractor these logarithms range from -1.03 to 3.95. The colors range from yellow (for -1.03) to red (for 3.95). Orbits converge near yellowish points and (at least some) orbits diverge near reddish points. Thus, red regions are "hot spots" where orbits are sensitive to slight perturbations.

FIGURE 5.2 I Color coding depicts the degree of sensitivity to initial conditions at points on the chaotic attractor in the treatment cm = 0.35 from the route-to-chaos experiment. The coloring scheme is based on the logarithm of largest moduli of the three eigenvalues of the Jacobian matrix of the LPA model evaluated at the point (using the parameter estimates in Table 4.1). On the attractor these logarithms range from -1.03 to 3.95. The colors range from yellow (for -1.03) to red (for 3.95). Orbits converge near yellowish points and (at least some) orbits diverge near reddish points. Thus, red regions are "hot spots" where orbits are sensitive to slight perturbations.

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